module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 75
} | {
"line": 613,
"column": 77
} | {
"line": 613,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 133,
"column": 64
} | {
"line": 133,
"column": 66
} | {
"line": 134,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\nv : V\nf : α →₀ W\ni : α\n⊢... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 617,
"column": 87
} | {
"line": 617,
"column": 89
} | {
"line": 618,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ AEStronglyMeasurable X μ",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestronglyMeasurable",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toO... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 138,
"column": 93
} | {
"line": 138,
"column": 95
} | {
"line": 139,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\ni : α\nv : V\nw : W\n⊢ (σ.f... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 625,
"column": 31
} | {
"line": 625,
"column": 33
} | {
"line": 626,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nY : Ω → ℝ\nh' : X =ᵐ[μ] Y\n⊢ HasSubgaussianMGF Y c μ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Unit.unit",
"MeasureThe... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 631,
"column": 41
} | {
"line": 631,
"column": 43
} | {
"line": 632,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Unit.unit",
"NormedCommRing.toSeminormedCommRing",
"Real... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 355,
"column": 39
} | {
"line": 355,
"column": 41
} | {
"line": 356,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\n⊢ ∑ k ∈ range K, ∫ (x : ℝ) ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 147,
"column": 88
} | {
"line": 147,
"column": 90
} | {
"line": 147,
"column": 91
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 635,
"column": 82
} | {
"line": 635,
"column": 84
} | {
"line": 636,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Pure.pure",
"MeasureTheory.ae",
"Unit.unit",
"Real.instLE",
"Rea... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 640,
"column": 85
} | {
"line": 640,
"column": 87
} | {
"line": 641,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\n⊢ HasSubgaussianMGF (fun x ↦ 0) 0 μ",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Unit.unit",
"Real",
"Real.instZero",
"NNReal",
"MeasureTheory.Measure.dirac... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 152,
"column": 39
} | {
"line": 152,
"column": 41
} | {
"line": 153,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nk : Type u\ninst✝ : CommSemiring k\nα : Type w'\ng : G\ni : α\nr s : k\n⊢ (leftRegularTensorTrivialIsoFree α) (MonoidAlgebra.single g r ⊗ₜ[k] MonoidAlgebra.single i s) =\n single i (MonoidAlgebra.single g (r * s))",
"ppTerm": "?m.50",
"assigned": true,
"use... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 646,
"column": 82
} | {
"line": 646,
"column": 84
} | {
"line": 647,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ HasSubgaussianMGF (-X) c μ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Real",
"Pi.instNeg",
"ProbabilityTheory.HasSubgau... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 159,
"column": 36
} | {
"line": 159,
"column": 38
} | {
"line": 160,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nk : Type u\ninst✝ : CommSemiring k\nα : Type w'\ni : α\ng : G\nr : k\n⊢ (leftRegularTensorTrivialIsoFree α).symm (single i (MonoidAlgebra.single g r)) =\n MonoidAlgebra.single g 1 ⊗ₜ[k] MonoidAlgebra.single i r",
"ppTerm": "?m.60",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 654,
"column": 57
} | {
"line": 654,
"column": 59
} | {
"line": 654,
"column": 60
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ≥0\nΩ' : Type u_2\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nX : Ω → ℝ\nhY : AEMeasurable Y μ\nh : HasSubgaussianMGF X c (Measure.map Y μ)\nt : ℝ\nh1 : Integrable (fun ω ↦ rexp (t * X ω)) (Measure.map Y μ)\n⊢ AEMeasurable Y μ",
"ppTerm": "?m.54... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 652,
"column": 26
} | {
"line": 652,
"column": 28
} | {
"line": 653,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ≥0\nΩ' : Type u_2\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nX : Ω → ℝ\nhY : AEMeasurable Y μ\nh : HasSubgaussianMGF X c (Measure.map Y μ)\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ∘ Y) ω)) μ",
"ppTerm": "?m.24",
"assigned": true,
"use... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 172,
"column": 60
} | {
"line": 172,
"column": 62
} | {
"line": 172,
"column": 63
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 655,
"column": 14
} | {
"line": 655,
"column": 16
} | {
"line": 656,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ≥0\nΩ' : Type u_2\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nX : Ω → ℝ\nhY : AEMeasurable Y μ\nh : HasSubgaussianMGF X c (Measure.map Y μ)\nt : ℝ\n⊢ mgf (X ∘ Y) μ t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.27",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 360,
"column": 47
} | {
"line": 360,
"column": 49
} | {
"line": 361,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\n⊢ ∫ (x : ℝ) in 0..↑K, x ∂ρ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 664,
"column": 32
} | {
"line": 664,
"column": 34
} | {
"line": 664,
"column": 35
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * id ω)) (Measure.map X μ)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Continuous.rexp",
"Co... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 314,
"column": 73
} | {
"line": 314,
"column": 75
} | {
"line": 315,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\n⊢ ∑ j ∈ range K, (↑j ^ 2)⁻¹ * ∫ (a : Ω), (truncation X ↑j ^ 2) a ≤ 2 * ∫ (a : Ω), X a",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"instWeakly... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 173,
"column": 16
} | {
"line": 173,
"column": 18
} | {
"line": 173,
"column": 19
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 174,
"column": 17
} | {
"line": 174,
"column": 19
} | {
"line": 174,
"column": 20
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 660,
"column": 66
} | {
"line": 660,
"column": 68
} | {
"line": 661,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\n⊢ HasSubgaussianMGF id c (Measure.map X μ) ↔ HasSubgaussianMGF X c μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Continuous.rexp",
"Eq.mpr",
"NormedCommRing.toSem... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 178,
"column": 62
} | {
"line": 178,
"column": 64
} | {
"line": 179,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝³ : Monoid G\nV : Type v'\ninst✝² : AddCommMonoid V\nk : Type u\ninst✝¹ : CommSemiring k\ninst✝ : Module k V\nσ : Representation k G V\ng : G\nv : V\n⊢ (σ.leftRegularMapEquiv.symm v) (MonoidAlgebra.single g 1) = (σ g) v",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 671,
"column": 32
} | {
"line": 671,
"column": 34
} | {
"line": 672,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nΩ' : Type u_2\nmΩ' : MeasurableSpace Ω'\nμ' : Measure Ω'\nY : Ω' → ℝ\nhX : HasSubgaussianMGF X c μ\nhXY : IdentDistrib X Y μ μ'\n⊢ HasSubgaussianMGF Y c μ'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 679,
"column": 43
} | {
"line": 679,
"column": 45
} | {
"line": 679,
"column": 46
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhm : m ≤ mΩ\nhXm : Measurable X\nhX : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ Measurable fun ω ↦ rexp (t * X ω)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNo... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 677,
"column": 26
} | {
"line": 677,
"column": 28
} | {
"line": 678,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhm : m ≤ mΩ\nhXm : Measurable X\nhX : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * X ω)) (μ.trim hm)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.HasSubgaussia... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 683,
"column": 45
} | {
"line": 683,
"column": 47
} | {
"line": 683,
"column": 48
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhm : m ≤ mΩ\nhXm : Measurable X\nhX : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ Measurable fun ω ↦ rexp (t * X ω)",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNo... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 680,
"column": 14
} | {
"line": 680,
"column": 16
} | {
"line": 681,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhm : m ≤ mΩ\nhXm : Measurable X\nhX : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ mgf X (μ.trim hm) t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.t... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 686,
"column": 72
} | {
"line": 686,
"column": 74
} | {
"line": 687,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nr : ℝ\n⊢ HasSubgaussianMGF (fun ω ↦ r * X ω) (⟨r ^ 2, ⋯⟩ * c) μ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Unit.unit",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 686,
"column": 72
} | {
"line": 688,
"column": 46
} | {
"line": 690,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nr : ℝ\n⊢ HasSubgaussianMGF (fun ω ↦ r * X ω) (⟨r ^ 2, ⋯⟩ * c) μ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Unit.unit",
... | [] | by
rw [HasSubgaussianMGF_iff_kernel] at h ⊢
exact Kernel.HasSubgaussianMGF.const_mul h r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.SubGaussian | {
"line": 691,
"column": 39
} | {
"line": 691,
"column": 41
} | {
"line": 692,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : HasSubgaussianMGF X c μ\n⊢ integrableExpSet X μ = Set.univ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"ProbabilityTheory.HasSubgaussianMGF.integrable_exp_mul",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 696,
"column": 42
} | {
"line": 696,
"column": 44
} | {
"line": 696,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : HasSubgaussianMGF X c μ\np : ℝ≥0\n⊢ 0 ∈ interior (integrableExpSet X μ)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"interior_univ",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 699,
"column": 47
} | {
"line": 699,
"column": 49
} | {
"line": 699,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : HasSubgaussianMGF X c μ\n⊢ 0 ∈ interior (integrableExpSet X μ)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"congrArg",
"interior_univ",
"Set.mem... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 705,
"column": 53
} | {
"line": 705,
"column": 55
} | {
"line": 706,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Pure.pure",
"MeasureTheory.ae",
"Unit.unit"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 713,
"column": 88
} | {
"line": 713,
"column": 90
} | {
"line": 714,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᵐ[μ] 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Pure.pure",
"MeasureTheory.ae",
"Unit.unit",
"Real",
"MeasureTheory.Measure",
"instCountabl... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 49
} | {
"line": 726,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ HasSubgaussianMGF ... | [] | simpa [HasSubgaussianMGF_iff_kernel] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.SubGaussian | {
"line": 722,
"column": 73
} | {
"line": 722,
"column": 75
} | {
"line": 723,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\n⊢ HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) μ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 729,
"column": 26
} | {
"line": 729,
"column": 28
} | {
"line": 730,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ω + Y ω))) μ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedComm... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 738,
"column": 55
} | {
"line": 738,
"column": 57
} | {
"line": 739,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ mgf X μ t * mgf Y μ t ≤ rexp (↑cX * t ^ 2 / 2) * rexp (↑cY * t ^ 2 / 2)",
"ppTerm": "?m.212",
"assigned": true,
"usedCons... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 403,
"column": 72
} | {
"line": 403,
"column": 74
} | {
"line": 404,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 743,
"column": 39
} | {
"line": 743,
"column": 41
} | {
"line": 743,
"column": 42
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ rexp (↑cX * t ^ 2 / 2) * rexp (↑cY * t ^ 2 / 2) = rexp ((↑cX + ↑cY) * t ^ 2 / 2)",
"ppTerm": "?m.247",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 43,
"column": 32
} | {
"line": 43,
"column": 34
} | {
"line": 43,
"column": 35
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 734,
"column": 14
} | {
"line": 734,
"column": 16
} | {
"line": 735,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ mgf (fun ω ↦ X ω + Y ω) μ t ≤ rexp (↑(cX + cY) * t ^ 2 / 2)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 399,
"column": 75
} | {
"line": 399,
"column": 77
} | {
"line": 400,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 747,
"column": 57
} | {
"line": 747,
"column": 59
} | {
"line": 748,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\n⊢ HasSubgaussianMGF (fun ω ↦ X ω - Y ω) (cX + cY) μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 46,
"column": 61
} | {
"line": 46,
"column": 63
} | {
"line": 47,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nf... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 53,
"column": 85
} | {
"line": 53,
"column": 87
} | {
"line": 54,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nf g : ρ.IntertwiningMap σ\nh : f.toLinearMap = g.toLinearMap\n... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
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} | {
"line": 60,
"column": 98
} | {
"line": 61,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\n⊢ Function.Injective fun f ↦ f.toFun",
"ppTerm": "?m.56",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 48
} | {
"line": 96,
"column": 49
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 103,
"column": 45
} | {
"line": 103,
"column": 47
} | {
"line": 104,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 114,
"column": 33
} | {
"line": 114,
"column": 35
} | {
"line": 114,
"column": 36
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 121,
"column": 71
} | {
"line": 121,
"column": 73
} | {
"line": 121,
"column": 74
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 127,
"column": 57
} | {
"line": 127,
"column": 59
} | {
"line": 128,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 138,
"column": 29
} | {
"line": 138,
"column": 31
} | {
"line": 138,
"column": 32
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 145,
"column": 85
} | {
"line": 145,
"column": 87
} | {
"line": 146,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nι : Type u_6\ns : Finset ι\nf : ι → ρ.IntertwiningMap σ\n⊢ (∑ ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 151,
"column": 41
} | {
"line": 151,
"column": 43
} | {
"line": 152,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nι : Type u_6\ns : Finset ι\nf : ι → ρ.IntertwiningMap σ\nv : V... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 755,
"column": 67
} | {
"line": 755,
"column": 69
} | {
"line": 756,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\nh_meas : ∀ (i : ι), AEMeasurable (X i) μ\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 161,
"column": 28
} | {
"line": 161,
"column": 30
} | {
"line": 161,
"column": 31
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW✝ : Type u_4\nU : Type u_5\ninst✝¹³ : Semiring A\ninst✝¹² : Monoid G\ninst✝¹¹ : AddCommMonoid V✝\ninst✝¹⁰ : AddCommMonoid W✝\ninst✝⁹ : AddCommMonoid U\ninst✝⁸ : Module A V✝\ninst✝⁷ : Module A W✝\ninst✝⁶ : Module A U\nρ✝ : Representation A G V✝\nσ✝ : Represent... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 167,
"column": 45
} | {
"line": 167,
"column": 47
} | {
"line": 168,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW✝ : Type u_4\nU : Type u_5\ninst✝¹³ : Semiring A\ninst✝¹² : Monoid G\ninst✝¹¹ : AddCommMonoid V✝\ninst✝¹⁰ : AddCommMonoid W✝\ninst✝⁹ : AddCommMonoid U\ninst✝⁸ : Module A V✝\ninst✝⁷ : Module A W✝\ninst✝⁶ : Module A U\nρ✝ : Representation A G V✝\nσ✝ : Represent... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 178,
"column": 33
} | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 36
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW✝ : Type u_4\nU : Type u_5\ninst✝¹³ : Semiring A\ninst✝¹² : Monoid G\ninst✝¹¹ : AddCommMonoid V✝\ninst✝¹⁰ : AddCommMonoid W✝\ninst✝⁹ : AddCommMonoid U\ninst✝⁸ : Module A V✝\ninst✝⁷ : Module A W✝\ninst✝⁶ : Module A U\nρ✝ : Representation A G V✝\nσ✝ : Represent... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 199,
"column": 21
} | {
"line": 199,
"column": 23
} | {
"line": 199,
"column": 24
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 771,
"column": 76
} | {
"line": 771,
"column": 78
} | {
"line": 772,
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} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i, X (↑i) ω) (∑ i, c ↑i) μ",
"ppTerm": "?m.42",
"assigned": true,
"usedCons... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 213,
"column": 21
} | {
"line": 213,
"column": 23
} | {
"line": 213,
"column": 24
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 770,
"column": 67
} | {
"line": 770,
"column": 69
} | {
"line": 771,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ",
"ppTerm": "?m.26",
"assigned": true,
"used... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
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"column": 48
} | {
"line": 224,
"column": 50
} | {
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"column": 51
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 791,
"column": 27
} | {
"line": 791,
"column": 29
} | {
"line": 791,
"column": 30
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\n⊢ ∀ i ∈ Finset.range n, HasSubgaussianMGF (X i) c μ",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 417,
"column": 8
} | {
"line": 418,
"column": 68
} | {
"line": 419,
"column": 8
} | [
{
"pp": "case e_f.e_a.hs\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 ... | [
"case e_f.e_a.h\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : ... | · intro j _
exact (hident j).aestronglyMeasurable_fst.memLp_truncation | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RepresentationTheory.Intertwining | {
"line": 227,
"column": 48
} | {
"line": 227,
"column": 50
} | {
"line": 227,
"column": 51
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 233,
"column": 40
} | {
"line": 233,
"column": 42
} | {
"line": 233,
"column": 43
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 239,
"column": 40
} | {
"line": 239,
"column": 42
} | {
"line": 239,
"column": 43
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 789,
"column": 81
} | {
"line": 789,
"column": 83
} | {
"line": 790,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑n * ↑c))",
"ppTerm": "?m.56",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 261,
"column": 40
} | {
"line": 261,
"column": 42
} | {
"line": 261,
"column": 43
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 802,
"column": 66
} | {
"line": 802,
"column": 68
} | {
"line": 803,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ\nhY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ\nhindep : X ⟂ᵢ[μ] Y\nh_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ\n⊢ μ.real {ω | X ω ≤ Y ω} =\n ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 342,
"column": 14
} | {
"line": 342,
"column": 16
} | {
"line": 342,
"column": 17
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nφ ψ : σ.Equiv ρ\nh : φ.toLinearEquiv = ψ.toLinearEquiv\n⊢ φ = ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 413,
"column": 76
} | {
"line": 413,
"column": 78
} | {
"line": 414,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 352,
"column": 30
} | {
"line": 352,
"column": 32
} | {
"line": 353,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 365,
"column": 61
} | {
"line": 365,
"column": 63
} | {
"line": 366,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nφ ψ : ρ.Equiv σ\nh : ⇑φ = ⇑ψ\n⊢ φ = ψ",
"ppTerm": "?m.62",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 373,
"column": 23
} | {
"line": 373,
"column": 25
} | {
"line": 373,
"column": 26
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 811,
"column": 9
} | {
"line": 811,
"column": 11
} | {
"line": 811,
"column": 12
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ\nhY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ\nhindep : X ⟂ᵢ[μ] Y\nh_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ\n⊢ Measurable fun x ↦ x - ∫ (x : ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 811,
"column": 23
} | {
"line": 811,
"column": 25
} | {
"line": 811,
"column": 26
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ\nhY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ\nhindep : X ⟂ᵢ[μ] Y\nh_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ\n⊢ Measurable fun x ↦ x - ∫ (x : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 397,
"column": 23
} | {
"line": 397,
"column": 25
} | {
"line": 398,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 404,
"column": 41
} | {
"line": 404,
"column": 43
} | {
"line": 405,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ne : V ≃ₗ[A] W\nhe : ∀ (g : G), ↑e ∘ₗ ρ g = σ g ∘ₗ ↑e\ng : G\n⊢... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 805,
"column": 58
} | {
"line": 805,
"column": 60
} | {
"line": 806,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ\nhY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ\nhindep : X ⟂ᵢ[μ] Y\nh_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ\n⊢ μ.real {ω | ∫ (x : Ω), X x ∂μ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 800,
"column": 81
} | {
"line": 800,
"column": 83
} | {
"line": 801,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) cX μ\nhY : HasSubgaussianMGF (fun ω ↦ Y ω - ∫ (x : Ω), Y x ∂μ) cY μ\nhindep : X ⟂ᵢ[μ] Y\nh_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ\n⊢ μ.real {ω | X ω ≤ Y ω} ≤ rexp ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 826,
"column": 61
} | {
"line": 826,
"column": 63
} | {
"line": 826,
"column": 64
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\n⊢ Set.Icc 0 t ⊆ interior (integrableExp... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 421,
"column": 97
} | {
"line": 421,
"column": 99
} | {
"line": 422,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 423,
"column": 23
} | {
"line": 423,
"column": 25
} | {
"line": 424,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : Semiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G W\nτ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 445,
"column": 63
} | {
"line": 445,
"column": 65
} | {
"line": 445,
"column": 66
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nφ : ρ.Equiv σ\n⊢ φ.trans φ.symm = refl ρ",
"ppTerm": "?m.8... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 448,
"column": 63
} | {
"line": 448,
"column": 65
} | {
"line": 448,
"column": 66
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\nφ : ρ.Equiv σ\n⊢ φ.symm.trans φ = refl σ",
"ppTerm": "?m.8... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 831,
"column": 36
} | {
"line": 831,
"column": 38
} | {
"line": 832,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\nhs : Set.Icc 0 t ⊆ interior (integrable... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 461,
"column": 78
} | {
"line": 461,
"column": 80
} | {
"line": 462,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ng : G\nφ : ρ.Equiv σ\n⊢ φ.toLinearEquiv.conj (ρ g) = σ g",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 473,
"column": 33
} | {
"line": 473,
"column": 35
} | {
"line": 473,
"column": 36
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 833,
"column": 27
} | {
"line": 833,
"column": 29
} | {
"line": 834,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\nhs : Set.Icc 0 t ⊆ interior (integrable... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 496,
"column": 23
} | {
"line": 496,
"column": 25
} | {
"line": 497,
"column": 8
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 837,
"column": 28
} | {
"line": 837,
"column": 30
} | {
"line": 837,
"column": 31
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\nhi : ∀ (u : ℝ), Integrable (fun ω ↦ rexp (u * X ω)) μ\nhs : Set.Icc 0 t ⊆ interior (integrable... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 824,
"column": 56
} | {
"line": 824,
"column": 58
} | {
"line": 825,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b t : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nht : 0 < t\n⊢ mgf X μ t ≤ rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)",
"ppTerm": "?m.83",
"assigned": ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 505,
"column": 20
} | {
"line": 505,
"column": 22
} | {
"line": 505,
"column": 23
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 506,
"column": 19
} | {
"line": 506,
"column": 21
} | {
"line": 506,
"column": 22
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 503,
"column": 25
} | {
"line": 503,
"column": 27
} | {
"line": 503,
"column": 28
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
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