module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 48
} | {
"line": 227,
"column": 49
} | [
{
"pp": "Ω : 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",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : 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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 57
} | {
"line": 218,
"column": 58
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaF... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaFinite (μ.tri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 98
} | {
"line": 219,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaF... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : SigmaFinite (μ.trim... | · simpa [condExp_of_not_sigmaFinite hm hsig, hp'] using integral_nonneg (fun x => by positivity) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 49
} | {
"line": 232,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ",
"ppTer... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α) in s, ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 57
} | {
"line": 364,
"column": 58
} | [
{
"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)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"_private.Mathlib.MeasureTheory.Measure.Finit... | [
"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) ≤ ∑' (a : ℕ), ↑μ (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 13
} | {
"line": 233,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ",
"ppTerm... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 233,
"column": 61
} | {
"line": 233,
"column": 72
} | {
"line": 233,
"column": 73
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖ ^ 1) μ",
"ppTerm": "?m.88",
"assigned": true,
... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 40
} | {
"line": 466,
"column": 41
} | [
{
"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)",
"ppTerm": "?m.128",
"assign... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 36
} | {
"line": 472,
"column": 2
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeasure.mass",
"Eq.mpr",
... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\n⊢ ∀ (x y : Ω →ᵇ ℝ≥0), dist (μ.testAgainstNN x) (μ.testAgainstNN y) ≤ ↑μ.mass * dist x y"
] | rw [lipschitzWith_iff_dist_le_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 29
} | {
"line": 480,
"column": 30
} | [
{
"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",
"p... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 15
} | {
"line": 484,
"column": 16
} | [
{
"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",
"... | [
"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.ProbabilityMeasure | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 66
} | {
"line": 134,
"column": 67
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 9
} | {
"line": 209,
"column": 10
} | [
{
"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)))",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"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)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 14
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\ns : Set Ω\n⊢ μ s ≤ 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\ns : Set Ω\n⊢ μ s ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 48
} | {
"line": 227,
"column": 49
} | [
{
"pp": "Ω : 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",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : 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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 57
} | {
"line": 275,
"column": 58
} | [
{
"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)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
... | [
"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) ≤ ∑' (a : ℕ), ↑μ (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 791,
"column": 8
} | {
"line": 791,
"column": 19
} | {
"line": 791,
"column": 20
} | [
{
"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... | [
"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 x‖₊}.indica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 793,
"column": 8
} | {
"line": 793,
"column": 19
} | {
"line": 793,
"column": 20
} | [
{
"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... | [
"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 x‖₊}.indica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 760,
"column": 2
} | {
"line": 764,
"column": 34
} | {
"line": 766,
"column": 0
} | [
{
"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": 123,
"column": 4
} | {
"line": 123,
"column": 35
} | {
"line": 123,
"column": 36
} | [
{
"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",
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 35
} | {
"line": 126,
"column": 36
} | [
{
"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... | [
"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\ni : ι\n⊢ (μs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 21
} | {
"line": 132,
"column": 22
} | [
{
"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 =... | [
"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 = 1 - μ E\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 35
} | {
"line": 147,
"column": 36
} | [
{
"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",
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 35
} | {
"line": 150,
"column": 36
} | [
{
"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... | [
"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\ni : ι\n⊢ (μs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 21
} | {
"line": 156,
"column": 22
} | [
{
"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 =... | [
"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 = 1 - μ E\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 829,
"column": 2
} | {
"line": 859,
"column": 96
} | {
"line": 861,
"column": 0
} | [
{
"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": 829,
"column": 2
} | {
"line": 859,
"column": 96
} | {
"line": 861,
"column": 0
} | [
{
"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.Tight | {
"line": 136,
"column": 2
} | {
"line": 146,
"column": 33
} | {
"line": 148,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝⁴ : TopologicalSpace 𝓧\ninst✝³ : TopologicalSpace 𝓨\ninst✝² : MeasurableSpace 𝓨\ninst✝¹ : OpensMeasurableSpace 𝓨\ninst✝ : T2Space 𝓨\nhS : IsTightMeasureSet S\nf : 𝓧 → 𝓨\nhf : Continuous[inst✝⁴, inst✝³] f\n⊢ IsTigh... | [] | rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le] at hS ⊢
simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
intro ε hε
obtain ⟨K, hK_compact, hKS⟩ := hS ε hε
refine ⟨f '' K, hK_compact.image hf, fun μ hμS ↦ ?_⟩
by_cases hf_meas : AEMeasurable f μ
swap; · simp [Measure.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 136,
"column": 2
} | {
"line": 146,
"column": 33
} | {
"line": 148,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝⁴ : TopologicalSpace 𝓧\ninst✝³ : TopologicalSpace 𝓨\ninst✝² : MeasurableSpace 𝓨\ninst✝¹ : OpensMeasurableSpace 𝓨\ninst✝ : T2Space 𝓨\nhS : IsTightMeasureSet S\nf : 𝓧 → 𝓨\nhf : Continuous[inst✝⁴, inst✝³] f\n⊢ IsTigh... | [] | rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le] at hS ⊢
simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
intro ε hε
obtain ⟨K, hK_compact, hKS⟩ := hS ε hε
refine ⟨f '' K, hK_compact.image hf, fun μ hμS ↦ ?_⟩
by_cases hf_meas : AEMeasurable f μ
swap; · simp [Measure.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 27
} | {
"line": 161,
"column": 28
} | [
{
"pp": "case h₁\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, ... | [
"case h₁\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 27
} | {
"line": 164,
"column": 28
} | [
{
"pp": "case h₂\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, ... | [
"case h₂\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 905,
"column": 4
} | {
"line": 905,
"column": 15
} | {
"line": 905,
"column": 16
} | [
{
"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 : ... | [
"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 : ℝ\nhε : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 21
} | [
{
"pp": "α : 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 =ᵐ[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✝¹ i)\na : α... | [
"α : 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 =ᵐ[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✝¹ i)\na : α\nha : κ a =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 21
} | [
{
"pp": "α : 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 =ᵐ[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✝¹ i)\na : α... | [
"α : 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 =ᵐ[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✝¹ i)\na : α\nha : κ a =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 357,
"column": 70
} | {
"line": 357,
"column": 81
} | {
"line": 357,
"column": 82
} | [
{
"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... | [
"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 (frontier ?m.34... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 914,
"column": 4
} | {
"line": 914,
"column": 15
} | {
"line": 914,
"column": 16
} | [
{
"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... | [
"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 : ℕ\nhn : n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | {
"line": 147,
"column": 21
} | [
{
"pp": "α : 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 =ᵐ[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 a) (a✝³ ∩ a... | [
"α : 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 =ᵐ[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 a) (a✝³ ∩ a✝²) = (κ a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 28
} | {
"line": 410,
"column": 29
} | [
{
"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... | [
"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 (Metric.thicken... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 941,
"column": 2
} | {
"line": 941,
"column": 13
} | {
"line": 941,
"column": 14
} | [
{
"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",
"ppTer... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | {
"line": 147,
"column": 21
} | [
{
"pp": "α : 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 =ᵐ[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 a) (a✝³ ∩ a... | [
"α : 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 =ᵐ[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 a) (a✝³ ∩ a✝²) = (η a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 23
} | {
"line": 179,
"column": 24
} | [
{
"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",
"ppTerm": "?m.54",
"assigned": false,
"usedConsta... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 970,
"column": 2
} | {
"line": 970,
"column": 13
} | {
"line": 970,
"column": 14
} | [
{
"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))\... | [
"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))\n⊢ MemLp g p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 16
} | [
{
"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}",
"ppTerm": "?m.37",
"a... | [
"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}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 85
} | {
"line": 222,
"column": 86
} | [
{
"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)\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 Ω := Fun... | [
"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)\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 Ω := Function.extend... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 270,
"column": 7
} | {
"line": 270,
"column": 22
} | {
"line": 270,
"column": 23
} | [
{
"pp": "α : 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",
"ppTer... | [
"α : 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) (t2 ∩ t1) = (κ a) t1 * (κ a) t2"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 17
} | {
"line": 170,
"column": 18
} | [
{
"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",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 13
} | {
"line": 586,
"column": 14
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 617,
"column": 2
} | {
"line": 617,
"column": 13
} | {
"line": 617,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 791,
"column": 6
} | {
"line": 791,
"column": 17
} | {
"line": 791,
"column": 18
} | [
{
"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",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 494,
"column": 38
} | {
"line": 494,
"column": 77
} | {
"line": 494,
"column": 77
} | [
{
"pp": "α : 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 : MeasurableSet... | [
"α : 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 : MeasurableSet t1\nh : IsM... | measure_iUnion hfd fun i ↦ h2 _ (hfm i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 859,
"column": 2
} | {
"line": 859,
"column": 18
} | {
"line": 859,
"column": 19
} | [
{
"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 (⋂... | [
"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 (⋂ i ∈ S, f i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 522,
"column": 6
} | {
"line": 522,
"column": 21
} | {
"line": 522,
"column": 22
} | [
{
"pp": "α : 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 : Indep... | [
"α : 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 : IndepSets p1 p2 κ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 527,
"column": 8
} | {
"line": 527,
"column": 23
} | {
"line": 527,
"column": 24
} | [
{
"pp": "α : 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 : Indep... | [
"α : 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 : IndepSets p1 p2 κ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 651,
"column": 67
} | {
"line": 651,
"column": 78
} | {
"line": 651,
"column": 79
} | [
{
"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... | [
"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).real F\n⊢ ∀ (F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 1081,
"column": 8
} | {
"line": 1081,
"column": 74
} | {
"line": 1082,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Basic | {
"line": 1081,
"column": 8
} | {
"line": 1081,
"column": 74
} | {
"line": 1082,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Basic | {
"line": 1081,
"column": 8
} | {
"line": 1081,
"column": 74
} | {
"line": 1082,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun ... | [] | exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Basic | {
"line": 1099,
"column": 2
} | {
"line": 1099,
"column": 13
} | {
"line": 1099,
"column": 14
} | [
{
"pp": "case a\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nt : ι → Set β\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ\nhy : ∀... | [
"case a\nι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nt : ι → Set β\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ\nhy : ∀ (i : ι), μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Density | {
"line": 330,
"column": 26
} | {
"line": 336,
"column": 77
} | {
"line": 338,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nG : Type u_2\nmΩ : MeasurableSpace Ω\nℙ : Measure Ω\ninst✝⁶ : Group G\nmG : MeasurableSpace G\ninst✝⁵ : MeasurableMul₂ G\ninst✝⁴ : MeasurableInv G\nμ : Measure G\ninst✝³ : μ.IsMulLeftInvariant\nX Y : Ω → G\ninst✝² : SFinite μ\ninst✝¹ : HasPDF X ℙ μ\ninst✝ : HasPDF Y ℙ μ\nσX : SigmaFinite ... | [] | by
have : AEMeasurable X ℙ := HasPDF.aemeasurable' μ
have : AEMeasurable Y ℙ := HasPDF.aemeasurable' μ
rw [hasPDF_iff_of_aemeasurable (by fun_prop),
hXY.map_mul_eq_map_mconv_map₀' (by fun_prop) (by fun_prop) σX σY]
refine ⟨?_, mconv_absolutelyContinuous HasPDF.absolutelyContinuous⟩
apply HaveLebesgueDecom... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Integrable | {
"line": 39,
"column": 62
} | {
"line": 39,
"column": 82
} | {
"line": 39,
"column": 83
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f... | [
"Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f : ∀ (c : ℝ≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integrable | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 15
} | {
"line": 43,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f... | [
"Ω : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : MeasurableSpace F\nf : Ω → E\ng : Ω → F\np : ℝ≥0∞\nhp : p ≠ 0\nhp' : p ≠ ∞\nhℒp : MemLp f p μ\nhindep : f ⟂ᵢ[μ] g\nh'f : ∀ (c : ℝ≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 21
} | {
"line": 769,
"column": 22
} | [
{
"pp": "case inl\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → Measure Ω\nν : Measure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))\nhν : ∀ s ∈ S, ν ... | [
"case inl\nΩ : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → Measure Ω\nν : Measure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i).real s) l (𝓝 (ν.real s))\nhν : ∀ s ∈ S, ν s ≠ ∞\nhμ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 783,
"column": 4
} | {
"line": 783,
"column": 15
} | {
"line": 783,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\n⊢ ∀ s ∈ S, Ten... | [
"Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\n⊢ ∀ s ∈ S, Tendsto (fun a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 784,
"column": 2
} | {
"line": 784,
"column": 13
} | {
"line": 784,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\nthis : Tendsto... | [
"Ω : Type u_1\nι : Type u_2\ninst✝ : MeasurableSpace Ω\nS : Set (Set Ω)\nhS : IsPiSystem S\nμ : ι → ProbabilityMeasure Ω\nν : ProbabilityMeasure Ω\nl : Filter ι\nt : Finset (Set Ω)\nht : ∀ s ∈ t, s ∈ S\nhmeas : ∀ s ∈ S, MeasurableSet s\nh : ∀ s ∈ S, Tendsto (fun i ↦ (μ i) s) l (𝓝 (ν s))\nthis : Tendsto (fun i ↦ (↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 76
} | {
"line": 803,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\n⊢ ∃ T ⊆ S, T.Countable ... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nthis : ∀ (x : ↑G), ∃ s ∈ S, s ∈ 𝓝 ... | have : ∀ (x : G), ∃ s ∈ S, s ∈ 𝓝 (x : Ω) ∧ s ⊆ G := fun x ↦ h G hG x x.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 810,
"column": 6
} | {
"line": 810,
"column": 17
} | {
"line": 810,
"column": 18
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\ns : ↑G → Set Ω\nhsS : ∀... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\ns : ↑G → Set Ω\nhsS : ∀ (x : ↑G), s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 22
} | {
"line": 826,
"column": 23
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nT : Set ... | [
"case refine_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : SecondCountableTopology Ω\nS : Set (Set Ω)\nν : ProbabilityMeasure Ω\nh : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u\nG : Set Ω\nhG : IsOpen[inst✝¹] G\nr : ℝ≥0\nhr : r < ν G\nT : Set (Set Ω)\nTS ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | {
"line": 130,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Mea... | [] | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 123,
"column": 2
} | {
"line": 128,
"column": 61
} | {
"line": 130,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nf g : Ω → ℝ≥0∞\nh_meas_f : AEMeasurable f μ\nh_meas_g : AEMeasurable g μ\nh_indep_fun : f ⟂ᵢ[μ] g\n⊢ ∫⁻ (ω : Ω), (f * g) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Mea... | [] | have fg_ae : f * g =ᵐ[μ] h_meas_f.mk _ * h_meas_g.mk _ := h_meas_f.ae_eq_mk.mul h_meas_g.ae_eq_mk
rw [lintegral_congr_ae h_meas_f.ae_eq_mk, lintegral_congr_ae h_meas_g.ae_eq_mk,
lintegral_congr_ae fg_ae]
apply lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun h_meas_f.measurable_mk
h_meas_g.measurable_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | {
"line": 822,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | {
"line": 822,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 821,
"column": 15
} | {
"line": 821,
"column": 46
} | {
"line": 822,
"column": 4
} | [
{
"pp": "case refine_1.empty\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₁ m₂ x✝ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh_indep : Indep m₁ m₂ κ μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\ns' t' : Set Ω\nht' : t' ∈ {s | MeasurableSet s}\n⊢ ∅ ∈ {s | MeasurableSet s}",
"... | [] | exact @MeasurableSet.empty _ m₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 15
} | {
"line": 208,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\nin... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\ninst✝¹ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.Variance | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 33
} | {
"line": 368,
"column": 33
} | [
{
"pp": "case pos.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ∫⁻ (ω : Ω), ↑‖X ω ^ 2‖₊ ∂μ",
"ppTerm": "?pos.e_a✝",
"assigned": true,
"usedConstants": ... | [
"case pos.e_a\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ ENNReal.ofReal (∫ (x : Ω), (X ^ 2) x ∂μ) = ENNReal.ofReal (∫ (a : Ω), ↑‖X a ^ 2‖₊ ∂μ)",
"case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Mea... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 17
} | {
"line": 370,
"column": 18
} | [
{
"pp": "case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ Integrable (fun x ↦ ↑‖X x ^ 2‖₊) μ",
"ppTerm": "?pos.e_a.hfi✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case pos.e_a.hfi\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : MemLp X 2 μ\n⊢ Integrable (fun x ↦ X x ^ 2) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 17
} | {
"line": 299,
"column": 18
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [
"Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.HasLaw | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX : Ω → 𝓧\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nx : 𝓧\nhX : X =ᵐ[P] fun x_1 ↦ x\n⊢ HasLaw X (dirac x) P",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX : Ω → 𝓧\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nx : 𝓧\nhX : X =ᵐ[P] fun x_1 ↦ x\n⊢ HasLaw X (dirac x) P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | {
"line": 312,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [] | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 283,
"column": 2
} | {
"line": 310,
"column": 37
} | {
"line": 312,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜... | [] | borelize E F
have hfXgY := (hXY.comp₀ hX hY hf.aemeasurable hg.aemeasurable)
have hfX := (hf.comp_aemeasurable hX)
have hgY := (hg.comp_aemeasurable hY)
by_cases h'X : ∀ᵐ ω ∂μ, f (X ω) = 0
· have h' : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h'X] with ω hω
simp [hω]
simp [integ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 17
} | {
"line": 94,
"column": 18
} | [
{
"pp": "case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX Y : (i : ι) → Ω... | [
"case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX Y : (i : ι) → Ω i → E\nZ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 14
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Mea... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 35
} | {
"line": 150,
"column": 36
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝³ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝² : TopologicalSpace E\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : OpensMeasurableSpace E\nX : ι → Ω' → E\nhX₁ : ∀ (i : ι), AEMeasu... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝³ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝² : TopologicalSpace E\ninst✝¹ : l.IsCountablyGenerated\ninst✝ : OpensMeasurableSpace E\nX : ι → Ω' → E\nhX₁ : ∀ (i : ι), AEMeasurable (X i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Floor | {
"line": 29,
"column": 4
} | {
"line": 29,
"column": 51
} | {
"line": 29,
"column": 52
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (floor ⁻¹' {⌊x⌋})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (Ico (↑⌊x⌋) (↑⌊x⌋ + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Floor | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 50
} | {
"line": 38,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : OpensMeasurableSpace R\nx : R\n⊢ MeasurableSet (Ioc (↑⌈x⌉ - 1) ↑⌈x⌉)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 366,
"column": 10
} | {
"line": 366,
"column": 52
} | {
"line": 366,
"column": 53
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | [
"case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 367,
"column": 10
} | {
"line": 367,
"column": 44
} | {
"line": 367,
"column": 45
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | [
"case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Filtration | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 27
} | {
"line": 369,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i... | [] | exact hle₁.trans hle₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Filtration | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 46
} | {
"line": 369,
"column": 47
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nhineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j\n⊢ ↑𝓕₊₊ i ≤ ↑𝓕₊ i",
"pp... | [
"case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nhineq : ⨅ j, ⨅ (_ : j > i), ↑𝓕₊ j ≤ ⨅ j, ⨅ (_ : j > i), ↑𝓕 j\n⊢ ∀ (i_1 : ι), i < i_1 → ⨅ j, ⨅ (_ : i <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 629,
"column": 14
} | {
"line": 629,
"column": 34
} | {
"line": 629,
"column": 34
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ns : Finset ι\n... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\ns : Finset ι\ni : ι\nhi : ... | Finset.prod_coe_sort | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 60
} | {
"line": 658,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns ... | [] | exact Finset.prod_congr rfl fun i hi ↦ (hω ⟨i, hi⟩).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"pp": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\... | [
"α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\nhN₀ : μ N =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 19
} | {
"line": 107,
"column": 20
} | [
{
"pp": "case refine_3.refine_1.refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u... | [
"case refine_3.refine_1.refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | {
"line": 44,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"ppTerm": "?refine_2",... | [] | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 100,
"column": 4
} | {
"line": 113,
"column": 80
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | [] | refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- This next block proves that a set of strictly positive natural density is infinite, mixed
-- with the fact that `{n | x ∈ s n}` has strictly positive natural density.
-- TODO: Separate it out to a lemma once we have a natural density API.
· refi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 100,
"column": 4
} | {
"line": 113,
"column": 80
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | [] | refine ⟨{n | x ∈ s n}, fun hxs ↦ ?_, fun u hux hu ↦ ?_⟩
-- This next block proves that a set of strictly positive natural density is infinite, mixed
-- with the fact that `{n | x ∈ s n}` has strictly positive natural density.
-- TODO: Separate it out to a lemma once we have a natural density API.
· refi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 291,
"column": 8
} | {
"line": 291,
"column": 47
} | {
"line": 291,
"column": 48
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTopology E\ninst✝¹ : BorelSpace E\ninst✝ : l.IsCountablyGenerated\nX : ι → ... | [
"ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTopology E\ninst✝¹ : BorelSpace E\ninst✝ : l.IsCountablyGenerated\nX : ι → Ω' → E\nh : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 43
} | {
"line": 310,
"column": 44
} | [
{
"pp": "case refine_2\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁸ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁷ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝⁶ : SeminormedAddCommGroup E\... | [
"case refine_2\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁸ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁷ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝⁶ : SeminormedAddCommGroup E\ninst✝⁵ : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 31,
"column": 17
} | {
"line": 31,
"column": 49
} | {
"line": 31,
"column": 50
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), Measurable (f i)\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (hs.piece... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), Measurable (f i)\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (⋃ i, s i ∩ f i ⁻¹' t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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