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.Probability.Independence.Basic | {
"line": 1080,
"column": 8
} | {
"line": 1080,
"column": 74
} | {
"line": 1081,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Basic | {
"line": 1080,
"column": 8
} | {
"line": 1080,
"column": 74
} | {
"line": 1081,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Basic | {
"line": 1080,
"column": 8
} | {
"line": 1080,
"column": 74
} | {
"line": 1081,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Basic | {
"line": 1098,
"column": 2
} | {
"line": 1098,
"column": 13
} | {
"line": 1098,
"column": 14
} | [
{
"pp": "case a\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nt : ι → Set β\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ\nhy : ∀... | [
"case a\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nt : ι → Set β\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ\nhy : ∀ (i : ι), μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 21
} | {
"line": 769,
"column": 22
} | [
{
"pp": "case inl\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → Measure Ω\nν : Measure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))\nhν : ∀ s ∈ S, ν ... | [
"case inl\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → Measure Ω\nν : Measure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))\nhν : ∀ s ∈ S, ν s ≠ ∞\nhμ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 783,
"column": 4
} | {
"line": 783,
"column": 15
} | {
"line": 783,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\n⊢ ∀ s ∈ S, Ten... | [
"Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\n⊢ ∀ s ∈ S, Tendsto (fun a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Density | {
"line": 330,
"column": 26
} | {
"line": 336,
"column": 77
} | {
"line": 338,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nG : Type u_2\nmΩ : MeasurableSpace Ω\nℙ : Measure Ω\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\nX Y : Ω → G\ninst✝² : SFinite μ\ninst✝¹ : HasPDF X ℙ μ\ninst✝ : HasPDF Y ℙ μ\nσX : SigmaFinite ... | [] | by
have : AEMeasurable X ℙ := HasPDF.aemeasurable' μ
have : AEMeasurable Y ℙ := HasPDF.aemeasurable' μ
rw [hasPDF_iff_of_aemeasurable (by fun_prop),
hXY.map_mul_eq_map_mconv_map₀' (by fun_prop) (by fun_prop) σX σY]
refine ⟨?_, mconv_absolutelyContinuous HasPDF.absolutelyContinuous⟩
apply HaveLebesgueDecom... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 784,
"column": 2
} | {
"line": 784,
"column": 13
} | {
"line": 784,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\nthis : Tendsto... | [
"Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\nthis : Tendsto (fun i ↦ (↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 801,
"column": 4
} | {
"line": 801,
"column": 76
} | {
"line": 802,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\n⊢ ∃ T ⊆ S, T.Countable ... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nthis : ∀ (x : ↑G), ∃ s ∈ S, s ∈ 𝓝 ... | have : ∀ (x : G), ∃ s ∈ S, s ∈ 𝓝 (x : Ω) ∧ s ⊆ G := fun x ↦ h G hG x x.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Integrable | {
"line": 39,
"column": 62
} | {
"line": 39,
"column": 82
} | {
"line": 39,
"column": 83
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f... | [
"Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f : ∀ (c : ℝ≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integrable | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | {
"line": 43,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f... | [
"Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f : ∀ (c : ℝ≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 809,
"column": 6
} | {
"line": 809,
"column": 17
} | {
"line": 809,
"column": 18
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\ns : ↑G → Set Ω\nhsS : ∀... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\ns : ↑G → Set Ω\nhsS : ∀ (x : ↑G), s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 825,
"column": 4
} | {
"line": 825,
"column": 22
} | {
"line": 825,
"column": 23
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nT : Set ... | [
"case refine_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nT : Set (Set Ω)\nTS ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 15
} | {
"line": 819,
"column": 46
} | {
"line": 820,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 15
} | {
"line": 819,
"column": 46
} | {
"line": 820,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 819,
"column": 15
} | {
"line": 819,
"column": 46
} | {
"line": 820,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | {
"line": 130,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Mea... | [] | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | {
"line": 130,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Mea... | [] | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 15
} | {
"line": 208,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\nin... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\ninst✝¹ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.Variance | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 33
} | {
"line": 366,
"column": 33
} | [
{
"pp": "case pos.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ∫⁻ (ω : Ω), ↑‖X ω ^ 2‖₊ ∂μ",
"ppTerm": "?pos.e_a✝",
"assigned": true,
"usedConstants": ... | [
"case pos.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ENNReal.ofReal (∫ (a : Ω), ↑‖X a ^ 2‖₊ ∂μ)",
"case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Mea... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 17
} | {
"line": 368,
"column": 18
} | [
{
"pp": "case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ Integrable (fun x ↦ ↑‖X x ^ 2‖₊) μ",
"ppTerm": "?pos.e_a.hfi✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ Integrable (fun x ↦ X x ^ 2) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 17
} | {
"line": 299,
"column": 18
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [
"Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.HasLaw | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX : Ω → 𝓧\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nx : 𝓧\nhX : X =ᵐ[P] fun x_1 ↦ x\n⊢ HasLaw X (dirac x) P",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX : Ω → 𝓧\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nx : 𝓧\nhX : X =ᵐ[P] fun x_1 ↦ x\n⊢ HasLaw X (dirac x) P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | {
"line": 312,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [] | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | {
"line": 312,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [] | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 17
} | {
"line": 96,
"column": 18
} | [
{
"pp": "case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX Y : (i : ι) → Ω... | [
"case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX Y : (i : ι) → Ω i → E\nZ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Mea... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 35
} | {
"line": 154,
"column": 36
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝³ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝² : TopologicalSpace E\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : OpensMeasurableSpace E\nX : ι → Ω' → E\nhX₁ : ∀ (i : ι), AEMeasu... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝³ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝² : TopologicalSpace E\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : OpensMeasurableSpace E\nX : ι → Ω' → E\nhX₁ : ∀ (i : ι), AEMeasurable (X i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.FactorsThrough | {
"line": 81,
"column": 2
} | {
"line": 83,
"column": 30
} | {
"line": 85,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝² : Nonempty Z\ninst✝¹ : MeasurableSpace Z\ninst✝ : StandardBorelSpace Z\nhg : Measurable g\n⊢ ∃ h, Measurable h ∧ g = h ∘ f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Topologi... | [] | let := upgradeStandardBorel Z
obtain ⟨h, mh, hh⟩ := hg.stronglyMeasurable.exists_eq_measurable_comp
exact ⟨h, mh.measurable, hh⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.FactorsThrough | {
"line": 81,
"column": 2
} | {
"line": 83,
"column": 30
} | {
"line": 85,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝² : Nonempty Z\ninst✝¹ : MeasurableSpace Z\ninst✝ : StandardBorelSpace Z\nhg : Measurable g\n⊢ ∃ h, Measurable h ∧ g = h ∘ f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Topologi... | [] | let := upgradeStandardBorel Z
obtain ⟨h, mh, hh⟩ := hg.stronglyMeasurable.exists_eq_measurable_comp
exact ⟨h, mh.measurable, hh⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 628,
"column": 14
} | {
"line": 628,
"column": 34
} | {
"line": 628,
"column": 34
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ns : Finset ι\n... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ns : Finset ι\ni : ι\nhi : ... | Finset.prod_coe_sort | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 656,
"column": 4
} | {
"line": 656,
"column": 60
} | {
"line": 657,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns ... | [] | exact Finset.prod_congr rfl fun i hi ↦ (hω ⟨i, hi⟩).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Floor | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 51
} | {
"line": 29,
"column": 52
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (floor ⁻¹' {⌊x⌋})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (Ico (↑⌊x⌋) (↑⌊x⌋ + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Floor | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 50
} | {
"line": 38,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (Ioc (↑⌈x⌉ - 1) ↑⌈x⌉)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 366,
"column": 10
} | {
"line": 366,
"column": 52
} | {
"line": 366,
"column": 53
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | [
"case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 367,
"column": 10
} | {
"line": 367,
"column": 44
} | {
"line": 367,
"column": 45
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | [
"case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 27
} | {
"line": 369,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i... | [] | exact hle₁.trans hle₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Filtration | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 46
} | {
"line": 369,
"column": 47
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nhineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j\n⊢ ↑𝓕₊₊ i ≤ ↑𝓕₊ i",
"pp... | [
"case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nhineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j\n⊢ ∀ (i_1 : ι), i < i_1 → ⨅ j, ⨅ (_ : i <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\... | [
"α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 296,
"column": 8
} | {
"line": 296,
"column": 47
} | {
"line": 296,
"column": 48
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTopology E\ninst✝¹ : BorelSpace E\ninst✝ : l.IsCountablyGenerated\nX : ι → ... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTopology E\ninst✝¹ : BorelSpace E\ninst✝ : l.IsCountablyGenerated\nX : ι → Ω' → E\nh : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 43
} | {
"line": 316,
"column": 44
} | [
{
"pp": "case refine_2\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁸ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁷ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝⁶ : SeminormedAddCommGroup E\... | [
"case refine_2\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁸ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁷ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝⁶ : SeminormedAddCommGroup E\ninst✝⁵ : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 19
} | {
"line": 107,
"column": 20
} | [
{
"pp": "case refine_3.refine_1.refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u... | [
"case refine_3.refine_1.refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 100,
"column": 4
} | {
"line": 113,
"column": 80
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | [] | refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- This next block proves that a set of strictly positive natural density is infinite, mixed
-- with the fact that `{n | x ∈ s n}` has strictly positive natural density.
-- TODO: Separate it out to a lemma once we have a natural density API.
· refi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 100,
"column": 4
} | {
"line": 113,
"column": 80
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | [] | refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- This next block proves that a set of strictly positive natural density is infinite, mixed
-- with the fact that `{n | x ∈ s n}` has strictly positive natural density.
-- TODO: Separate it out to a lemma once we have a natural density API.
· refi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 31,
"column": 17
} | {
"line": 31,
"column": 49
} | {
"line": 31,
"column": 50
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), Measurable (f i)\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (hs.piece... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), Measurable (f i)\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (⋃ i, s i ∩ f i ⁻¹' t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 67,
"column": 9
} | {
"line": 67,
"column": 41
} | {
"line": 67,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ μ ∅ ≠ ∞",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [
"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ 0 ≠ ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 43
} | {
"line": 140,
"column": 44
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns : Set X\ni : ι\nhi : μ (F i) ≠ 0\nhi' : μ (F i) ≠ ∞\n⊢ μ (s ∩ F i) / μ (F i) ∈ Icc 0 1",
"ppTerm": "?m.92",... | [
"G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns : Set X\ni : ι\nhi : μ (F i) ≠ 0\nhi' : μ (F i) ≠ ∞\n⊢ μ (s ∩ F i) ≤ μ (F i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 27
} | {
"line": 166,
"column": 28
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng h : G\n⊢ Tendsto (fun i ↦ μ ((g • F i) ∆ (h • F i)) / μ (F i)) (↑u) (𝓝 0)... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng h : G\n⊢ Tendsto (fun i ↦ μ ((g • F i) ∆ (h • F i)) / μ (F i)) (↑u) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 69
} | {
"line": 182,
"column": 70
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng✝ h✝ : G\ns : Set X\ng h : G\ni : ι\nhi : μ (F i) ≠ 0\n⊢ μ (g • s ∩ F i) - ... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng✝ h✝ : G\ns : Set X\ng h : G\ni : ι\nhi : μ (F i) ≠ 0\n⊢ μ (s ∩ g⁻¹ • F i) ≤ μ ((s ∩ g⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"column": 14
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng : G\ns : Set X\n⊢ mean μ u F (g • s) = mean μ u F s",
"ppTerm": "?m.22... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng : G\ns : Set X\n⊢ mean μ u F (g • s) = mean μ u F s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 67
} | {
"line": 197,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ≥0\nhε : 0 < ε\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg : ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ≥0\nhε : 0 < ε\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg : ∀ (i : Fin n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 97,
"column": 76
} | {
"line": 97,
"column": 92
} | {
"line": 97,
"column": 93
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | [
"E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L\nfund : IsA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 32
} | {
"line": 103,
"column": 33
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hg.hf",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 27
} | {
"line": 115,
"column": 28
} | [
{
"pp": "R r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty\nthis :\n ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],\n IsMaxOn (... | [
"R r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty\nthis :\n ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],\n IsMaxOn ((fun x ↦ ‖x‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 83
} | {
"line": 137,
"column": 84
} | [
{
"pp": "R : ℝ\nhR : 0 < R\nz x : ℂ\nf : ℂ → ℂ\nhx : x ∈ ball z R\nhf : ContinuousOn f (sphere z R)\nr : ℝ\nhr : r < R\nhrx : x ∈ ball z r\nε' : ℝ\nhε' : ε' > 0\nH : ball x ε' ⊆ ball z r\na : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nhb : (a, b).2 ∈ [[0, 2 * π]]\nhab :\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]]... | [
"R : ℝ\nhR : 0 < R\nz x : ℂ\nf : ℂ → ℂ\nhx : x ∈ ball z R\nhf : ContinuousOn f (sphere z R)\nr : ℝ\nhr : r < R\nhrx : x ∈ ball z r\nε' : ℝ\nhε' : ε' > 0\nH : ball x ε' ⊆ ball z r\na : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nhb : (a, b).2 ∈ [[0, 2 * π]]\nhab :\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ‖ci... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 45
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case pos\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ✝ : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\nγ : Type u_5\ninst✝¹ : TopologicalSpace γ\ninst✝ : Zero γ\ng : C(β, γ)\nf : α →C_c β\nhg : g 0 = 0\n⊢ HasCompactSupport (g.comp ↑f).toFun",
"ppTerm": "?pos✝"... | [
"case neg\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ✝ : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\nγ : Type u_5\ninst✝¹ : TopologicalSpace γ\ninst✝ : Zero γ\ng : C(β, γ)\nf : α →C_c β\nhg : ¬g 0 = 0\n⊢ HasCompactSupport (ContinuousMap.toFun 0)"
] | · exact f.hasCompactSupport'.comp_left hg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 281,
"column": 38
} | {
"line": 281,
"column": 79
} | {
"line": 281,
"column": 80
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C_c β\n⊢ HasCompactSupport (-⇑f.toContinuousMap)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants":... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C_c β\n⊢ IsCompact (closure (Function.support ⇑f))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 294,
"column": 17
} | {
"line": 294,
"column": 45
} | {
"line": 294,
"column": 46
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C_c β\n⊢ HasCompactSupport (⇑f.toContinuousMap - ⇑g.toContinuousMap)",
"ppTerm": "?m.59",
"assigned": tru... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C_c β\n⊢ HasCompactSupport (⇑f + -⇑g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 683,
"column": 4
} | {
"line": 683,
"column": 34
} | {
"line": 683,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nf₁ f₂ : α →C_c ℝ≥0\nh : f₁ ≤ f₂\nx : α\n⊢ ↑((f₁ + { toContinuousMap := f₂.toContinuousMap - f₁.toContinuousMap, hasCompactSupport' := ⋯ }) x) = ↑(f₂ x)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NNReal.instTo... | [
"case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nf₁ f₂ : α →C_c ℝ≥0\nh : f₁ ≤ f₂\nx : α\n⊢ f₁ x + (f₂ x - f₁ x) = f₂ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 13
} | {
"line": 698,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\nhf : 0 ≤ f\nx : α\n⊢ ↑((-f).nnrealPart x) = ↑(0 x)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\nhf : 0 ≤ f\nx : α\n⊢ 0 ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 721,
"column": 2
} | {
"line": 721,
"column": 13
} | {
"line": 721,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nf g : α →C_c ℝ\nx : α\n⊢ (f + g).nnrealPart x ≤ (f.nnrealPart + g.nnrealPart) x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"NonU... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\nf g : α →C_c ℝ\nx : α\n⊢ (f x + g x).toNNReal ≤ (f x).toNNReal + (g x).toNNReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 854,
"column": 4
} | {
"line": 854,
"column": 51
} | {
"line": 854,
"column": 52
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : Group α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : R1Space β\ninst✝² : Group β\ninst✝¹ : ContinuousMul β\ninst✝ : NormedAddCommGroup γ\nφ : α →* β\nhφ : Topology.IsClosedEmbedding ⇑φ\nf : β →C_... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : Group α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : R1Space β\ninst✝² : Group β\ninst✝¹ : ContinuousMul β\ninst✝ : NormedAddCommGroup γ\nφ : α →* β\nhφ : Topology.IsClosedEmbedding ⇑φ\nf : β →C_c γ\nb : β\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 64,
"column": 60
} | {
"line": 67,
"column": 64
} | {
"line": 69,
"column": 0
} | [
{
"pp": "p : ℝ\nhp : 0 < p\n⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x ^ p) = Gamma (1 / p + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Real.instPow",
"Real.parti... | [] | by
convert! (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 79
} | {
"line": 58,
"column": 80
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [
"case refine_4\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i : ι) in L, As... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 210,
"column": 14
} | {
"line": 210,
"column": 25
} | {
"line": 210,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ.symm\n⊢ CurveIntegrable ω γ",
"ppTerm": "?m.49",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ.symm\n⊢ CurveIntegrable ω γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 69
} | {
"line": 62,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [i... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [inst : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 39
} | {
"line": 63,
"column": 40
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Ty... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 31
} | {
"line": 399,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ CurveIntegrable (-ω) γ",
"ppTerm": "?m.62",
"as... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ IntervalIntegrable (-curveIntegralFun ω γ) volume 0 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 403,
"column": 14
} | {
"line": 403,
"column": 25
} | {
"line": 403,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable (-ω) γ\n⊢ CurveIntegrable ω γ",
"ppTerm": "?m.65",
"as... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable (-ω) γ\n⊢ CurveIntegrable ω γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 31
} | {
"line": 468,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommCla... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommClass 𝕜 𝕝 F\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 476,
"column": 4
} | {
"line": 476,
"column": 20
} | {
"line": 476,
"column": 21
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : S... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommClass... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 30
} | {
"line": 534,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x : E) in... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x : E) in 𝓝[s] a, Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 85
} | {
"line": 136,
"column": 86
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 62,
"column": 37
} | {
"line": 62,
"column": 53
} | {
"line": 62,
"column": 54
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\n⊢ IsConnected s",
"ppTerm": "?m.193",
"assigned": true... | [
"a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\n⊢ IsConnected (Ioc a b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 61
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh✝ : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\nhs_conn : IsConnected s\nhfg : IntegrableOn (fun x ↦ f x * g ... | [
"a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh✝ : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\nhs_conn : IsConnected s\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nc : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 144,
"column": 40
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 80
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 34,
"column": 6
} | {
"line": 34,
"column": 26
} | {
"line": 35,
"column": 6
} | [
{
"pp": "case mp.ht\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g ... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 36,
"column": 36
} | {
"line": 36,
"column": 78
} | {
"line": 36,
"column": 79
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 24
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 45
} | {
"line": 49,
"column": 46
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 21
} | {
"line": 89,
"column": 22
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countab... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 19
} | {
"line": 205,
"column": 20
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nhf₀ : f a = 0\n⊢ f =O[𝓝 ... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nhf₀ : f a = 0\n⊢ f =O[𝓝 a] fun x ↦ ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 31,
"column": 16
} | {
"line": 31,
"column": 50
} | {
"line": 32,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a N) ⊆ Ioc (a N) (a (N + 1)) ∪ ⋃ x, ⋃ (_ : x < N), Ioc (a x) (a (x + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 41,
"column": 16
} | {
"line": 41,
"column": 50
} | {
"line": 42,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))\n⊢ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ico (a i) (a (i + 1))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))\n⊢ Ico (a 0) (a N) ⊆ Ico (a N) (a (N + 1)) ∪ ⋃ x, ⋃ (_ : x < N), Ico (a x) (a (x + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 32
} | {
"line": 114,
"column": 33
} | [
{
"pp": "case inl\nf : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\nh_f_int : IntervalIntegrable f volume a (a + ↑N * h)\nk✝ : ℕ\nhk✝ : k✝ < N\nh_neg : h ≤ 0\nk : ℕ\nhk : ↑k ≤ ↑N\n⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.parti... | [
"case inl\nf : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\nh_f_int : IntervalIntegrable f volume a (a + ↑N * h)\nk✝ : ℕ\nhk✝ : k✝ < N\nh_neg : h ≤ 0\nk : ℕ\nhk : ↑k ≤ ↑N\n⊢ 0 ≤ ↑k * h ∧ ↑k * h ≤ ↑N * h ∨ ↑N * h ≤ ↑k * h ∧ ↑k * h ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 39
} | {
"line": 152,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | rw [iteratedDerivWithin_eq_iterate] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 11
} | {
"line": 188,
"column": 12
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : Disjoint s₀ s₁\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fu... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : Disjoint s₀ s₁\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fun j ↦ if j =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 38
} | {
"line": 167,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 34
} | {
"line": 83,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ↑{ carrier :... | refine (Λ.mono ?_).trans hg.2.le | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)... | [
"case pos\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ↑{... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 43
} | {
"line": 124,
"column": 43
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : (Fin n → ℂ) → E\nc : Fin n → ℂ\n⊢ IntegrableOn (fun θ ↦ f (torusMap c 0 θ)) (Icc 0 fun x ↦ 2 * π) volume",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Pi.preorder",
"Real.pi",
... | [
"n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : (Fin n → ℂ) → E\nc : Fin n → ℂ\n⊢ IntegrableOn (fun θ ↦ f (const (Fin n → ℝ) c θ)) (Icc 0 fun x ↦ 2 * π) volume"
] | torusMap_zero_radius | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 58
} | {
"line": 158,
"column": 4
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x... | [
"n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ ∫ (θ : Fin n → ℝ) in Icc 0 fun x ↦ 2 * π,\n (∏ i, ↑(R i) * cexp (↑(θ i) * I) * I) • f (torusMap c R θ) +\n (∏ i, ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | {
"line": 83,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"ppTerm": "?m.42... | [] | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | {
"line": 83,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"ppTerm": "?m.42... | [] | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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