module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 67,
"column": 9
} | {
"line": 67,
"column": 41
} | {
"line": 67,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ μ ∅ ≠ ∞",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [
"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ 0 ≠ ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 43
} | {
"line": 140,
"column": 44
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns : Set X\ni : ι\nhi : μ (F i) ≠ 0\nhi' : μ (F i) ≠ ∞\n⊢ μ (s ∩ F i) / μ (F i) ∈ Icc 0 1",
"ppTerm": "?m.92",... | [
"G : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns : Set X\ni : ι\nhi : μ (F i) ≠ 0\nhi' : μ (F i) ≠ ∞\n⊢ μ (s ∩ F i) ≤ μ (F i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 27
} | {
"line": 166,
"column": 28
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng h : G\n⊢ Tendsto (fun i ↦ μ ((g • F i) ∆ (h • F i)) / μ (F i)) (↑u) (𝓝 0)... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng h : G\n⊢ Tendsto (fun i ↦ μ ((g • F i) ∆ (h • F i)) / μ (F i)) (↑u) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 69
} | {
"line": 182,
"column": 70
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng✝ h✝ : G\ns : Set X\ng h : G\ni : ι\nhi : μ (F i) ≠ 0\n⊢ μ (g • s ∩ F i) - ... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng✝ h✝ : G\ns : Set X\ng h : G\ni : ι\nhi : μ (F i) ≠ 0\n⊢ μ (s ∩ g⁻¹ • F i) ≤ μ ((s ∩ g⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"column": 14
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng : G\ns : Set X\n⊢ mean μ u F (g • s) = mean μ u F s",
"ppTerm": "?m.22... | [
"G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng : G\ns : Set X\n⊢ mean μ u F (g • s) = mean μ u F s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 32
} | {
"line": 103,
"column": 33
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hg.hf",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun i ↦ ((circleMap z R i.2 - i.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 97,
"column": 76
} | {
"line": 97,
"column": 92
} | {
"line": 97,
"column": 93
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | [
"E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L\nfund : IsA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 67
} | {
"line": 197,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ≥0\nhε : 0 < ε\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg : ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ≥0\nhε : 0 < ε\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg : ∀ (i : Fin n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 27
} | {
"line": 115,
"column": 28
} | [
{
"pp": "R r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty\nthis :\n ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],\n IsMaxOn (... | [
"R r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty\nthis :\n ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],\n IsMaxOn ((fun x ↦ ‖x‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 83
} | {
"line": 137,
"column": 84
} | [
{
"pp": "R : ℝ\nhR : 0 < R\nz x : ℂ\nf : ℂ → ℂ\nhx : x ∈ ball z R\nhf : ContinuousOn f (sphere z R)\nr : ℝ\nhr : r < R\nhrx : x ∈ ball z r\nε' : ℝ\nhε' : ε' > 0\nH : ball x ε' ⊆ ball z r\na : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nhb : (a, b).2 ∈ [[0, 2 * π]]\nhab :\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]]... | [
"R : ℝ\nhR : 0 < R\nz x : ℂ\nf : ℂ → ℂ\nhx : x ∈ ball z R\nhf : ContinuousOn f (sphere z R)\nr : ℝ\nhr : r < R\nhrx : x ∈ ball z r\nε' : ℝ\nhε' : ε' > 0\nH : ball x ε' ⊆ ball z r\na : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nhb : (a, b).2 ∈ [[0, 2 * π]]\nhab :\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]])),\n ‖ci... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 64,
"column": 60
} | {
"line": 67,
"column": 64
} | {
"line": 69,
"column": 0
} | [
{
"pp": "p : ℝ\nhp : 0 < p\n⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x ^ p) = Gamma (1 / p + 1)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Real.instPow",
"Real.parti... | [] | by
convert! (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 45
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case pos\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ✝ : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\nγ : Type u_5\ninst✝¹ : TopologicalSpace γ\ninst✝ : Zero γ\ng : C(β, γ)\nf : α →C_c β\nhg : g 0 = 0\n⊢ HasCompactSupport (g.comp ↑f).toFun",
"ppTerm": "?pos✝"... | [
"case neg\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ✝ : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\nγ : Type u_5\ninst✝¹ : TopologicalSpace γ\ninst✝ : Zero γ\ng : C(β, γ)\nf : α →C_c β\nhg : ¬g 0 = 0\n⊢ HasCompactSupport (ContinuousMap.toFun 0)"
] | · exact f.hasCompactSupport'.comp_left hg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 281,
"column": 38
} | {
"line": 281,
"column": 79
} | {
"line": 281,
"column": 80
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C_c β\n⊢ HasCompactSupport (-⇑f.toContinuousMap)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants":... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C_c β\n⊢ IsCompact (closure (Function.support ⇑f))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 294,
"column": 17
} | {
"line": 294,
"column": 45
} | {
"line": 294,
"column": 46
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C_c β\n⊢ HasCompactSupport (⇑f.toContinuousMap - ⇑g.toContinuousMap)",
"ppTerm": "?m.59",
"assigned": tru... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C_c β\n⊢ HasCompactSupport (⇑f + -⇑g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 210,
"column": 14
} | {
"line": 210,
"column": 25
} | {
"line": 210,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ.symm\n⊢ CurveIntegrable ω γ",
"ppTerm": "?m.49",
"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ.symm\n⊢ CurveIntegrable ω γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 34
} | {
"line": 682,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nf₁ f₂ : α →C_c ℝ≥0\nh : f₁ ≤ f₂\nx : α\n⊢ ↑((f₁ + { toContinuousMap := f₂.toContinuousMap - f₁.toContinuousMap, hasCompactSupport' := ⋯ }) x) = ↑(f₂ x)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NNReal.instTo... | [
"case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nf₁ f₂ : α →C_c ℝ≥0\nh : f₁ ≤ f₂\nx : α\n⊢ f₁ x + (f₂ x - f₁ x) = f₂ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 13
} | {
"line": 697,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\nhf : 0 ≤ f\nx : α\n⊢ ↑((-f).nnrealPart x) = ↑(0 x)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\nhf : 0 ≤ f\nx : α\n⊢ 0 ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 720,
"column": 2
} | {
"line": 720,
"column": 13
} | {
"line": 720,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nf g : α →C_c ℝ\nx : α\n⊢ (f + g).nnrealPart x ≤ (f.nnrealPart + g.nnrealPart) x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"NonU... | [
"α : Type u_2\ninst✝ : TopologicalSpace α\nf g : α →C_c ℝ\nx : α\n⊢ (f x + g x).toNNReal ≤ (f x).toNNReal + (g x).toNNReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 852,
"column": 4
} | {
"line": 852,
"column": 51
} | {
"line": 852,
"column": 52
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : Group α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : R1Space β\ninst✝² : Group β\ninst✝¹ : ContinuousMul β\ninst✝ : NormedAddCommGroup γ\nφ : α →* β\nhφ : Topology.IsClosedEmbedding ⇑φ\nf : β →C_... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : Group α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : R1Space β\ninst✝² : Group β\ninst✝¹ : ContinuousMul β\ninst✝ : NormedAddCommGroup γ\nφ : α →* β\nhφ : Topology.IsClosedEmbedding ⇑φ\nf : β →C_c γ\nb : β\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [] | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 79
} | {
"line": 58,
"column": 80
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | [
"case refine_4\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i : ι) in L, As... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 69
} | {
"line": 62,
"column": 70
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [i... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [inst : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 39
} | {
"line": 63,
"column": 40
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Ty... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 31
} | {
"line": 395,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ CurveIntegrable (-ω) γ",
"ppTerm": "?m.62",
"as... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ IntervalIntegrable (-curveIntegralFun ω γ) volume 0 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 399,
"column": 14
} | {
"line": 399,
"column": 25
} | {
"line": 399,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable (-ω) γ\n⊢ CurveIntegrable ω γ",
"ppTerm": "?m.65",
"as... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable (-ω) γ\n⊢ CurveIntegrable ω γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 31
} | {
"line": 464,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommCla... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommClass 𝕜 𝕝 F\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 472,
"column": 4
} | {
"line": 472,
"column": 20
} | {
"line": 472,
"column": 21
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : S... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommClass... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 85
} | {
"line": 135,
"column": 86
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 143,
"column": 40
} | {
"line": 143,
"column": 79
} | {
"line": 143,
"column": 80
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Ioo 0 1 → ‖lin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 530,
"column": 4
} | {
"line": 530,
"column": 30
} | {
"line": 530,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x : E) in... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x : E) in 𝓝[s] a, Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 21
} | {
"line": 89,
"column": 22
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countab... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 62,
"column": 37
} | {
"line": 62,
"column": 53
} | {
"line": 62,
"column": 54
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\n⊢ IsConnected s",
"ppTerm": "?m.193",
"assigned": true... | [
"a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\n⊢ IsConnected (Ioc a b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 61
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh✝ : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\nhs_conn : IsConnected s\nhfg : IntegrableOn (fun x ↦ f x * g ... | [
"a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh✝ : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\nhs_conn : IsConnected s\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nc : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 34,
"column": 6
} | {
"line": 34,
"column": 26
} | {
"line": 35,
"column": 6
} | [
{
"pp": "case mp.ht\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g ... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 36,
"column": 36
} | {
"line": 36,
"column": 78
} | {
"line": 36,
"column": 79
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 24
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 45
} | {
"line": 49,
"column": 46
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 19
} | {
"line": 203,
"column": 20
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nhf₀ : f a = 0\n⊢ f =O[𝓝 ... | [
"E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nhf₀ : f a = 0\n⊢ f =O[𝓝 a] fun x ↦ ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 31,
"column": 16
} | {
"line": 31,
"column": 50
} | {
"line": 32,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a N) ⊆ Ioc (a N) (a (N + 1)) ∪ ⋃ x, ⋃ (_ : x < N), Ioc (a x) (a (x + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 41,
"column": 16
} | {
"line": 41,
"column": 50
} | {
"line": 42,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))\n⊢ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ico (a i) (a (i + 1))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))\n⊢ Ico (a 0) (a N) ⊆ Ico (a N) (a (N + 1)) ∪ ⋃ x, ⋃ (_ : x < N), Ico (a x) (a (x + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 32
} | {
"line": 114,
"column": 33
} | [
{
"pp": "case inl\nf : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\nh_f_int : IntervalIntegrable f volume a (a + ↑N * h)\nk✝ : ℕ\nhk✝ : k✝ < N\nh_neg : h ≤ 0\nk : ℕ\nhk : ↑k ≤ ↑N\n⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.parti... | [
"case inl\nf : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\nh_f_int : IntervalIntegrable f volume a (a + ↑N * h)\nk✝ : ℕ\nhk✝ : k✝ < N\nh_neg : h ≤ 0\nk : ℕ\nhk : ↑k ≤ ↑N\n⊢ 0 ≤ ↑k * h ∧ ↑k * h ≤ ↑N * h ∨ ↑N * h ≤ ↑k * h ∧ ↑k * h ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 39
} | {
"line": 152,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | rw [iteratedDerivWithin_eq_iterate] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 38
} | {
"line": 167,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 11
} | {
"line": 188,
"column": 12
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : Disjoint s₀ s₁\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fu... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : Disjoint s₀ s₁\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fun j ↦ if j =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | {
"line": 83,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"ppTerm": "?m.42... | [] | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | {
"line": 83,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"ppTerm": "?m.42... | [] | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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.TorusIntegral | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 43
} | {
"line": 124,
"column": 43
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : (Fin n → ℂ) → E\nc : Fin n → ℂ\n⊢ IntegrableOn (fun θ ↦ f (torusMap c 0 θ)) (Icc 0 fun x ↦ 2 * π) volume",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Pi.preorder",
"Real.pi",
... | [
"n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : (Fin n → ℂ) → E\nc : Fin n → ℂ\n⊢ IntegrableOn (fun θ ↦ f (const (Fin n → ℝ) c θ)) (Icc 0 fun x ↦ 2 * π) volume"
] | torusMap_zero_radius | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 58
} | {
"line": 158,
"column": 4
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x... | [
"n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ ∫ (θ : Fin n → ℝ) in Icc 0 fun x ↦ 2 * π,\n (∏ i, ↑(R i) * cexp (↑(θ i) * I) * I) • f (torusMap c R θ) +\n (∏ i, ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 34
} | {
"line": 82,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ↑{ carrier :... | refine (Λ.mono ?_).trans hg.2.le | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)... | [
"case pos\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ↑{... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 241,
"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.Integral.RieszMarkovKakutani.Real | {
"line": 126,
"column": 8
} | {
"line": 126,
"column": 19
} | {
"line": 126,
"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.MeasurableSpace.Card | {
"line": 142,
"column": 2
} | {
"line": 152,
"column": 45
} | {
"line": 154,
"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.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.MeasurableSpace.Card | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 30
} | {
"line": 217,
"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": 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.Integral.RieszMarkovKakutani.Real | {
"line": 164,
"column": 4
} | {
"line": 166,
"column": 21
} | {
"line": 167,
"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": 164,
"column": 4
} | {
"line": 166,
"column": 21
} | {
"line": 167,
"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.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": 87,
"column": 47
} | {
"line": 87,
"column": 58
} | {
"line": 87,
"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 = (innerSL ℝ).toLinearMap₁₂\nhi... | [
"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 = (innerSL ℝ).toLinearMap₁₂\nhint' : ∀ (k :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 65
} | {
"line": 264,
"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": 262,
"column": 4
} | {
"line": 262,
"column": 65
} | {
"line": 264,
"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": 262,
"column": 4
} | {
"line": 262,
"column": 65
} | {
"line": 264,
"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.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.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": 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.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.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.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.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 291,
"column": 38
} | {
"line": 291,
"column": 49
} | {
"line": 291,
"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": 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.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.CharacteristicFunction.Basic | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 65
} | {
"line": 540,
"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": 538,
"column": 4
} | {
"line": 538,
"column": 65
} | {
"line": 540,
"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": 538,
"column": 4
} | {
"line": 538,
"column": 65
} | {
"line": 540,
"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.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.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.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.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.Integral.RieszMarkovKakutani.Real | {
"line": 337,
"column": 4
} | {
"line": 338,
"column": 11
} | {
"line": 338,
"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.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.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.RieszMarkovKakutani.Real | {
"line": 348,
"column": 48
} | {
"line": 348,
"column": 59
} | {
"line": 348,
"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": 383,
"column": 93
} | {
"line": 385,
"column": 56
} | {
"line": 386,
"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.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 |
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