module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 142,
"column": 8
} | {
"line": 142,
"column": 33
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 792,
"column": 8
} | {
"line": 792,
"column": 19
} | [
{
"pp": "case pos.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 794,
"column": 8
} | {
"line": 794,
"column": 19
} | [
{
"pp": "case neg.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 830,
"column": 2
} | {
"line": 860,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nhfu : UniformIntegrable f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indic... | obtain ⟨-, hfu, M, hM⟩ := hfu
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ { x | C ≤ ‖f i x‖₊ } ≤ ENNReal.ofReal δ := by
by_contra! hcon
choose ℐ hℐ using hcon
lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ eLpNorm (f (ℐ C)) p μ := by
int... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 830,
"column": 2
} | {
"line": 860,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nhfu : UniformIntegrable f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indic... | obtain ⟨-, hfu, M, hM⟩ := hfu
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ { x | C ≤ ‖f i x‖₊ } ≤ ENNReal.ofReal δ := by
by_contra! hcon
choose ℐ hℐ using hcon
lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ eLpNorm (f (ℐ C)) p μ := by
int... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 35
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 35
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 21
} | [
{
"pp": "case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 35
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 35
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 21
} | [
{
"pp": "case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 66
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 66
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\ns✝ t : Set Ω\nμ ν : FiniteMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 9
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : ProbabilityMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 13
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\ns : Set Ω\n⊢ μ s ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 48
} | [
{
"pp": "case h\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 187,
"column": 2
} | {
"line": 188,
"column": 9
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : FiniteMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 239,
"column": 50
} | {
"line": 239,
"column": 61
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nh : μ.mass = 1\n⊢ (↑μ).real univ = 1",
"usedConstants": [
"MeasureTheory.FiniteMeasure",
"Eq.mpr",
"Real",
"Set.univ",
"MeasureTheory.Measure.real",
"id",
"NNReal",
"MeasureTheory.FiniteMea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ns : Set Ω\n⊢ μ s ≤ μ.mass",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 48
} | [
{
"pp": "case h\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : FiniteMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 57
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"MeasureTheory.Measure",
"ENNReal.ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 906,
"column": 4
} | {
"line": 906,
"column": 15
} | [
{
"pp": "case refine_2.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nhf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nε : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 57
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"_private.Mathlib.MeasureTheory.Measure.FiniteMeasure.0.MeasureTheory.FiniteMeasure.apply_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 15
} | [
{
"pp": "case refine_3.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nC : ℝ≥0\nhC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 29
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : TopologicalSpace Ω\nμ : FiniteMeasure Ω\n⊢ μ.testAgainstNN 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 942,
"column": 2
} | {
"line": 942,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : TendstoInMeasure μ f u g\n⊢ MemLp g p μ",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᶠ[ae μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (κ a) (a✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᶠ[ae μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (η a) (a✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 467,
"column": 4
} | {
"line": 467,
"column": 40
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nle_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)\nω : Ω\nle' : f ω ≤ g ω + nndist f g\n⊢ ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)",
"usedConstants": [
"ENNRe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᶠ[ae μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ * (κ a) a✝²\na : α\nha : κ a = η a\nh'a : (κ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 354,
"column": 70
} | {
"line": 354,
"column": 81
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : HasOuterApproxClosed Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : μ (frontier E) = 0\n⊢ ↑μ (fr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 36
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\n⊢ LipschitzWith μ.mass fun f ↦ μ.testAgainstNN f",
"usedConstants": [
"MeasureTheory.FiniteMeasure.mass",
"Eq.mpr",
"Real.instLE",
"Real",
"Lipsc... | rw [lipschitzWith_iff_dist_le_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᶠ[ae μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (η a) (a✝³ ∩ a✝²) = (η a) a✝³ * (η a) a✝²\na : α\nha : κ a = η a\nh'a : (η ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 481,
"column": 4
} | {
"line": 481,
"column": 29
} | [
{
"pp": "case left\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₂ ≤ μ.testAgainstNN f₁ + μ.mass * nndist f₂ f₁\n⊢ ↑(μ.testAgainstNN f₂) ≤ ↑(μ.testAgainstNN f₁) + ↑μ.mass * dist f₁ f₂",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 23
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : iIndepSets π κ μ\na : α\nha : (κ a) (⋂ i ∈ ∅, univ) = ∏ i ∈ ∅, (κ a) univ\n⊢ (κ a) univ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 15
} | [
{
"pp": "case right\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₁ ≤ μ.testAgainstNN f₂ + μ.mass * nndist f₁ f₂\n⊢ ↑(μ.testAgainstNN f₁) ≤ ↑(μ.testAgainstNN f₂) + ↑μ.mass * dist f₁ f₂",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 28
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : SFinite μ\ns : Set Ω\na b : ℝ\nhab : a < b\nmbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Function.onFun Disjoint fun r ↦ frontier (Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nμ : Measure α\ninst✝¹ : Subsingleton ι\nm : ι → Set (Set Ω)\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\ns : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ s, f i ∈ m i\n⊢ s = ∅ ∨ ∃ i, s = {i}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 85
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nι' : Type u_5\ng : ι' → ι\nhg : Function.Injective g\nh : iIndepSets π κ μ\ns : Finset ι'\nf : ι' → Set Ω\nhf : ∀ i ∈ s, f i ∈ (π ∘ g) i\nf' : ι → Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 13
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_3\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), (μs i).testAgainstNN f ≤ nndist f 0 * (μs i).mass\nlim_pair : Te... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 270,
"column": 7
} | {
"line": 270,
"column": 22
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns₁ s₂ : Set (Set Ω)\nh : IndepSets s₁ s₂ κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\na : α\nha : (κ a) (t2 ∩ t1) = (κ a) t2 * (κ a) t1\n⊢ (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2",
... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 723,
"column": 2
} | {
"line": 723,
"column": 34
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\nh : ∀ (f : Ω →ᵇ ℝ), Tendsto (fun i ↦ ∫ (x : Ω), f x ∂↑(μs i)) F (𝓝 (∫ (x : Ω), f x ∂↑μ))\nf : Ω →ᵇ ℝ≥0\nlip : LipschitzWith ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 494,
"column": 38
} | {
"line": 494,
"column": 77
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : Measu... | measure_iUnion hfd fun i ↦ h2 _ (hfm i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 761,
"column": 2
} | {
"line": 765,
"column": 34
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nγ : Type u_2\n𝕜 : Type u_3\ninst✝ : RCLike 𝕜\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\nh : ∀ (f : Ω →ᵇ 𝕜), Tendsto (fun i ↦ ∫ (ω : Ω), f ω ∂↑(μs i)) F (𝓝 (∫ (ω ... | · specialize h ((RCLike.ofRealAm (K := 𝕜)).compLeftContinuousBounded ℝ
RCLike.lipschitzWith_ofReal f)
simp only [AlgHom.compLeftContinuousBounded_apply_apply, RCLike.ofRealAm_coe,
integral_ofReal] at h
exact tendsto_ofReal_iff'.mp h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 525,
"column": 6
} | {
"line": 525,
"column": 84
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)\nf : Ω →ᵇ ℝ\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\nsame : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 522,
"column": 6
} | {
"line": 522,
"column": 21
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 527,
"column": 8
} | {
"line": 527,
"column": 23
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 648,
"column": 67
} | {
"line": 648,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nh : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (↑(μs i)).real F) L ≤ (↑μ).rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 672,
"column": 4
} | {
"line": 672,
"column": 15
} | [
{
"pp": "case mp\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nf : Ω → ℝ\nhf_bounded : ∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C\nhf_lip : ∃ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 17
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nπ : ι → Set (Set Ω)\nx✝ : MeasurableSpace Ω\nμ : Measure Ω\nh : iIndepSets π μ\n⊢ μ univ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 13
} | [
{
"pp": "Ω : Type u_1\n_mΩ : MeasurableSpace Ω\ns t : Set Ω\nμ : Measure Ω\nh : IndepSet s t μ\n⊢ μ (s ∩ t) = μ s * μ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 617,
"column": 2
} | {
"line": 617,
"column": 13
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : ι → Set Ω\nh : iIndepSet f μ\ns : Finset ι\n⊢ μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 701,
"column": 6
} | {
"line": 701,
"column": 17
} | [
{
"pp": "case refine_2\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh✝ :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 791,
"column": 6
} | {
"line": 791,
"column": 17
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nβ : ι → Type u_10\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh : iIndepFun f μ\n⊢ μ univ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 21
} | [
{
"pp": "case hf.inl\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → Measure Ω\nν : Measure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))\nhν : ∀ s ∈ S,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 752,
"column": 4
} | {
"line": 752,
"column": 15
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\n⊢ ∀ s ∈ S, Ten... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 753,
"column": 2
} | {
"line": 753,
"column": 13
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\nthis : Tendsto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 771,
"column": 4
} | {
"line": 771,
"column": 76
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\n⊢ ∃ T ⊆ S, T.Countable ... | have : ∀ (x : G), ∃ s ∈ S, s ∈ 𝓝 (x : Ω) ∧ s ⊆ G := fun x ↦ h G hG x x.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Basic | {
"line": 859,
"column": 2
} | {
"line": 859,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : Fintype ι\nβ : ι → Type u_11\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 779,
"column": 6
} | {
"line": 779,
"column": 17
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\ns : ↑G → Set Ω\nhsS : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 1075,
"column": 8
} | {
"line": 1075,
"column": 74
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Basic | {
"line": 1075,
"column": 8
} | {
"line": 1075,
"column": 74
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Basic | {
"line": 1075,
"column": 8
} | {
"line": 1075,
"column": 74
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Basic | {
"line": 1093,
"column": 2
} | {
"line": 1093,
"column": 13
} | [
{
"pp": "case h.e'_3.a\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nt : ι → Set β\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 795,
"column": 4
} | {
"line": 795,
"column": 22
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nT : Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integrable | {
"line": 39,
"column": 62
} | {
"line": 39,
"column": 82
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integrable | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | [
{
"pp": "case h\nΩ : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Density | {
"line": 336,
"column": 26
} | {
"line": 342,
"column": 77
} | [
{
"pp": "Ω : Type u_1\nG : Type u_2\nmΩ : MeasurableSpace Ω\nℙ : Measure Ω\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\nX Y : Ω → G\ninst✝² : SFinite μ\ninst✝¹ : HasPDF X ℙ μ\ninst✝ : HasPDF Y ℙ μ\nσX : SigmaFinite ... | by
have : AEMeasurable X ℙ := HasPDF.aemeasurable' μ
have : AEMeasurable Y ℙ := HasPDF.aemeasurable' μ
rw [hasPDF_iff_of_aemeasurable (by fun_prop),
hXY.map_mul_eq_map_mconv_map₀' (by fun_prop) (by fun_prop) σX σY]
refine ⟨?_, mconv_absolutelyContinuous HasPDF.absolutelyContinuous⟩
apply HaveLebesgueDecom... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Measur... | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Measur... | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 15
} | [
{
"pp": "case h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 17
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 17
} | [
{
"pp": "case h.e'_3.h\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX Y : (i : ι)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Mea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | [
{
"pp": "case inr.inl\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁴ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 523,
"column": 2
} | {
"line": 523,
"column": 13
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : Mul β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 13
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : Mul β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 13
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : Div β\ninst✝ : MeasurableDiv₂ β\nf : ι → Ω → β\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 13
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : Div β\ninst✝ : MeasurableDiv₂ β\nf : ι → Ω → β\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 629,
"column": 14
} | {
"line": 629,
"column": 34
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ns : Fi... | Finset.prod_coe_sort | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 60
} | [
{
"pp": "case h\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘... | exact Finset.prod_congr rfl fun i hi ↦ (hω ⟨i, hi⟩).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Variance | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 33
} | [
{
"pp": "case pos.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ∫⁻ (ω : Ω), ↑‖X ω ^ 2‖₊ ∂μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNorm... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 369,
"column": 6
} | {
"line": 369,
"column": 17
} | [
{
"pp": "case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ Integrable (fun x ↦ ↑‖X x ^ 2‖₊) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 181,
"column": 34
} | {
"line": 181,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nx✝ : Set (Filtration ι m)\n⊢ IsLUB (seq '' x✝) ↑(sSup x✝)",
"usedConstants": [
"Set.image_image",
"Eq.mpr",
"Pi.preorder",
"MeasureTheory.Filtration.seq",
"congrArg",
"MeasurableSpace.instPart... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 50
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nx✝ : Set (Filtration ι m)\nhn : x✝.Nonempty\n⊢ IsGLB (seq '' x✝) ↑{ seq := fun i ↦ sInf ((fun f ↦ ↑f i) '' x✝), mono' := ⋯, le' := ⋯ }",
"usedConstants": [
"Set.image_image",
"Eq.mpr",
"Pi.preorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 22
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nx✝ : Set (Filtration ι m)\nhn✝ : ¬x✝.Nonempty\nhn : x✝ = ∅\n⊢ IsGLB x✝ { seq := fun i ↦ m, mono' := ⋯, le' := ⋯ }",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.instLE",
"MeasureTheory.Filtration.mk"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 365,
"column": 10
} | {
"line": 365,
"column": 52
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 366,
"column": 10
} | {
"line": 366,
"column": 44
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 367,
"column": 6
} | {
"line": 367,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i... | exact hle₁.trans hle₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Filtration | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 46
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nhineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j\n⊢ ↑𝓕₊₊ i ≤ ↑𝓕₊ i",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 49
} | [
{
"pp": "case refine_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁴ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Floor | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (floor ⁻¹' {⌊x⌋})",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Floor | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 50
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Set.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"usedConstants": [
... | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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