module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 333,
"column": 70
} | {
"line": 414,
"column": 42
} | {
"line": 416,
"column": 0
} | [
{
"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 : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\n⊢... | [] | by
rcases eq_or_ne μ 0 with rfl | hμ
· simp
obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
exists_ae_eq_isMarkovKernel hf_Indep.ae_isProbabilityMeasure hμ
apply IndepFun.congr (Filter.EventuallyEq.symm η_eq)
-- We introduce π-systems, built from the π-system of boxes which gene... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 10
} | {
"line": 238,
"column": 6
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁴ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : Secon... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁴ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTo... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Process.Filtration | {
"line": 330,
"column": 27
} | {
"line": 333,
"column": 40
} | {
"line": 335,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝¹ : LinearOrder ι\ninst✝ : DenselyOrdered ι\n𝓕 : Filtration ι m\ni : ι\nhi : ¬IsMax i\n⊢ ↑𝓕₊ i = ⨅ j, ⨅ (_ : j > i), ↑𝓕 j",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Preorder.topology",
"Set.Ioi",
"P... | [] | by
letI := Preorder.topology ι; haveI : OrderTopology ι := ⟨rfl⟩
have : (𝓝[>] i).NeBot := nhdsGT_neBot_of_exists_gt (not_isMax_iff.mp hi)
exact rightCont_eq_of_neBot_nhdsGT _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 275,
"column": 6
} | {
"line": 275,
"column": 30
} | {
"line": 276,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁴ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 731,
"column": 70
} | {
"line": 733,
"column": 69
} | {
"line": 734,
"column": 8
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nβ : Type u_4\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\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... | [] | by
congr 1
simp only [Set.iInter_ite, Set.iInter_univ, Set.inter_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 52
} | {
"line": 325,
"column": 2
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝¹¹ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝¹⁰ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝⁹ : SeminormedAddCommGroup E\n... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝¹¹ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝¹⁰ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝⁹ : SeminormedAddCommGroup E\ninst✝⁸ : Sec... | refine TendstoInDistribution.continuous_comp hg ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 11
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case neg\nι : 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 : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\ninst✝ : TopologicalSpace β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nFi : Infinite ι\ne : ℕ ≃ ι\ng : (n : ℕ)... | [] | · simp [hb]
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 143,
"column": 34
} | {
"line": 143,
"column": 51
} | {
"line": 143,
"column": 51
} | [
{
"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\nmem_Icc : ∀ᶠ (i : ι) in ↑u, μ (s ∩ F i) / μ (F i) ∈ Icc 0 1\nx : ℝ≥0∞\nhx : x ∈ Icc 0 1 ∧ ↑(Ultrafilte... | [] | use x; exact hx.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 143,
"column": 34
} | {
"line": 143,
"column": 51
} | {
"line": 143,
"column": 51
} | [
{
"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\nmem_Icc : ∀ᶠ (i : ι) in ↑u, μ (s ∩ F i) / μ (F i) ∈ Icc 0 1\nx : ℝ≥0∞\nhx : x ∈ Icc 0 1 ∧ ↑(Ultrafilte... | [] | use x; exact hx.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 49
} | {
"line": 107,
"column": 2
} | [
{
"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... | let Z : ℕ → Set E := fun n => (S n) ∩ (L \ {0}) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Group.ModularCharacter | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 6
} | {
"line": 116,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\ng h : G\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nmul_g_meas : Measurable fun x ↦ x * g\nmul_h_meas : Measurable fun x ↦ x * h\nν : Measure G := haar\n⊢ modular... | [
"G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\ng h : G\nthis✝¹ : MeasurableSpace G := ⋯\nthis✝ : BorelSpace G\nmul_g_meas : Measurable fun x ↦ x * g\nmul_h_meas : Measurable fun x ↦ x * h\nν : Measure G := ⋯\n⊢ modularCharacterFun g * modu... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 67
} | {
"line": 260,
"column": 2
} | [
{
"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 c : E\nω : E → E →L[𝕜] F\nγab : Path a b\nh : CurveIntegrable ω γab\nγbc : Path b c\n⊢ IntervalIntegrable (curveIntegral... | [
"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 c : E\nω : E → E →L[𝕜] F\nγab : Path a b\nh : CurveIntegrable ω γab\nγbc : Path b c\n⊢ IntervalIntegrable (fun t ↦ 2 • curveIntegral... | refine .congr_ae ?_ (curveIntegralFun_trans_aeeq_left _ _ _).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 70
} | {
"line": 52,
"column": 4
} | [
{
"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... | have hgc : ContinuousOn g (Icc a b) :=
(UniformSpace.Completion.continuous_coe E).comp_continuousOn hfc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 74
} | {
"line": 31,
"column": 4
} | [
{
"pp": "case mp\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' : g =ᵐ[μ.wi... | [
"case mp\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' : g =ᵐ[μ.withDensity fu... | refine ⟨fun x => (f x : ℝ) • g' x, hf.coe_nnreal_real.smul g'meas, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 100,
"column": 42
} | {
"line": 100,
"column": 48
} | {
"line": 100,
"column": 48
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 100,
"column": 42
} | {
"line": 100,
"column": 48
} | {
"line": 100,
"column": 48
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 100,
"column": 42
} | {
"line": 100,
"column": 48
} | {
"line": 100,
"column": 48
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 52
} | {
"line": 107,
"column": 52
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 52
} | {
"line": 107,
"column": 52
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 52
} | {
"line": 107,
"column": 52
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 72
} | {
"line": 158,
"column": 2
} | [
{
"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 /... | [] | grw [fpp_bound x, abs_of_nonneg (sub_nonneg.mpr hx.1), div_mul_comm] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 42
} | {
"line": 168,
"column": 43
} | [
{
"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 : s₀ ⊆ s₁ᶜ\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fun j ↦ if j = 0 the... | ← subset_compl_iff_disjoint_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 49
} | {
"line": 127,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 49
} | {
"line": 127,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 127,
"column": 43
} | {
"line": 127,
"column": 49
} | {
"line": 127,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 254,
"column": 6
} | {
"line": 257,
"column": 55
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK₁ K₂ : Compacts X\ndisj : Disjoint ↑K₁ ↑K₂\nb : ℝ≥0\nf : X →C_c ℝ≥0\nhf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}\nΛf_eq_b : Λ f = b\nhsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f\nh... | [] | simp only [CompactlySupportedContinuousMap.coe_add, ContinuousMap.toFun_eq_coe,
CompactlySupportedContinuousMap.coe_toContinuousMap] at sum_g
rw [sum_g (mem_of_subset_of_mem subset_closure (mem_support.mpr h))]
simp only [Pi.one_apply, NNReal.coe_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 254,
"column": 6
} | {
"line": 257,
"column": 55
} | {
"line": 258,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK₁ K₂ : Compacts X\ndisj : Disjoint ↑K₁ ↑K₂\nb : ℝ≥0\nf : X →C_c ℝ≥0\nhf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}\nΛf_eq_b : Λ f = b\nhsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f\nh... | [] | simp only [CompactlySupportedContinuousMap.coe_add, ContinuousMap.toFun_eq_coe,
CompactlySupportedContinuousMap.coe_toContinuousMap] at sum_g
rw [sum_g (mem_of_subset_of_mem subset_closure (mem_support.mpr h))]
simp only [Pi.one_apply, NNReal.coe_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 236,
"column": 2
} | {
"line": 265,
"column": 74
} | {
"line": 267,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK₁ K₂ : Compacts X\ndisj : Disjoint ↑K₁ ↑K₂\n⊢ rieszContentAux Λ (K₁ ⊔ K₂) = rieszContentAux Λ K₁ + rieszContentAux Λ K₂",
"ppTerm": "?m.33",
"assigned": true,
"usedCo... | [] | refine le_antisymm (rieszContentAux_sup_le Λ K₁ K₂) ?_
refine le_csInf (rieszContentAux_image_nonempty Λ (K₁ ⊔ K₂)) ?_
intro b ⟨f, ⟨hf, Λf_eq_b⟩⟩
have hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support f := by
intro x hx
rw [mem_support]
exact ne_of_gt <| lt_of_lt_of_le (zero_lt_one' ℝ≥0) (hf x hx)
have hsubsuppf ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 236,
"column": 2
} | {
"line": 265,
"column": 74
} | {
"line": 267,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK₁ K₂ : Compacts X\ndisj : Disjoint ↑K₁ ↑K₂\n⊢ rieszContentAux Λ (K₁ ⊔ K₂) = rieszContentAux Λ K₁ + rieszContentAux Λ K₂",
"ppTerm": "?m.33",
"assigned": true,
"usedCo... | [] | refine le_antisymm (rieszContentAux_sup_le Λ K₁ K₂) ?_
refine le_csInf (rieszContentAux_image_nonempty Λ (K₁ ⊔ K₂)) ?_
intro b ⟨f, ⟨hf, Λf_eq_b⟩⟩
have hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support f := by
intro x hx
rw [mem_support]
exact ne_of_gt <| lt_of_lt_of_le (zero_lt_one' ℝ≥0) (hf x hx)
have hsubsuppf ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 139,
"column": 43
} | {
"line": 139,
"column": 49
} | {
"line": 139,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 139,
"column": 43
} | {
"line": 139,
"column": 49
} | {
"line": 139,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 139,
"column": 43
} | {
"line": 139,
"column": 49
} | {
"line": 139,
"column": 49
} | [
{
"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.Countable... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 99,
"column": 11
} | {
"line": 99,
"column": 28
} | {
"line": 99,
"column": 29
} | [
{
"pp": "case h\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\nK : Set X\nhK : IsCompact K\nhfK : ∀ x ∈ K, f x = 1\np : NNReal\nhp : Λ f = ↑p\n⊢ f.nnr... | [
"case h\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\nK : Set X\nhK : IsCompact K\nhfK : ∀ x ∈ K, f x = 1\np : NNReal\nhp : Λ f = ↑p\n⊢ (∀ x ∈ { carrier ... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 44
} | {
"line": 200,
"column": 44
} | [
{
"pp": "E : Type u_2\nmE : MeasurableSpace E\nμ : Measure E\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : BorelSpace E\nr : ℝ\nt : E\n⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂Measure.map (fun x ↦ r • x) μ = ∫ (x : E), cexp (↑⟪x, r • t⟫ * I) ∂μ",
"ppTerm": "?m.44",
"assigned": true... | [
"E : Type u_2\nmE : MeasurableSpace E\nμ : Measure E\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : BorelSpace E\nr : ℝ\nt : E\n⊢ ∫ (x : E), cexp (↑⟪r • x, t⟫ * I) ∂μ = ∫ (x : E), cexp (↑⟪x, r • t⟫ * I) ∂μ"
] | integral_map (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 11
} | {
"line": 144,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nhxU : ∀ (i : ↥I'), x ∈ U i\nhU : ∀ (i : ↥I'), ∃ f, Continuous[t ↑i, _] f ∧ f x = 0 ∧ EqOn f 1 (... | [
"ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nhU : ∀ (i : ↥I'), ∃ f, Continuous[t ↑i, _] f ∧ f x = 0 ∧ EqOn f 1 (U i)ᶜ\n⊢ ∃ f, Continuous[⨅ i, t i, _] f ... | clear hxU | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.MeasureTheory.Measure.DiracProba | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 64
} | {
"line": 40,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nK : Set X\nK_closed : IsClosed[inst✝¹] K\nx✝ : X\nx_notin_K : x✝ ∉ K\ng : X → ↑unitInterval\ng_cont : Continuous[inst✝¹, _] g\ngx_zero : g x✝ = 0\ng_one_on_K : EqOn g 1 K\nx y : X\n⊢ ↑1 * dist ↑(g x) ↑(g y) ≤ 1",
"ppTerm":... | [] | simpa using Real.dist_le_of_mem_Icc_01 (g x).prop (g y).prop | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.FiniteMeasurePi | {
"line": 105,
"column": 4
} | {
"line": 106,
"column": 37
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case hmeas\nι : Type u_1\nα : ι → Type u_2\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → MeasurableSpace (α i)\ninst✝³ : (i : ι) → TopologicalSpace (α i)\ninst✝² : ∀ (i : ι), SecondCountableTopology (α i)\ninst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)\ninst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)\nμ : (i ... | [] | rintro - ⟨s, rfl, smeas, hs⟩
exact MeasurableSet.univ_pi smeas | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.FiniteMeasurePi | {
"line": 105,
"column": 4
} | {
"line": 106,
"column": 37
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case hmeas\nι : Type u_1\nα : ι → Type u_2\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → MeasurableSpace (α i)\ninst✝³ : (i : ι) → TopologicalSpace (α i)\ninst✝² : ∀ (i : ι), SecondCountableTopology (α i)\ninst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)\ninst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)\nμ : (i ... | [] | rintro - ⟨s, rfl, smeas, hs⟩
exact MeasurableSet.univ_pi smeas | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 26
} | {
"line": 134,
"column": 27
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\n⊢ ε < ∞ →\n MeasurableSet B →\n μ B ≤\n... | [
"case neg\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\nε_lt_top : ε < ∞\n⊢ MeasurableSet B →\n μ B ≤\n κ... | ε_lt_top | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 162,
"column": 63
} | {
"line": 162,
"column": 85
} | {
"line": 164,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\nε_lt_top : ε < ∞\nB_mble : MeasurableSet B\nhalf_ε_pos : ... | [] | rw [add_comm r.toReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 162,
"column": 63
} | {
"line": 162,
"column": 85
} | {
"line": 164,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\nε_lt_top : ε < ∞\nB_mble : MeasurableSet B\nhalf_ε_pos : ... | [] | rw [add_comm r.toReal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 162,
"column": 63
} | {
"line": 162,
"column": 85
} | {
"line": 164,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν κ : Measure Ω\nLPμν_finite : ¬levyProkhorovEDist μ ν = ∞\nLPνκ_finite : ¬levyProkhorovEDist ν κ = ∞\nε : ℝ≥0∞\nB : Set Ω\nε_pos : 0 < ε\nε_lt_top : ε < ∞\nB_mble : MeasurableSet B\nhalf_ε_pos : ... | [] | rw [add_comm r.toReal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 226,
"column": 17
} | {
"line": 226,
"column": 25
} | {
"line": 226,
"column": 26
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ≥0∞\nh : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε\nε : ... | [
"Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ≥0∞\nh : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε\nε : ℝ≥0∞\nB : Se... | ε_lt_top | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 247,
"column": 17
} | {
"line": 247,
"column": 25
} | {
"line": 247,
"column": 26
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ\nδ_nn : 0 ≤ δ\nh : ∀ (ε : ℝ) (B : Set Ω), δ < ε → MeasurableSet B → μ B ≤ ν (thickening ε B) + ENNReal.ofReal... | [
"Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoEMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : IsProbabilityMeasure ν\nδ : ℝ\nδ_nn : 0 ≤ δ\nh : ∀ (ε : ℝ) (B : Set Ω), δ < ε → MeasurableSet B → μ B ≤ ν (thickening ε B) + ENNReal.ofReal ε\nε : ℝ≥0∞... | ε_lt_top | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 100,
"column": 6
} | {
"line": 106,
"column": 52
} | {
"line": 107,
"column": 6
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹³ : Group A\ninst✝¹² : Group B\ninst✝¹¹ : Group C\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : TopologicalSpace B\ninst✝⁸ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁷ : IsTopologicalGroup A\ninst✝⁶ : IsTopologicalGroup B\ni... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹³ : Group A\ninst✝¹² : Group B\ninst✝¹¹ : Group C\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : TopologicalSpace B\ninst✝⁸ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁷ : IsTopologicalGroup A\ninst✝⁶ : IsTopologicalGroup B\ninst✝⁵ : Norm... | obtain ⟨δ, hδ0, hδ⟩ : ∃ δ > 0, ENNReal.ofReal δ * μA S < ENNReal.ofReal ε := by
rw [← ENNReal.ofReal_toReal hSc.measure_ne_top, ← measureReal_def]
by_cases hS' : μA.real S = 0
· simp [hS', hε, exists_gt]
· refine ⟨ε / 2 / μA.real S, by positivity, ?_⟩
rwa [← ENNReal.ofReal_mul'... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 215,
"column": 62
} | {
"line": 215,
"column": 68
} | {
"line": 215,
"column": 68
} | [
{
"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.Countable\n... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 215,
"column": 62
} | {
"line": 215,
"column": 68
} | {
"line": 215,
"column": 68
} | [
{
"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.Countable\n... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 215,
"column": 62
} | {
"line": 215,
"column": 68
} | {
"line": 215,
"column": 68
} | [
{
"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.Countable\n... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 120,
"column": 11
} | {
"line": 120,
"column": 28
} | {
"line": 120,
"column": 29
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹³ : Group A\ninst✝¹² : Group B\ninst✝¹¹ : Group C\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : TopologicalSpace B\ninst✝⁸ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁷ : IsTopologicalGroup A\ninst✝⁶ : IsTopologicalGroup B\ni... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹³ : Group A\ninst✝¹² : Group B\ninst✝¹¹ : Group C\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : TopologicalSpace B\ninst✝⁸ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁷ : IsTopologicalGroup A\ninst✝⁶ : IsTopologicalGroup B\ninst✝⁵ : Norm... | Set.mem_setOf_eq, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 122,
"column": 41
} | {
"line": 122,
"column": 56
} | {
"line": 122,
"column": 56
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ ψ : G ≃ₜ* G\nhφ : Measurable ⇑φ\nhψ : Measurable ⇑ψ\n⊢ haar.haarScalarFactor (map (⇑φ ∘ ⇑ψ) haar) = haar.haarScalarFactor (map ... | [
"G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ ψ : G ≃ₜ* G\nhφ : Measurable ⇑φ\nhψ : Measurable ⇑ψ\n⊢ haar.haarScalarFactor (map (⇑φ) (map (⇑ψ) haar)) = haar.haarScalarFactor (map (⇑ψ) h... | ← map_map hφ hψ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 119,
"column": 2
} | {
"line": 125,
"column": 39
} | {
"line": 127,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ ψ : G ≃ₜ* G\n⊢ mulEquivHaarChar (ψ.trans φ) = mulEquivHaarChar ψ * mulEquivHaarChar φ",
"ppTerm": "?m.49",
"assigned": ... | [] | rw [mulEquivHaarChar_eq haar ψ, mulEquivHaarChar_eq haar (ψ.trans φ)]
have hφ : Measurable φ := by fun_prop
have hψ : Measurable ψ := by fun_prop
simp_rw [ContinuousMulEquiv.coe_trans, ← map_map hφ hψ]
have h_reg : (haar.map ψ).Regular := Regular.map ψ.toHomeomorph
rw [MeasureTheory.Measure.haarScalarFactor_e... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 119,
"column": 2
} | {
"line": 125,
"column": 39
} | {
"line": 127,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ ψ : G ≃ₜ* G\n⊢ mulEquivHaarChar (ψ.trans φ) = mulEquivHaarChar ψ * mulEquivHaarChar φ",
"ppTerm": "?m.49",
"assigned": ... | [] | rw [mulEquivHaarChar_eq haar ψ, mulEquivHaarChar_eq haar (ψ.trans φ)]
have hφ : Measurable φ := by fun_prop
have hψ : Measurable ψ := by fun_prop
simp_rw [ContinuousMulEquiv.coe_trans, ← map_map hφ hψ]
have h_reg : (haar.map ψ).Regular := Regular.map ψ.toHomeomorph
rw [MeasureTheory.Measure.haarScalarFactor_e... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 6
} | {
"line": 131,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ : G ≃ₜ* G\n⊢ mulEquivHaarChar φ.symm = (mulEquivHaarChar φ)⁻¹",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants... | [
"G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ : G ≃ₜ* G\n⊢ (mulEquivHaarChar φ)⁻¹ = mulEquivHaarChar φ.symm"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 205,
"column": 8
} | {
"line": 205,
"column": 48
} | {
"line": 205,
"column": 48
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹⁸ : Group A\ninst✝¹⁷ : Group B\ninst✝¹⁶ : Group C\ninst✝¹⁵ : TopologicalSpace A\ninst✝¹⁴ : TopologicalSpace B\ninst✝¹³ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹² : IsTopologicalGroup A\ninst✝¹¹ : IsTopologicalGroup ... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹⁸ : Group A\ninst✝¹⁷ : Group B\ninst✝¹⁶ : Group C\ninst✝¹⁵ : TopologicalSpace A\ninst✝¹⁴ : TopologicalSpace B\ninst✝¹³ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹² : IsTopologicalGroup A\ninst✝¹¹ : IsTopologicalGroup B\ninst✝¹⁰ :... | integral_map (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 174,
"column": 8
} | {
"line": 174,
"column": 48
} | {
"line": 174,
"column": 48
} | [
{
"pp": "case e'_4\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\na : E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)\nx : ℝ\n⊢ ∫ (x_1 : E... | [
"case e'_4\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\na : E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)\nx : ℝ\n⊢ ∫ (x_1 : E), cexp (↑x ... | integral_map (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 99,
"column": 11
} | {
"line": 110,
"column": 16
} | {
"line": 111,
"column": 2
} | [] | [] | ∫ x, g x ∂μ
_ ≤ ∫ (x : E), ‖g.toBoundedContinuousFunction‖ ∂μ := by
gcongr
· exact g.continuous.integrable_of_hasCompactSupport g.hasCompactSupport
· simp
· intro x
apply le_of_abs_le
exact g.toBoundedContinuousFunction.norm_coe_le_norm x
_ ≤ C * ‖g.toBoun... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 383,
"column": 4
} | {
"line": 385,
"column": 73
} | {
"line": 386,
"column": 2
} | [
{
"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\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nω : E → E →L[𝕜] F\ninst✝ : CompleteSpace F\nhs : Convex ℝ s\... | [] | refine hs.exists_forall_hasFDerivWithinAt_of_fderivWithin_symmetric hω fun a ha x _ y _ ↦ ?_
rw [fderivWithin_eq_fderiv, hdω a ha]
exacts [hso.uniqueDiffOn a ha, hω.differentiableAt (hso.mem_nhds ha)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 383,
"column": 4
} | {
"line": 385,
"column": 73
} | {
"line": 386,
"column": 2
} | [
{
"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\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ns : Set E\nω : E → E →L[𝕜] F\ninst✝ : CompleteSpace F\nhs : Convex ℝ s\... | [] | refine hs.exists_forall_hasFDerivWithinAt_of_fderivWithin_symmetric hω fun a ha x _ y _ ↦ ?_
rw [fderivWithin_eq_fderiv, hdω a ha]
exacts [hso.uniqueDiffOn a ha, hω.differentiableAt (hso.mem_nhds ha)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 432,
"column": 42
} | {
"line": 432,
"column": 61
} | {
"line": 432,
"column": 61
} | [
{
"pp": "a : ℂ\nr : ℝ\n⊢ volume (Metric.ball a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ENNReal.ofNNReal",
"MeasureTheory.Measur... | [
"a : ℂ\nr : ℝ\n⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi = ENNReal.ofReal r ^ 2 * ↑NNReal.pi"
] | Complex.volume_ball | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 221,
"column": 8
} | {
"line": 221,
"column": 39
} | {
"line": 221,
"column": 39
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (... | [
"E : Type u_1\nmE : MeasurableSpace E\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r ... | Measure.map_apply (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 112,
"column": 27
} | {
"line": 112,
"column": 31
} | {
"line": 112,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 32
} | {
"line": 411,
"column": 8
} | [
{
"pp": "case h\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict... | [
"case h\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed... | filter_upwards [] with x | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 56
} | {
"line": 161,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : Set α\nhs : MeasurableSet s\nε✝ : ℝ≥0∞\nhε : 0 < ε✝\nf : ℕ → Set α\nf_disj : Pairwise (Function.onFun Disjoint ... | [] | gcongr; exact ENNReal.sum_le_tsum (Finset.range n) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 56
} | {
"line": 161,
"column": 4
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : Set α\nhs : MeasurableSet s\nε✝ : ℝ≥0∞\nhε : 0 < ε✝\nf : ℕ → Set α\nf_disj : Pairwise (Function.onFun Disjoint ... | [] | gcongr; exact ENNReal.sum_le_tsum (Finset.range n) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 157,
"column": 10
} | {
"line": 157,
"column": 56
} | {
"line": 158,
"column": 10
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝ : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜 : Set (Set X)\nh𝒜 : μ.MeasureDense 𝒜\nc : E\nhc : c ≠ 0\np_pos : 0 < p\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nhε : ε > 0\naux... | [
"X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝ : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜 : Set (Set X)\nh𝒜 : μ.MeasureDense 𝒜\nc : E\nhc : c ≠ 0\np_pos : 0 < p\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nhε : ε > 0\naux : 0 < (ε / ... | have := toReal_pos p_pos.ne.symm p_ne_top.elim | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 85
} | {
"line": 208,
"column": 8
} | [
{
"pp": "case iUnion\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\ninst✝ : IsFiniteMeasure μ\nh𝒜 : IsSetAlgebra 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nf : ℕ → Set X\nhs✝ : ∀ (n : ℕ), MeasurableSet (f n)\nhf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈... | [
"case iUnion\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\ninst✝ : IsFiniteMeasure μ\nh𝒜 : IsSetAlgebra 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nf : ℕ → Set X\nhs✝ : ∀ (n : ℕ), MeasurableSet (f n)\nhf : ∀ (n : ℕ), MeasurableSet (f n) ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real ... | rcases Metric.tendsto_atTop.1 this (ε / 2) (by linarith [ε_pos]) with ⟨N, hN⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.WithDensityFinite | {
"line": 64,
"column": 2
} | {
"line": 68,
"column": 36
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\n⊢ ae μ.toFiniteAux = ae μ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.ae.congr_simp",
"Exists.choose_spec",
"MeasureTheory.Measure... | [] | rw [Measure.toFiniteAux]
split_ifs
· simp
· obtain ⟨_, h₁, h₂⟩ := (exists_isFiniteMeasure_absolutelyContinuous μ).choose_spec
exact h₂.ae_le.antisymm h₁.ae_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensityFinite | {
"line": 64,
"column": 2
} | {
"line": 68,
"column": 36
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\n⊢ ae μ.toFiniteAux = ae μ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.ae.congr_simp",
"Exists.choose_spec",
"MeasureTheory.Measure... | [] | rw [Measure.toFiniteAux]
split_ifs
· simp
· obtain ⟨_, h₁, h₂⟩ := (exists_isFiniteMeasure_absolutelyContinuous μ).choose_spec
exact h₂.ae_le.antisymm h₁.ae_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensityFinite | {
"line": 86,
"column": 67
} | {
"line": 87,
"column": 70
} | {
"line": 89,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\n⊢ μ.toFinite = 0 ↔ μ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.toFinite_apply_eq_zero_iff._simp_1",
"Set.un... | [] | by
simp_rw [← Measure.measure_univ_eq_zero, toFinite_apply_eq_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 55
} | {
"line": 189,
"column": 6
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\ninst✝ : IsFiniteMeasure μ\nh𝒜 : IsSetAlgebra 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\n⊢ MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ.real (s ∆ t) < ε",
"ppTerm": "?m.53",
"assigned... | [] | induction s, ms using generateFrom_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 83
} | {
"line": 289,
"column": 6
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\nh𝒜 : IsSetAlgebra 𝒜\nS : μ.FiniteSpanningSetsIn 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nε_pos : 0 < ε\nT : ℕ → Set X := accumulate S.set\nT_mem : ∀ (n : ℕ), T n ∈ 𝒜\nT... | [
"X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\nh𝒜 : IsSetAlgebra 𝒜\nS : μ.FiniteSpanningSetsIn 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nε_pos : 0 < ε\nT : ℕ → Set X := accumulate S.set\nT_mem : ∀ (n : ℕ), T n ∈ 𝒜\nT_finite : ∀ ... | rcases Metric.tendsto_atTop.1 this (ε / 2) (by linarith [ε_pos]) with ⟨N, hN⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 309,
"column": 14
} | {
"line": 309,
"column": 38
} | {
"line": 310,
"column": 14
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\nh𝒜 : IsSetAlgebra 𝒜\nS : μ.FiniteSpanningSetsIn 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nε_pos : 0 < ε\nT : ℕ → Set X := accumulate S.set\nT_mem : ∀ (n : ℕ), T n ∈ 𝒜\nT... | [
"case ac\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\n𝒜 : Set (Set X)\nh𝒜 : IsSetAlgebra 𝒜\nS : μ.FiniteSpanningSetsIn 𝒜\nhgen : m = MeasurableSpace.generateFrom 𝒜\ns : Set X\nms : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ\nε_pos : 0 < ε\nT : ℕ → Set X := ⋯\nT_mem : ∀ (n : ℕ), T n ∈ 𝒜\nT_finite : ∀ (n : ℕ... | apply ENNReal.add_lt_add | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 74,
"column": 4
} | {
"line": 79,
"column": 93
} | {
"line": 80,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x ... | [
"case refine_2\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume ... | calc
ENNReal.ofReal (4⁻¹ ^ Fintype.card ι)
= volume (closedBall (f (ε n) (hε' n)) (ε n / 4)) / volume (closedBall x (ε n)) := ?_
_ ≤ volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n)) := by
gcongr
exact subset_inter ((hf₁ _ <| hε' n).trans interior_subset_closure) ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 386,
"column": 6
} | {
"line": 386,
"column": 30
} | {
"line": 387,
"column": 6
} | [
{
"pp": "case refine_1\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set (Se... | [
"case refine_1\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set (Set X)\ncount_... | rintro - ⟨s, s_mem, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.OuterMeasure.OfAddContent | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 89
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns : Set α\nhC : IsSetSemiring C\nm : AddContent ℝ≥0∞ C\nm_sigma_subadd : m.IsSigmaSubadditive\nm_top : ∀ s ∉ C, m s = ∞\nhs : s ∈ C\nf : ℕ → Set α\nhf : ∀ (i : ℕ), f i ∈ C\nhs_subset : s ⊆ ⋃ i, f i\ni : ℕ\n⊢ m (s ∩ f i) ≤ m (f i)",
"ppTerm": "?m.151",
"assigned": ... | [] | exact addContent_mono hC (hC.inter_mem _ hs _ (hf i)) (hf i) Set.inter_subset_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 409,
"column": 12
} | {
"line": 409,
"column": 23
} | {
"line": 409,
"column": 24
} | [
{
"pp": "case refine_2.inr\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set... | [
"case refine_2.inr\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set (Set X)\nco... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SpecificCodomains.WithLp | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 48
} | {
"line": 58,
"column": 49
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\np q : ℝ≥0∞\ninst✝² : Fact (1 ≤ q)\nE : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → WithLp q (E × F)\n⊢ MemLp f p μ ↔ MemLp (fun x ↦ (f x).ofLp.1) p μ ∧ MemLp (fun x ↦ (f x).snd) p μ",
"ppTerm": "?m... | [
"X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\np q : ℝ≥0∞\ninst✝² : Fact (1 ≤ q)\nE : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → WithLp q (E × F)\n⊢ MemLp f p μ ↔ MemLp (fun x ↦ (f x).ofLp.1) p μ ∧ MemLp (fun x ↦ (f x).ofLp.2) p μ"
] | ← WithLp.ofLp_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 20
} | {
"line": 81,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ ν : Measure X\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ns : Set X\nhs : IsHahnDecomposition μ ν s\nhsc : IsHahnDecomposition ν μ sᶜ\nh₁ :\n ((ν - μ).restrict s).toSignedMeasure =\n VectorMeasure.restrict ν.toSignedMeasure s - VectorMeasure.restr... | [
"X : Type u_1\nmX : MeasurableSpace X\nμ ν : Measure X\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ns : Set X\nhs : IsHahnDecomposition μ ν s\nhsc : IsHahnDecomposition ν μ sᶜ\nh₁ :\n ((ν - μ).restrict s).toSignedMeasure =\n VectorMeasure.restrict ν.toSignedMeasure s - VectorMeasure.restrict μ.toSign... | have partition₁ := VectorMeasure.restrict_add_restrict_compl (v := (μ - ν).toSignedMeasure)
hs.measurableSet | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 31
} | {
"line": 223,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [] | exact tendsto_nhds_unique A B | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 31
} | {
"line": 258,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [] | exact tendsto_nhds_unique A B | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 72
} | {
"line": 164,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : μ.ennrealVariation s = 0\nhsm : MeasurableSet s\n⊢ μ s = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [] | suffices ‖μ s‖ₑ ≤ 0 by simp_all
grw [enorm_measure_le_variation, ← ennrealVariation_apply _ hsm, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 72
} | {
"line": 164,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : μ.ennrealVariation s = 0\nhsm : MeasurableSet s\n⊢ μ s = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConsta... | [] | suffices ‖μ s‖ₑ ≤ 0 by simp_all
grw [enorm_measure_le_variation, ← ennrealVariation_apply _ hsm, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 39
} | {
"line": 288,
"column": 2
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetSemiring C\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : hα = generateFrom C\nh''C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 303,
"column": 14
} | {
"line": 303,
"column": 49
} | {
"line": 304,
"column": 2
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetSemiring C\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : hα = generateFrom C\ns : Set α\nhs : s ∈ C\n⊢ MeasurableSet... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 36
} | {
"line": 271,
"column": 6
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : ENormedAddCommMonoid V\ninst✝¹ : T2Space V\nμ : VectorMeasure X V\nY : Type u_3\ninst✝ : MeasurableSpace Y\nφ : X → Y\nhφ : MeasurableEmbedding φ\ns : Set Y\nhs : MeasurableSet s\nt : Set Y\nht : t ⊆ s \\ Set.rang... | [
"X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : ENormedAddCommMonoid V\ninst✝¹ : T2Space V\nμ : VectorMeasure X V\nY : Type u_3\ninst✝ : MeasurableSpace Y\nφ : X → Y\nhφ : MeasurableEmbedding φ\ns : Set Y\nhs : MeasurableSet s\nt : Set Y\nht : t ⊆ s \\ Set.range φ\nt_meas ... | have : φ ⁻¹' t = ∅ := by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 336,
"column": 57
} | {
"line": 336,
"column": 96
} | {
"line": 336,
"column": 96
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup V\nμ ν : VectorMeasure X V\ninst✝ : Finite X\nthis : Fintype X := Fintype.ofFinite X\nb : Finpartition ⟨Set.univ, ⋯⟩\nhb : b ∈ Finset.univ\n⊢ ∑ x ∈ b.parts, ‖μ ↑x‖ₑ < ∞",
"ppTerm": "?m.36",
"assigned": true,
"us... | [] | simp [ENNReal.sum_lt_top, enorm_lt_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 336,
"column": 57
} | {
"line": 336,
"column": 96
} | {
"line": 336,
"column": 96
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup V\nμ ν : VectorMeasure X V\ninst✝ : Finite X\nthis : Fintype X := Fintype.ofFinite X\nb : Finpartition ⟨Set.univ, ⋯⟩\nhb : b ∈ Finset.univ\n⊢ ∑ x ∈ b.parts, ‖μ ↑x‖ₑ < ∞",
"ppTerm": "?m.36",
"assigned": true,
"us... | [] | simp [ENNReal.sum_lt_top, enorm_lt_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 336,
"column": 57
} | {
"line": 336,
"column": 96
} | {
"line": 336,
"column": 96
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup V\nμ ν : VectorMeasure X V\ninst✝ : Finite X\nthis : Fintype X := Fintype.ofFinite X\nb : Finpartition ⟨Set.univ, ⋯⟩\nhb : b ∈ Finset.univ\n⊢ ∑ x ∈ b.parts, ‖μ ↑x‖ₑ < ∞",
"ppTerm": "?m.36",
"assigned": true,
"us... | [] | simp [ENNReal.sum_lt_top, enorm_lt_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 109,
"column": 2
} | {
"line": 131,
"column": 32
} | {
"line": 133,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ μ.semivariation t = ∞ ∧ 1 ≤ ‖μ (s \\ t)‖ₑ",
"ppTerm": "?m.41",
"assig... | [] | obtain ⟨t, ts, t_meas, ht⟩ : ∃ t ⊆ s, MeasurableSet t ∧ 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ := by
apply exists_subset_lt_enorm_apply_of_lt_semivariation hs
rw [h's]
finiteness
have h't : 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ := by
apply (ENNReal.mul_le_mul_iff_right (a := 2) (by simp) (by simp)).1
rw [mul_add, add_comm, mu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 109,
"column": 2
} | {
"line": 131,
"column": 32
} | {
"line": 133,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\n⊢ ∃ t, MeasurableSet t ∧ t ⊆ s ∧ μ.semivariation t = ∞ ∧ 1 ≤ ‖μ (s \\ t)‖ₑ",
"ppTerm": "?m.41",
"assig... | [] | obtain ⟨t, ts, t_meas, ht⟩ : ∃ t ⊆ s, MeasurableSet t ∧ 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ := by
apply exists_subset_lt_enorm_apply_of_lt_semivariation hs
rw [h's]
finiteness
have h't : 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ := by
apply (ENNReal.mul_le_mul_iff_right (a := 2) (by simp) (by simp)).1
rw [mul_add, add_comm, mu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 424,
"column": 31
} | {
"line": 424,
"column": 41
} | {
"line": 424,
"column": 41
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : VectorMeasure X ℝ≥0∞\ns : Set X\nhs : MeasurableSet s\n⊢ ⨆ P, μ s = μ s",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"ENNReal.instAddCommMonoid",
"congrArg",
"iSup",
"CompletelyDistribLatti... | [] | iSup_const | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 38
} | {
"line": 207,
"column": 4
} | [
{
"pp": "case mp\nM : Type u_1\nN : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ns : Set M\nF : Type u_6\ninst✝¹ : EquivLike F M N\ninst✝ : AddEquivClass F M N\nf : F\nh : IsSemilinearSet (⇑f '' s)\n⊢ IsSemilinearSet s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"A... | [
"M : Type u_1\nN : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ns : Set M\nF : Type u_6\ninst✝¹ : EquivLike F M N\ninst✝ : AddEquivClass F M N\nf : F\nh : IsSemilinearSet (⇑f '' s)\n⊢ s = ⇑(↑f).symm '' ⇑f '' s"
] | convert! h.image (f : M ≃+ N).symm | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 89
} | {
"line": 233,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\na : M\nt : Set M\nht : t.Finite\nx : M\n⊢ x ∈ ↑(closure (a +ᵥ ↑(closure t))) ↔ x ∈ {0} ∪ (a +ᵥ ↑(closure ({a} ∪ t)))",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.ModelTheory.Arithmetic.Presburger.Se... | [
"M : Type u_1\ninst✝ : AddCommMonoid M\na : M\nt : Set M\nht : t.Finite\nx : M\n⊢ x ∈ closure (a +ᵥ ↑(closure t)) ↔ x = 0 ∨ ∃ y ∈ closure (insert a t), a + y = x"
] | simp only [SetLike.mem_coe, singleton_union, mem_insert_iff, mem_vadd_set, vadd_eq_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 66
} | {
"line": 345,
"column": 2
} | [
{
"pp": "case pos\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nhindep : LinearIndepOn ℕ id ↑t\n⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))",
"ppTerm": "?pos✝",
"... | [] | exact IsProperLinearSet.isProperSemilinearSet ⟨a, t, by simpa⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 66
} | {
"line": 345,
"column": 2
} | [
{
"pp": "case pos\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nhindep : LinearIndepOn ℕ id ↑t\n⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))",
"ppTerm": "?pos✝",
"... | [] | exact IsProperLinearSet.isProperSemilinearSet ⟨a, t, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 66
} | {
"line": 345,
"column": 2
} | [
{
"pp": "case pos\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nhindep : LinearIndepOn ℕ id ↑t\n⊢ IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))",
"ppTerm": "?pos✝",
"... | [] | exact IsProperLinearSet.isProperSemilinearSet ⟨a, t, by simpa⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 410,
"column": 8
} | {
"line": 411,
"column": 57
} | {
"line": 412,
"column": 6
} | [
{
"pp": "case inl\nS : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑∅\n⊢ ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ a +ᵥ ↑(closure ↑∅) ↔ x + p ∈ a +ᵥ ↑(closure ↑∅)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Fin... | [] | refine ⟨a + 1, 1, zero_lt_one, fun x hx => ?_⟩
simp [(by grind : x ≠ a), (by grind : x + 1 ≠ a)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 410,
"column": 8
} | {
"line": 411,
"column": 57
} | {
"line": 412,
"column": 6
} | [
{
"pp": "case inl\nS : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑∅\n⊢ ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ a +ᵥ ↑(closure ↑∅) ↔ x + p ∈ a +ᵥ ↑(closure ↑∅)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Fin... | [] | refine ⟨a + 1, 1, zero_lt_one, fun x hx => ?_⟩
simp [(by grind : x ≠ a), (by grind : x + 1 ≠ a)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 191,
"column": 11
} | {
"line": 191,
"column": 45
} | {
"line": 191,
"column": 46
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\n⊢ IsLinearSet s ↔ ∃ a n f, s = a +ᵥ range ⇑f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instVAddOfAdd",
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"Exists",
... | [
"M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\n⊢ (∃ a P, P.FG ∧ s = a +ᵥ ↑P) ↔ ∃ a n f, s = a +ᵥ range ⇑f"
] | isLinearSet_iff_exists_fg_eq_vadd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 6
} | {
"line": 319,
"column": 2
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [
"X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ] F →L[ℝ] ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.