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.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 89,
"column": 9
} | {
"line": 89,
"column": 20
} | {
"line": 89,
"column": 21
} | [
{
"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 : integralLinearMap μ = integralLinearMap ν\nf : X →C_c ℝ≥0\n⊢ ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ... | [
"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 : integralLinearMap μ = integralLinearMap ν\nf : X →C_c ℝ≥0\n⊢ ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 127,
"column": 8
} | {
"line": 127,
"column": 19
} | {
"line": 127,
"column": 20
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nhy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m\nE : Fin N → Set X := f... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nhy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m\nE : Fin N → Set X := fun n ↦ ⇑f ⁻¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | {
"line": 242,
"column": 14
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : (Fin (n + 1) → ℂ) → E\nc : Fin (n + 1) → ℂ\nR : Fin (n + 1) → ℝ\nhf : TorusIntegrable f c R\n⊢ (∯ (x : Fin (n + 1) → ℂ) in T(c, R), f x) =\n ∮ (x : ℂ) in C(c 0, R 0), ∯ (y : Fin n → ℂ) in T(c ∘ Fin.succ, R ∘ Fin.succ), ... | [
"n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : (Fin (n + 1) → ℂ) → E\nc : Fin (n + 1) → ℂ\nR : Fin (n + 1) → ℝ\nhf : TorusIntegrable f c R\n⊢ (∯ (x : Fin (n + 1) → ℂ) in T(c, R), f x) =\n ∮ (x : ℂ) in C(c 0, R 0), ∯ (y : Fin n → ℂ) in T(c ∘ Fin.succ, R ∘ Fin.succ), f (Fin.cons ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 139,
"column": 2
} | {
"line": 149,
"column": 45
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set (Set α)\nt : Set α\nht : t ∈ {t | GenerateMeasurable s t}\n⊢ t ∈ generateMeasurableRec s (ω_ 1)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Order.succ",
"Ordinal.partialOrder",
"MeasurableSpace.generateMeasurableRec... | [] | induction ht with
| basic u hu => exact self_subset_generateMeasurableRec s _ hu
| empty => exact empty_mem_generateMeasurableRec s _
| compl u _ IH =>
rw [generateMeasurableRec_omega_one, mem_iUnion₂] at IH
obtain ⟨i, hi, hi'⟩ := IH
exact generateMeasurableRec_mono _ ((isSuccLimit_omega 1).succ_lt hi... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 165,
"column": 4
} | {
"line": 167,
"column": 21
} | {
"line": 168,
"column": 2
} | [
{
"pp": "case h.refine_1\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\nV₁ : Opens ... | [] | intro x hx
suffices ∀ x ∈ V₂.carrier, f x < c from this x (mem_of_mem_inter_right hx)
exact fun _ a ↦ a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 165,
"column": 4
} | {
"line": 167,
"column": 21
} | {
"line": 168,
"column": 2
} | [
{
"pp": "case h.refine_1\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : T2Space X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\nε : ℝ\nhε : 0 < ε\nE : Set X\nμ : Content X\nhμ : μ.outerMeasure E ≠ ∞\nhμ' : MeasurableSet E\nc : ℝ\nhfE : ∀ x ∈ E, f x < c\nhε' : ε.toNNReal ≠ 0\nV₁ : Opens ... | [] | intro x hx
suffices ∀ x ∈ V₂.carrier, f x < c from this x (mem_of_mem_inter_right hx)
exact fun _ a ↦ a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 30
} | {
"line": 211,
"column": 30
} | [
{
"pp": "α : Type u\ns : Set (Set α)\nhs : #↑s ≤ 𝔠\n⊢ max (#↑s) 2 ^ ℵ₀ ≤ 𝔠",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Nat.i... | [
"α : Type u\ns : Set (Set α)\nhs : #↑s ≤ 𝔠\n⊢ max (#↑s) 2 ^ ℵ₀ ≤ 𝔠 ^ ℵ₀"
] | ← continuum_power_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 72
} | {
"line": 154,
"column": 73
} | [
{
"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... | [
"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 : ∀ a_1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 72
} | {
"line": 156,
"column": 73
} | [
{
"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... | [
"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 : ∀ a_1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | {
"line": 67,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\n⊢ MemLp id (↑0) μ",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Ch... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\n⊢ AEStronglyMeasurable id μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 86,
"column": 47
} | {
"line": 86,
"column": 58
} | {
"line": 86,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = ContinuousLinearMap.toLinearM... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = ContinuousLinearMap.toLinearMap₁₂ (innerS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 266,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁶ : MeasurableSpace E\nμ ν : Measure E\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nt : E\n⊢ Integrable (fun x ↦ cexp (↑⟪x, t⟫ * I)) (μ ∗ ν)",
"ppT... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 266,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁶ : MeasurableSpace E\nμ ν : Measure E\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nt : E\n⊢ Integrable (fun x ↦ cexp (↑⟪x, t⟫ * I)) (μ ∗ ν)",
"ppT... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 65
} | {
"line": 266,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁶ : MeasurableSpace E\nμ ν : Measure E\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nt : E\n⊢ Integrable (fun x ↦ cexp (↑⟪x, t⟫ * I)) (μ ∗ ν)",
"ppT... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 44
} | {
"line": 148,
"column": 45
} | [
{
"pp": "case h.right.left\nι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⋯\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'... | [
"case h.right.left\nι : 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\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'), fs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 19
} | {
"line": 176,
"column": 20
} | [
{
"pp": "case refine_1.a\nE : 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\nh... | [
"case refine_1.a\nE : 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.Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 292,
"column": 38
} | {
"line": 292,
"column": 49
} | {
"line": 292,
"column": 50
} | [
{
"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\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b\... | [
"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\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b\nN : ℕ\nhN :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 173,
"column": 2
} | {
"line": 186,
"column": 8
} | {
"line": 188,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\n⊢ IsInducing stoneCechUnit",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Cond... | [] | rw [isInducing_iff_nhds]
intro x
apply le_antisymm
· rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt
· simp_rw [le_nhds_iff, ((nhds_basis_opens _).comap _).mem_iff, and_assoc]
intro U hxU hU
obtain ⟨f, hf, efx, hfU⟩ :=
CompletelyRegularSpace.completely_regular_isOpen x U hU... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 173,
"column": 2
} | {
"line": 186,
"column": 8
} | {
"line": 188,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\n⊢ IsInducing stoneCechUnit",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Cond... | [] | rw [isInducing_iff_nhds]
intro x
apply le_antisymm
· rw [← map_le_iff_le_comap]; exact continuous_stoneCechUnit.continuousAt
· simp_rw [le_nhds_iff, ((nhds_basis_opens _).comap _).mem_iff, and_assoc]
intro U hxU hU
obtain ⟨f, hf, efx, hfU⟩ :=
CompletelyRegularSpace.completely_regular_isOpen x U hU... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 19
} | {
"line": 216,
"column": 20
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx : X\ns : Set X\nhxs : x ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\n⊢ lift.{u, 0} #↑R < lift.{0, u} 𝔠",
"ppTerm": "?m.53",
... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx : X\ns : Set X\nhxs : x ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\n⊢ #↑(range f) < 𝔠"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 19
} | {
"line": 177,
"column": 20
} | [
{
"pp": "case refine_1.a\nE : 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\nh... | [
"case refine_1.a\nE : 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.Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 220,
"column": 38
} | {
"line": 220,
"column": 53
} | {
"line": 220,
"column": 54
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx : X\ns : Set X\nhxs : x ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\nhR : #↑R < #↑I\nr : ↑I\nhr : r ∈ Rᶜ\n⊢ ∀ (x : X), f x ≠ r",
... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx : X\ns : Set X\nhxs : x ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\nhR : #↑R < #↑I\nr : ↑I\nhr : r ∈ Rᶜ\n⊢ ∀ (x : X), ¬f x = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 227,
"column": 26
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx✝ : X\ns : Set X\nhxs✝ : x✝ ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x✝ = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\nhR : #↑R < #↑I\nr : ↑I\nhr : r ∈ Rᶜ\nhr' ... | [
"case refine_2\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx✝ : X\ns : Set X\nhxs✝ : x✝ ∈ s\nhs : IsOpen[inst✝¹] s\nf : X → ↑I\nhfc : Continuous[inst✝¹, _] f\nhf₀ : f x✝ = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\nhR : #↑R < #↑I\nr : ↑I\nhr : r ∈ Rᶜ\nhr' : ∀ (x : X),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 29
} | {
"line": 178,
"column": 30
} | [
{
"pp": "case refine_1.a\nE : 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\nh... | [
"case refine_1.a\nE : 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.Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.DiracProba | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 15
} | {
"line": 39,
"column": 16
} | [
{
"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":... | [
"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⊢ dist ↑(g x) ↑(g y) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.DiracProba | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 46
} | {
"line": 59,
"column": 47
} | [
{
"pp": "X : Type u_2\ninst✝¹ : MeasurableSpace X\ninst✝ : MeasurableSpace.SeparatesPoints X\nx y : X\nx_eq_y : (fun x ↦ diracProba x) x = (fun x ↦ diracProba x) y\n⊢ x = y",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_2\ninst✝¹ : MeasurableSpace X\ninst✝ : MeasurableSpace.SeparatesPoints X\nx y : X\nx_eq_y : (fun x ↦ diracProba x) x = (fun x ↦ diracProba x) y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.DiracProba | {
"line": 87,
"column": 42
} | {
"line": 87,
"column": 78
} | {
"line": 87,
"column": 79
} | [
{
"pp": "X : Type u_1\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\nL : Filter X\nh : ¬Tendsto id L (𝓝 x)\nU : Set X\nU_nhds : U ∈ 𝓝 x\nhU : ∃ᶠ (x : X) in L, x ∉ U\n⊢ interior U ∈ 𝓝 x",
"ppTerm": "?m.62",
"assigned"... | [
"X : Type u_1\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\nL : Filter X\nh : ¬Tendsto id L (𝓝 x)\nU : Set X\nU_nhds : U ∈ 𝓝 x\nhU : ∃ᶠ (x : X) in L, x ∉ U\n⊢ U ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.DiracProba | {
"line": 90,
"column": 10
} | {
"line": 90,
"column": 51
} | {
"line": 90,
"column": 52
} | [
{
"pp": "X : Type u_1\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\nL : Filter X\nh : ¬Tendsto id L (𝓝 x)\nU : Set X\nU_nhds : U ∈ 𝓝 x\nhU : ∃ᶠ (x : X) in L, x ∉ U\nUint_nhds : interior U ∈ 𝓝 x\n⊢ ?m.81 ∉ (interior ?m.77)ᶜ"... | [
"X : Type u_1\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\nL : Filter X\nh : ¬Tendsto id L (𝓝 x)\nU : Set X\nU_nhds : U ∈ 𝓝 x\nhU : ∃ᶠ (x : X) in L, x ∉ U\nUint_nhds : interior U ∈ 𝓝 x\n⊢ ?m.81 ∈ interior ?m.77"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 30
} | {
"line": 92,
"column": 31
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nδ : ℝ≥0∞\nh :\n ∀ (ε : ℝ≥0∞) (B : Set Ω),\n 0 < ε →\n ε < ∞ →\n MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε\nε : ℝ≥0\nhε... | [
"case neg\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nδ : ℝ≥0∞\nh :\n ∀ (ε : ℝ≥0∞) (B : Set Ω),\n 0 < ε →\n ε < ∞ →\n MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε\nε : ℝ≥0\nhε : 0 < ε\na✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 32
} | {
"line": 103,
"column": 33
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nδ : ℝ≥0∞\nh :\n ∀ (ε : ℝ≥0∞) (B : Set Ω),\n δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε\nδ_top : ¬δ = ∞\nx : ℝ≥0∞\nB : Set Ω\nx_pos : 0 ... | [
"case neg\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace Ω\nμ ν : Measure Ω\nδ : ℝ≥0∞\nh :\n ∀ (ε : ℝ≥0∞) (B : Set Ω),\n δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε\nδ_top : ¬δ = ∞\nx : ℝ≥0∞\nB : Set Ω\nx_pos : 0 < x\nx_lt_to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 65
} | {
"line": 542,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nμ ν : Measure E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nL : StrongDual ℝ E\n⊢ Integrable (fun v ↦ cexp (↑(L v) * I)) (μ ∗ ν)",
"p... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 65
} | {
"line": 542,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nμ ν : Measure E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nL : StrongDual ℝ E\n⊢ Integrable (fun v ↦ cexp (↑(L v) * I)) (μ ∗ ν)",
"p... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 65
} | {
"line": 542,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nμ ν : Measure E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nL : StrongDual ℝ E\n⊢ Integrable (fun v ↦ cexp (↑(L v) * I)) (μ ∗ ν)",
"p... | [] | exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasurePi | {
"line": 115,
"column": 45
} | {
"line": 115,
"column": 56
} | {
"line": 115,
"column": 57
} | [
{
"pp": "ι : 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 : ι) → Proba... | [
"ι : 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 : ι) → ProbabilityMeasur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 190,
"column": 39
} | {
"line": 190,
"column": 60
} | {
"line": 190,
"column": 61
} | [
{
"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.Measure.LevyProkhorovMetric | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 56
} | {
"line": 251,
"column": 57
} | [
{
"pp": "case refine_2\nΩ : 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) +... | [
"case refine_2\nΩ : 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.ofR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 86
} | {
"line": 344,
"column": 87
} | [
{
"pp": "case refine_2.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\nA : Set Ω\nhA : A ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s... | [
"case refine_2.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\nA : Set Ω\nhA : A ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\n⊢ levyPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasurePi | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 19
} | {
"line": 118,
"column": 20
} | [
{
"pp": "case hx\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 : ι... | [
"case hx\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 : ι) → Probabil... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 338,
"column": 4
} | {
"line": 339,
"column": 11
} | {
"line": 339,
"column": 12
} | [
{
"pp": "case calc_7\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\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ... | [
"case calc_7\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\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ⇑f ⊆ Ioo a b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 349,
"column": 48
} | {
"line": 349,
"column": 59
} | {
"line": 349,
"column": 60
} | [
{
"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\n⊢ ∫ (x : X), f x ∂rieszMeasure Λ = -∫ (x : X), (-f) x ∂rieszMeasure Λ",
"ppTerm": "?m.60",
"assigned": true,
... | [
"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\n⊢ ∫ (x : X), f x ∂rieszMeasure Λ = -∫ (x : X), -f x ∂rieszMeasure Λ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 384,
"column": 93
} | {
"line": 386,
"column": 56
} | {
"line": 387,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : T2Space X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nμ ν : Measure X\ninst✝³ : LocallyCompactSpace X\ninst✝² : ν.OuterRegular\ninst✝¹ : IsFiniteMeasureOnCompacts ν\ninst✝ : IsFiniteMeasureOnCompacts μ\nhμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂... | [] | by
obtain ⟨f, hf1, hf2, hf3⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK pV2 pV1
exact ⟨⟨f, hasCompactSupport_def.mpr hf2⟩, hf1, hf3⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 467,
"column": 17
} | {
"line": 467,
"column": 28
} | {
"line": 467,
"column": 29
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityMeasure Ω := toMea... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityMeasure Ω := toMeasure ∘ μs\nf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 64,
"column": 72
} | {
"line": 66,
"column": 42
} | {
"line": 68,
"column": 0
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝⁹ : Group G\ninst✝⁸ : AddCommGroup A\ninst✝⁷ : DistribMulAction G A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : IsTopologicalAddGroup A\ninst✝⁴ : LocallyCompactSpace A\ninst✝³ : ContinuousConstSMul G A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ ... | [] | by
borelize A
exact addHaarScalarFactor_smul_congr' .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 23
} | {
"line": 214,
"column": 24
} | [
{
"pp": "case refine_1\nE : 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\nh... | [
"case refine_1\nE : 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.Counta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 548,
"column": 6
} | {
"line": 548,
"column": 35
} | {
"line": 548,
"column": 36
} | [
{
"pp": "case inl.refine_1\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : IsEmpty Ω\nn : ℕ\n⊢ diam ((fun x ↦ ∅) n) ≤ ε",
"ppTerm": "?inl.refine_1",
"assigned": true,
"usedConstants": [
... | [
"case inl.refine_1\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : IsEmpty Ω\nn : ℕ\n⊢ 0 ≤ ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 28
} | {
"line": 564,
"column": 29
} | [
{
"pp": "case inr.refine_4\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : Nonempty Ω\nxs : ℕ → Ω\nxs_dense : DenseRange xs\nhalf_ε_pos : 0 < ε / 2\nBs : ℕ → Set Ω := fun n ↦ ball (xs n) (ε / 2)\nAs : ... | [
"case inr.refine_4\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : Nonempty Ω\nxs : ℕ → Ω\nxs_dense : DenseRange xs\nhalf_ε_pos : 0 < ε / 2\nBs : ℕ → Set Ω := fun n ↦ ball (xs n) (ε / 2)\nAs : ℕ → Set Ω :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 594,
"column": 6
} | {
"line": 594,
"column": 52
} | {
"line": 595,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n)\nEs_bdd : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 49
} | {
"line": 596,
"column": 50
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n)\nEs_bdd : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 644,
"column": 26
} | {
"line": 644,
"column": 60
} | {
"line": 644,
"column": 61
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nP : ProbabilityMeasure Ω\nε : ℝ\nε_pos : ε > 0\nthird_ε_pos : 0 < ε / 3\nthird_ε_pos' : 0 < ENNReal.ofReal (ε / 3)\nEs : ℕ → Set Ω\nEs_mble : ∀ (n : ℕ), MeasurableSet (Es n)\nEs_bdd : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 54
} | {
"line": 265,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : Me... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : MeasurableSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 166,
"column": 95
} | {
"line": 175,
"column": 31
} | {
"line": 177,
"column": 0
} | [
{
"pp": "E : 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\n⊢ μ.real {x | r < |⟪a, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun μ (t • ... | [] | by
have : IsProbabilityMeasure (μ.map (fun x ↦ ⟪a, x⟫)) :=
Measure.isProbabilityMeasure_map (by fun_prop)
convert! measureReal_abs_gt_le_integral_charFun (μ := μ.map (fun x ↦ ⟪a, x⟫)) hr with x
· rw [map_measureReal_apply (by fun_prop)]
· simp
· exact MeasurableSet.preimage measurableSet_Ioi (by fun_p... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 174,
"column": 2
} | {
"line": 177,
"column": 36
} | {
"line": 178,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\neq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p)... | [
"case e'_2\nι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\neq_zero : ∀ (x : ι → ℝ), (∑ i, |x i| ^ p) ... | convert!
(measure_lt_one_eq_integral_div_gamma (volume : Measure (ι → ℝ)) (g := fun x =>
(∑ i, |x i| ^ p) ^ (1 / p)) nm_zero nm_neg nm_add (eq_zero _).mp (fun r x => nm_smul r x)
(by linarith : 0 < p)) using 4 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 13
} | {
"line": 179,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : MeasurableSpace E\ninst✝³ : TopologicalSpace E\ninst✝² : T2Space E\ninst✝¹ : BorelSpace E\ninst✝ : CompactSpace E\n⊢ IsCompact (ProbabilityMeasure.toFiniteMeasure '' univ)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeasure.ins... | [
"E : Type u_1\ninst✝⁴ : MeasurableSpace E\ninst✝³ : TopologicalSpace E\ninst✝² : T2Space E\ninst✝¹ : BorelSpace E\ninst✝ : CompactSpace E\n⊢ IsCompact {μ | μ.mass = 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 25
} | {
"line": 193,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := Subtype.val\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f\nT : Set (FiniteMeasure... | [
"case h₁\nE : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := ⋯\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := ⋯\nT : Set (FiniteMeasure ↑K) := ⋯\n⊢ {μ | μ.mass ≤ ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 43
} | {
"line": 251,
"column": 44
} | [
{
"pp": "E : 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)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [
"E : 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)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 121,
"column": 4
} | {
"line": 122,
"column": 41
} | {
"line": 123,
"column": 4
} | [
{
"pp": "case pos\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topol... | [
"case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → TopologicalSpace ... | · simp only [Set.mem_pi, mem_univ, forall_const] at hy
exact Finset.prod_eq_one (by simpa) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 43
} | {
"line": 123,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topol... | [
"case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → TopologicalSpace ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 128,
"column": 4
} | {
"line": 129,
"column": 41
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case pos\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topol... | [
"case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → TopologicalSpace ... | · simp only [Set.mem_pi, mem_univ, forall_const] at hy
exact Finset.prod_eq_one (by simpa) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 43
} | {
"line": 130,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topol... | [
"case neg\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → TopologicalSpace ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 63,
"column": 61
} | {
"line": 63,
"column": 72
} | {
"line": 63,
"column": 73
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝¹ : PseudoMetricSpace E\ninst✝ : ProperSpace E\nx : E\nh✝ : ∀ ε > 0, ∃ N, ∀ n ≥ N, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε\nε : ℝ≥0∞\nhε : 0 < ε\nr : ℝ\nh : ∀ n ≥ r, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε\n⊢ ∀ μ ∈ S, μ (Metric.closedBall ... | [
"E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝¹ : PseudoMetricSpace E\ninst✝ : ProperSpace E\nx : E\nh✝ : ∀ ε > 0, ∃ N, ∀ n ≥ N, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε\nε : ℝ≥0∞\nhε : 0 < ε\nr : ℝ\nh : ∀ n ≥ r, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε\n⊢ ∀ μ ∈ S, μ (Metric.closedBall x r)ᶜ ≤ ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 13
} | {
"line": 115,
"column": 14
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : ProperSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)\nn : ℕ\nh_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {μ n}... | [
"E : Type u_1\nmE : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : ProperSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖x‖}) atTop) atTop (𝓝 0)\nn : ℕ\nh_tight : Tendsto (fun r ↦ ⨆ μ_1 ∈ {μ n}, μ_1 {x | r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 27
} | {
"line": 353,
"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\na b : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ] E →L[𝕜] F\nins... | [
"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\na b : E\ns : Set E\nω : E → E →L[𝕜] F\ndω : E → E →L[ℝ] E →L[𝕜] F\ninst✝ : Complet... | refine .const_add _ <| ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 146,
"column": 26
} | {
"line": 146,
"column": 60
} | {
"line": 146,
"column": 61
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_2\nι : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype ι\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\nh : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r ... | [
"E : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_2\nι : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype ι\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\nh : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 15
} | {
"line": 405,
"column": 16
} | [
{
"pp": "case hdω\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : 𝕜 → E\ns : Set 𝕜\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\na : 𝕜\nha : a ∈ s\nx y : 𝕜... | [
"case hdω\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : 𝕜 → E\ns : Set 𝕜\nhs : Convex ℝ s\nhf : DifferentiableOn 𝕜 f s\nthis : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E\na : 𝕜\nha : a ∈ s\nx y : 𝕜\n⊢ x • y • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 21
} | {
"line": 224,
"column": 22
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 386,
"column": 8
} | {
"line": 386,
"column": 19
} | {
"line": 386,
"column": 20
} | [
{
"pp": "case h₂\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)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈... | [
"case h₂\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)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 52
} | {
"line": 272,
"column": 53
} | [
{
"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), ‖y‖ = 1 → Tendsto (fun r ↦ limsup ... | [
"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), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 120,
"column": 28
} | {
"line": 120,
"column": 39
} | {
"line": 120,
"column": 40
} | [
{
"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✝\ns : Set α\nhs : MeasurableSet s\nh's : ∀ (ε : ℝ≥0∞), 0 ... | [
"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✝\ns : Set α\nhs : MeasurableSet s\nh's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 413,
"column": 36
} | {
"line": 413,
"column": 47
} | {
"line": 413,
"column": 48
} | [
{
"pp": "E : 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 (disjoi... | [
"E : 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 K i) = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 414,
"column": 31
} | {
"line": 414,
"column": 42
} | {
"line": 414,
"column": 43
} | [
{
"pp": "E : 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 (disjoi... | [
"E : 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 K i) = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 112,
"column": 33
} | {
"line": 112,
"column": 44
} | {
"line": 112,
"column": 45
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 48
} | {
"line": 82,
"column": 49
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : Set X\nhs : MeasurableSet s\nP : Finpartition ⟨s, hs⟩\n⊢ ∑ p ∈ P.parts, f ↑p ≤ preVariationFun f s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"MeasurableSet",
... | [
"X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : Set X\nhs : MeasurableSet s\nP : Finpartition ⟨s, hs⟩\n⊢ ∀ (b : ℝ≥0∞), (∀ (i : Finpartition ⟨s, ⋯⟩), ∑ p ∈ i.parts, f ↑p ≤ b) → ∑ p ∈ P.parts, f ↑p ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 17
} | {
"line": 222,
"column": 18
} | [
{
"pp": "ι : Type u_1\nT : Type u_4\nX : ι → Type u_5\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmT : MeasurableSpace T\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : BorelSpace T\ninst✝³ : HasOu... | [
"ι : Type u_1\nT : Type u_4\nX : ι → Type u_5\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmT : MeasurableSpace T\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : BorelSpace T\ninst✝³ : HasOuterApproxClo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 425,
"column": 10
} | {
"line": 425,
"column": 26
} | {
"line": 425,
"column": 27
} | [
{
"pp": "case pos\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.restri... | [
"case pos\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 (disjoint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 149,
"column": 44
} | {
"line": 149,
"column": 55
} | {
"line": 149,
"column": 56
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : Set X\nhs : MeasurableSet s\nh : preVariationFun f s ≠ ∞\nε : ℝ≥0\nhε : 0 < ↑ε\nh'ε : ↑ε < ∞\n⊢ 0 < ε",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\ns : Set X\nhs : MeasurableSet s\nh : preVariationFun f s ≠ ∞\nε : ℝ≥0\nhε : 0 < ↑ε\nh'ε : ↑ε < ∞\n⊢ 0 < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 429,
"column": 10
} | {
"line": 429,
"column": 21
} | {
"line": 429,
"column": 22
} | [
{
"pp": "case neg\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.restri... | [
"case neg\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 (disjoint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Support | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 62
} | {
"line": 79,
"column": 63
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\n⊢ μ.support = Set.univ",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.to... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\n⊢ ∀ (x : X), ∀ U ∈ 𝓝 x, 0 < μ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Support | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 9
} | {
"line": 193,
"column": 10
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : MeasurableSpace X\nμ : Measure X\nh : IsLindelof μ.supportᶜ\ns : X\nhs : s ∈ μ.supportᶜ\n⊢ ∃ t ∈ 𝓝[μ.supportᶜ] s, tᶜ ∈ ae μ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : MeasurableSpace X\nμ : Measure X\nh : IsLindelof μ.supportᶜ\ns : X\nhs : s ∈ μ.supportᶜ\n⊢ ∃ t ∈ 𝓝 s, μ t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ResolventTransform | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 13
} | {
"line": 71,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : MeasurableSpace 𝕜\na : A\ninst✝⁵ : OpensMeasurableSpace 𝕜\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : CompleteSpace A\ninst✝¹ : MeasurableSpace A\ninst✝ : BorelSpace A\nh1 : ContinuousOn (resolvent a) (resolv... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : MeasurableSpace 𝕜\na : A\ninst✝⁵ : OpensMeasurableSpace 𝕜\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : CompleteSpace A\ninst✝¹ : MeasurableSpace A\ninst✝ : BorelSpace A\nh1 : ContinuousOn (resolvent a) (resolventSet 𝕜 a)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 19
} | {
"line": 144,
"column": 20
} | [
{
"pp": "case hbc.refine_1.refine_2\nE : 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), ... | [
"case hbc.refine_1.refine_2\nE : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 166,
"column": 47
} | {
"line": 166,
"column": 58
} | {
"line": 166,
"column": 59
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : PolishSpace E\ninst✝ : BorelSpace E\nι : Type u_4\n𝓕 : Filter ι\nμ : ι → ProbabilityMeasure E\nh_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}\nμ₀ : ProbabilityMeasure E\nA : StarSubalg... | [
"𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : PolishSpace E\ninst✝ : BorelSpace E\nι : Type u_4\n𝓕 : Filter ι\nμ : ι → ProbabilityMeasure E\nh_tight : IsTightMeasureSet {x | ∃ n, ↑(μ n) = x}\nμ₀ : ProbabilityMeasure E\nA : StarSubalgebra 𝕜 (E →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ResolventTransform | {
"line": 161,
"column": 10
} | {
"line": 161,
"column": 56
} | {
"line": 161,
"column": 57
} | [
{
"pp": "case hcd\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉ ⇑... | [
"case hcd\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉ ⇑(algebraMap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ResolventTransform | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 43
} | {
"line": 170,
"column": 44
} | [
{
"pp": "case right\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉... | [
"case right\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉ ⇑(algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 46
} | {
"line": 196,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nι : Type u_2\n𝓕 : Filter ι\nμ₀ : ProbabilityMeasure E\nμ : ι → ProbabilityMeasure E\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))\ng : E →ᵇ ℂ\... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nι : Type u_2\n𝓕 : Filter ι\nμ₀ : ProbabilityMeasure E\nμ : ι → ProbabilityMeasure E\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (↑(μ n)) t) 𝓕 (𝓝 (charFun (↑μ₀) t))\ng : E →ᵇ ℂ\nw : AddMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 234,
"column": 18
} | {
"line": 243,
"column": 60
} | {
"line": 244,
"column": 16
} | [
{
"pp": "case h₁\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,... | [] | rw [measure_sdiff (h_fin := measure_ne_top _ _),
toReal_sub_of_le (ha := measure_ne_top _ _)]
· apply lt_of_le_of_lt (sub_le_dist ..)
simp only [Finset.mem_range, Nat.lt_add_one_iff]
exact (dist_comm (α := ℝ) .. ▸ hN N (le_refl N))
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 234,
"column": 18
} | {
"line": 243,
"column": 60
} | {
"line": 244,
"column": 16
} | [
{
"pp": "case h₁\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,... | [] | rw [measure_sdiff (h_fin := measure_ne_top _ _),
toReal_sub_of_le (ha := measure_ne_top _ _)]
· apply lt_of_le_of_lt (sub_le_dist ..)
simp only [Finset.mem_range, Nat.lt_add_one_iff]
exact (dist_comm (α := ℝ) .. ▸ hN N (le_refl N))
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 57,
"column": 2
} | {
"line": 65,
"column": 19
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\n⊢ (∃ s, μ s = 1) ↔ μ univ = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"False",
"MeasureTheory.Measure",
"congrArg",
"instIsBo... | [] | constructor
· rintro ⟨s, h⟩
rcases μ.zero_one univ with (h₀ | h₁)
· have := measure_mono (μ := μ) <| subset_univ s
rw [h] at this
simp_all
· exact h₁
· intro h
exact ⟨univ, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne | {
"line": 57,
"column": 2
} | {
"line": 65,
"column": 19
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsZeroOneMeasure μ\n⊢ (∃ s, μ s = 1) ↔ μ univ = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"False",
"MeasureTheory.Measure",
"congrArg",
"instIsBo... | [] | constructor
· rintro ⟨s, h⟩
rcases μ.zero_one univ with (h₀ | h₁)
· have := measure_mono (μ := μ) <| subset_univ s
rw [h] at this
simp_all
· exact h₁
· intro h
exact ⟨univ, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 541,
"column": 35
} | {
"line": 541,
"column": 46
} | {
"line": 541,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nS : Set (ProbabilityMeasure E)\nhS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}\nu : ℕ → ℝ≥0\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nn : ℕ\nK : Set E\nK_comp : IsCompact K\... | [
"E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nS : Set (ProbabilityMeasure E)\nhS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}\nu : ℕ → ℝ≥0\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nn : ℕ\nK : Set E\nK_comp : IsCompact K\nhK : ∀ μ ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 595,
"column": 6
} | {
"line": 595,
"column": 73
} | {
"line": 595,
"column": 74
} | [
{
"pp": "case h\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\... | [
"case h\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\nhcomp : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 371,
"column": 6
} | {
"line": 378,
"column": 74
} | {
"line": 380,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\nhgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m\n𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpann... | [] | induction hs with
| base t t_mem =>
rcases t_mem with t_mem | ⟨n, rfl⟩
· exact hgen ▸ measurableSet_generateFrom t_mem
· exact μ.toFiniteSpanningSetsIn.set_mem n
| empty => exact MeasurableSet.empty
| compl t _ t_mem => exact MeasurableSet.compl t_mem
| union t u _ _ t_me... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 371,
"column": 6
} | {
"line": 378,
"column": 74
} | {
"line": 380,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\nhgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m\n𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpann... | [] | induction hs with
| base t t_mem =>
rcases t_mem with t_mem | ⟨n, rfl⟩
· exact hgen ▸ measurableSet_generateFrom t_mem
· exact μ.toFiniteSpanningSetsIn.set_mem n
| empty => exact MeasurableSet.empty
| compl t _ t_mem => exact MeasurableSet.compl t_mem
| union t u _ _ t_me... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 371,
"column": 6
} | {
"line": 378,
"column": 74
} | {
"line": 380,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\nhgen : MeasurableSpace.generateFrom (countableGeneratingSet X) = m\n𝒜 : Set (Set X) := countableGeneratingSet X ∪ {x | ∃ n, μ.toFiniteSpann... | [] | induction hs with
| base t t_mem =>
rcases t_mem with t_mem | ⟨n, rfl⟩
· exact hgen ▸ measurableSet_generateFrom t_mem
· exact μ.toFiniteSpanningSetsIn.set_mem n
| empty => exact MeasurableSet.empty
| compl t _ t_mem => exact MeasurableSet.compl t_mem
| union t u _ _ t_me... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 607,
"column": 8
} | {
"line": 607,
"column": 75
} | {
"line": 608,
"column": 12
} | [
{
"pp": "case h\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\... | [
"case h\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝² : PseudoMetricSpace 𝓧\ninst✝¹ : OpensMeasurableSpace 𝓧\ninst✝ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\nU : ℕ → Set 𝓧\nO : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nCov : ⋃ i, U i = univ\nhcomp : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Defs | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 57
} | {
"line": 48,
"column": 58
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nV : Type u_2\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : ℕ → { t // MeasurableSet t }\nhs : Pairwise (Function.onFun Disjoint (Subtype.val ∘ s))\nhmeas : ∀ (i : ℕ), MeasurableSet ↑(s i)\n⊢ (fun x ↦ ‖μ... | [
"X : Type u_1\nmX : MeasurableSpace X\nV : Type u_2\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : ℕ → { t // MeasurableSet t }\nhs : Pairwise (Function.onFun Disjoint (Subtype.val ∘ s))\nhmeas : ∀ (i : ℕ), MeasurableSet ↑(s i)\n⊢ ‖∑' (i : ℕ), μ ↑(s i)‖ₑ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 31
} | {
"line": 105,
"column": 4
} | [
{
"pp": "case refine_2\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 : MeasurableSet s\na : ℝ≥0∞\nha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s\nP : Finpartition ⟨s, hs⟩\nhP : a < ∑ p ∈ P.p... | [
"case refine_2\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 : MeasurableSet s\na : ℝ≥0∞\nha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s\nP : Finpartition ⟨s, hs⟩\nhP : a < ∑ p ∈ P.parts, (fun x... | rcases hi with ⟨h'i, i_mem⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 107,
"column": 34
} | {
"line": 107,
"column": 45
} | {
"line": 107,
"column": 46
} | [
{
"pp": "X : 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 : MeasurableSet s\na : ℝ≥0∞\nha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s\nP : Finpartition ⟨s, hs⟩\nhP : a < ∑ p ∈ P.parts, (fun x ↦ ... | [
"X : 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 : MeasurableSet s\na : ℝ≥0∞\nha : a < preVariationFun (fun x ↦ ‖μ x‖ₑ) s\nP : Finpartition ⟨s, hs⟩\nhP : a < ∑ p ∈ P.parts, (fun x ↦ ‖μ x‖ₑ) ↑p\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 658,
"column": 6
} | {
"line": 658,
"column": 17
} | {
"line": 658,
"column": 18
} | [
{
"pp": "case refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝³ : PseudoMetricSpace 𝓧\ninst✝² : OpensMeasurableSpace 𝓧\ninst✝¹ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\ninst✝ : CompleteSpace 𝓧\nhcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)\nhnonempty : No... | [
"case refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝³ : PseudoMetricSpace 𝓧\ninst✝² : OpensMeasurableSpace 𝓧\ninst✝¹ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\ninst✝ : CompleteSpace 𝓧\nhcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)\nhnonempty : Nonempty 𝓧\nD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 31
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case refine_2\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 : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\nhμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞\nP : Finpartition ⟨s, hs⟩\nhP : p... | [
"case refine_2\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 : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\nhμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞\nP : Finpartition ⟨s, hs⟩\nhP : preVariationF... | rcases hi with ⟨h'i, i_mem⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 132,
"column": 34
} | {
"line": 132,
"column": 45
} | {
"line": 132,
"column": 46
} | [
{
"pp": "X : 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 : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\nhμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞\nP : Finpartition ⟨s, hs⟩\nhP : preVariationFun ... | [
"X : 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 : MeasurableSet s\nε : ℝ≥0∞\nhε : 0 < ε\nhμ : preVariationFun (fun x ↦ ‖μ x‖ₑ) s ≠ ∞\nP : Finpartition ⟨s, hs⟩\nhP : preVariationFun (fun x ↦ ‖μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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