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 | {
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"column": 77
} | {
"line": 237,
"column": 79
} | {
"line": 238,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)\nh_mgf : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ),... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 244,
"column": 46
} | {
"line": 244,
"column": 48
} | {
"line": 244,
"column": 49
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh_int✝¹ : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)\nh_mgf✝ : ∀ᵐ (a : Ω') ∂ν, ∀ (i : ℚ), mgf X (κ a) ↑i ≤ rexp (↑c * ↑i ^ 2 / 2)\nh_int✝ : ∀ᵐ (ω' : Ω') ∂ν, ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 235,
"column": 12
} | {
"line": 235,
"column": 14
} | {
"line": 236,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)\nh_mgf : ∀ (q : ℚ), ∀ᵐ (ω' : Ω') ∂ν, mgf X (κ ω') ↑q ≤ rexp (↑c * ↑q ^ 2 / 2)\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 249,
"column": 24
} | {
"line": 249,
"column": 26
} | {
"line": 249,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsZeroOrMarkovKernel κ\n⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * 0)) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 250,
"column": 12
} | {
"line": 250,
"column": 14
} | {
"line": 250,
"column": 15
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsZeroOrMarkovKernel κ\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (fun x ↦ 0) (κ ω') t ≤ rexp (↑0 * t ^ 2 / 2)",
"ppTerm": "?m.28",
"assigned": true,
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 256,
"column": 65
} | {
"line": 256,
"column": 67
} | {
"line": 257,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nX : Ω → ℝ\nc : ℝ≥0\n⊢ HasSubgaussianMGF X c 0 ν",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 262,
"column": 66
} | {
"line": 262,
"column": 68
} | {
"line": 262,
"column": 69
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\n⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ 0)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 262,
"column": 75
} | {
"line": 262,
"column": 77
} | {
"line": 262,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\n⊢ ∀ᵐ (ω' : Ω') ∂0, ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real.instLE",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 265,
"column": 26
} | {
"line": 265,
"column": 28
} | {
"line": 265,
"column": 29
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (-X) ω)) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 60
} | {
"line": 266,
"column": 62
} | {
"line": 266,
"column": 63
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.72",... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 12
} | {
"line": 266,
"column": 14
} | {
"line": 266,
"column": 15
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 272,
"column": 40
} | {
"line": 272,
"column": 42
} | {
"line": 272,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nY : Ω → ℝ\nh : HasSubgaussianMGF X c κ ν\nh' : X =ᵐ[⇑κ ∘ₘ ν] Y\nt : ℝ\nω : Ω\nhω : X ω = Y ω\n⊢ rexp (t * Y ω) = rexp (t * X ω)",
"ppTerm": "?m.70",
"assigned": tr... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 270,
"column": 26
} | {
"line": 270,
"column": 28
} | {
"line": 271,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nY : Ω → ℝ\nh : HasSubgaussianMGF X c κ ν\nh' : X =ᵐ[⇑κ ∘ₘ ν] Y\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * Y ω)) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.32",
"assigned": true,
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 273,
"column": 12
} | {
"line": 273,
"column": 14
} | {
"line": 274,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nY : Ω → ℝ\nh : HasSubgaussianMGF X c κ ν\nh' : X =ᵐ[⇑κ ∘ₘ ν] Y\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.34",
"assigned... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 288,
"column": 84
} | {
"line": 288,
"column": 86
} | {
"line": 288,
"column": 87
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nκ : Kernel Ω' Ω''\nY : Ω'' → Ω\nX : Ω → ℝ\nhY : Measurable Y\nh : HasSubgaussianMGF X c (κ.map Y) ν\nt : ℝ\nh1 : Integrable (fun ω ↦ rexp (t * X ω)) (Measu... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 286,
"column": 26
} | {
"line": 286,
"column": 28
} | {
"line": 287,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nκ : Kernel Ω' Ω''\nY : Ω'' → Ω\nX : Ω → ℝ\nhY : Measurable Y\nh : HasSubgaussianMGF X c (κ.map Y) ν\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ∘ Y) ω)) (⇑κ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 290,
"column": 12
} | {
"line": 290,
"column": 14
} | {
"line": 291,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nc : ℝ≥0\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nκ : Kernel Ω' Ω''\nY : Ω'' → Ω\nX : Ω → ℝ\nhY : Measurable Y\nh : HasSubgaussianMGF X c (κ.map Y) ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (X ∘ Y) (κ ω') t ≤ r... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 304,
"column": 65
} | {
"line": 304,
"column": 67
} | {
"line": 304,
"column": 68
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : Measurable X\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\n⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * id ω)) (Measure.map X (⇑κ ∘ₘ ν))",
"ppTerm": "?m.101",
"assigned... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 299,
"column": 70
} | {
"line": 299,
"column": 72
} | {
"line": 300,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : Measurable X\n⊢ HasSubgaussianMGF id c (κ.map X) ν ↔ HasSubgaussianMGF X c κ ν",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureThe... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 310,
"column": 26
} | {
"line": 310,
"column": 28
} | {
"line": 311,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nr t : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (r * X ω))) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.44",
"assigned": true,
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"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 313,
"column": 12
} | {
"line": 313,
"column": 14
} | {
"line": 314,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nr : ℝ\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (fun ω ↦ r * X ω) (κ ω') t ≤ rexp (↑(NNReal.mk (r ^ 2) ⋯ * c) * t ^ 2 / 2)",
"ppTerm": "?m.46",... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 71,
"column": 22
} | {
"line": 71,
"column": 24
} | {
"line": 71,
"column": 25
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : Preorder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\n𝓕 : Filtration ι mΩ\nP : Measure Ω\nn : ℕ\n⊢ IsStoppingTime 𝓕 fun x ↦ ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"False",
"Measurable... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 72,
"column": 36
} | {
"line": 72,
"column": 38
} | {
"line": 72,
"column": 39
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : Preorder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\n𝓕 : Filtration ι mΩ\nP : Measure Ω\nx✝³ : Ω\nx✝² x✝¹ : ℕ\nx✝ : x✝² ≤ x✝¹\n⊢ (fun x ↦ ⊤) x✝² ≤ (fun x ↦ ⊤) x✝¹",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 85,
"column": 17
} | {
"line": 85,
"column": 19
} | {
"line": 86,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝² : LinearOrder ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nτ σ : ℕ → Ω → WithTop ι\nhτ : IsLocalizingSequence 𝓕 τ P\nhσ : IsLocalizingSequence 𝓕 σ P\n⊢ ∀ᵐ (ω : Ω) ∂P, Tendsto (fun x ↦ (τ ⊓ σ) x... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 84,
"column": 10
} | {
"line": 84,
"column": 12
} | {
"line": 84,
"column": 13
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝² : LinearOrder ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nτ σ : ℕ → Ω → WithTop ι\nhτ : IsLocalizingSequence 𝓕 τ P\nhσ : IsLocalizingSequence 𝓕 σ P\n⊢ ∀ᵐ (ω : Ω) ∂P, Monotone fun x ↦ (τ ⊓ σ) x... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 118,
"column": 52
} | {
"line": 118,
"column": 54
} | {
"line": 118,
"column": 55
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : Zero E\nhp : p X\n⊢ ∀ (n : ℕ), p (stoppedProcess (fun... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 328,
"column": 40
} | {
"line": 329,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nε : ℝ\nω' : Ω'\nh1 : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nh2 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\na✝ : IsF... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 324,
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} | {
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"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nε : ℝ\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑c * t ^ 2 / 2)",
"ppTerm": "?m.88",
"assign... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
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} | {
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} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nε : ℝ\nhε : 0 ≤ ε\nhc0 : ¬c = 0\nω' : Ω'\nh : ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑c * t ^ 2 / 2)\n⊢ 0 ≤ ε / ↑c",... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 346,
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} | {
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} | {
"line": 346,
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} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nε : ℝ\nhε : 0 ≤ ε\nhc0 : ¬c = 0\nω' : Ω'\nh : ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-t * ε + ↑c * t ^ 2 / 2)\n⊢ rexp (-(ε / ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
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} | {
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} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nε : ℝ\nhε : 0 ≤ ε\n⊢ ∀ᵐ (ω' : Ω') ∂ν, (κ ω').real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))",
"ppTerm": "?m.61",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 354,
"column": 69
} | {
"line": 354,
"column": 71
} | {
"line": 355,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\n⊢ {ω | 0 < X ω} = ⋃ ε, {ω | ↑↑ε ≤ X ω}",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Rat.instOfNat",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 154,
"column": 44
} | {
"line": 154,
"column": 46
} | {
"line": 155,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\np : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\n⊢ IsStable 𝓕 fun Y ↦ Locally p 𝓕 Y P",... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 366,
"column": 43
} | {
"line": 366,
"column": 45
} | {
"line": 366,
"column": 46
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\nhs : {ω | 0 < X ω} = ⋃ ε, {ω | ↑↑ε ≤ X ω}\nε : ℚ\nω' : Ω'\na✝ : IsFiniteMeasure (κ ω')\nhε : 0 < ε\nhm : ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ↑ε ≤ X ω... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 367,
"column": 77
} | {
"line": 367,
"column": 79
} | {
"line": 368,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\nhs : {ω | 0 < X ω} = ⋃ ε, {ω | ↑↑ε ≤ X ω}\nε : ℚ\nω' : Ω'\na✝ : IsFiniteMeasure (κ ω')\nhε : 0 < ε\nhm : ∀ (t : ℝ), 0 ≤ t → (κ ω').real {ω | ↑ε ≤ X ω... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 168,
"column": 41
} | {
"line": 168,
"column": 43
} | {
"line": 169,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np q : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\nhq : IsStable 𝓕 q\nx✝ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 363,
"column": 66
} | {
"line": 363,
"column": 68
} | {
"line": 364,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\nhs : {ω | 0 < X ω} = ⋃ ε, {ω | ↑↑ε ≤ X ω}\nε : ℚ\n⊢ ∀ᵐ (ω' : Ω') ∂ν, 0 < ε → (κ ω') {ω | ↑ε ≤ X ω} = 0",
"ppTerm": "?m.159",
"assigned": true... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 44,
"column": 78
} | {
"line": 44,
"column": 80
} | {
"line": 45,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\ninst✝³ : Semiring A\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid W\ninst✝ : Module A W\nρ : Representation A G W\n⊢ Function.Injective toSubmodule",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Monoid",
"Submodule",
"Repr... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 353,
"column": 42
} | {
"line": 353,
"column": 44
} | {
"line": 354,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, (κ ω') {ω | 0 < X ω} = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 57,
"column": 14
} | {
"line": 57,
"column": 16
} | {
"line": 57,
"column": 17
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ' : Subrepresentation ρ\n⊢ (ρ 1).restrict ⋯ = 1",
"ppTerm": "?m.61",
"assign... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 58,
"column": 18
} | {
"line": 58,
"column": 20
} | {
"line": 58,
"column": 21
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ' : Subrepresentation ρ\nx y : G\n⊢ (ρ (x * y)).restrict ⋯ = (ρ x).restrict ⋯ * (ρ y... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 386,
"column": 40
} | {
"line": 386,
"column": 42
} | {
"line": 386,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\nω' : Ω'\nh2 : (κ ω') {ω | 0 < X ω} = 0\nh1 : (κ ω') {ω | X ω < 0} = 0\n⊢ (κ ω') {ω | X ω ≠ 0} = (κ ω') {ω | X ω < 0 ∨ 0 < X ω}",
"ppTerm": "?m.11... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 388,
"column": 11
} | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\nω' : Ω'\nh2 : (κ ω') {ω | 0 < X ω} = 0\nh1 : (κ ω') {ω | X ω < 0} = 0\n⊢ (κ ω') {ω | X ω < 0} + (κ ω') {ω | 0 < X ω} = 0",
"ppTerm": "?m.152",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 379,
"column": 30
} | {
"line": 379,
"column": 32
} | {
"line": 380,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, X =ᵐ[κ ω'] 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 394,
"column": 73
} | {
"line": 394,
"column": 75
} | {
"line": 394,
"column": 76
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nhX : Measurable X\nh : HasSubgaussianMGF X 0 κ ν\n⊢ Measurable 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"R... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 393,
"column": 22
} | {
"line": 393,
"column": 24
} | {
"line": 394,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nhX : Measurable X\nh : HasSubgaussianMGF X 0 κ ν\n⊢ X =ᵐ[⇑κ ∘ₘ ν] 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 398,
"column": 22
} | {
"line": 398,
"column": 24
} | {
"line": 399,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 κ ν\n⊢ X =ᵐ[⇑κ ∘ₘ ν] 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real",
"MeasureTheory.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 61,
"column": 56
} | {
"line": 61,
"column": 58
} | {
"line": 62,
"column": 6
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ρ₁.toSubmodule ⊔ ρ₂.toSubmodule... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 413,
"column": 79
} | {
"line": 413,
"column": 81
} | {
"line": 413,
"column": 82
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhY : HasSubgaussianMGF Y cY κ ν\nhX0✝ : cX = 0\nhX : HasSubgaussianMGF X 0 κ ν\nω : Ω\nhX0 : X ω = 0 ω\n⊢ Y ω = X ω + Y ω",
"ppTerm": "?m.89",
"assigned": tr... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 68,
"column": 56
} | {
"line": 68,
"column": 58
} | {
"line": 69,
"column": 6
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nρ₁ ρ₂ : Subrepresentation ρ\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ρ₁.toSubmodule ⊓ ρ₂.toSubmodule... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 15
} | {
"line": 93,
"column": 16
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ⊤ → (ρ g) v ∈ ⊤",
"ppTerm": "?m.56",
"assigned": tru... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 95,
"column": 13
} | {
"line": 95,
"column": 15
} | {
"line": 95,
"column": 16
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\n⊢ ∀ (g : G) ⦃v : W⦄, v ∈ ⊥ → (ρ g) v ∈ ⊥",
"ppTerm": "?m.57",
"assigned": tru... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 417,
"column": 79
} | {
"line": 417,
"column": 81
} | {
"line": 417,
"column": 82
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhX0 : ¬cX = 0\nhY0✝ : cY = 0\nhY : HasSubgaussianMGF Y 0 κ ν\nω : Ω\nhY0 : Y ω = 0 ω\n⊢ X ω = X ω + Y ω",
"ppTerm": "?m.140",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 178,
"column": 39
} | {
"line": 178,
"column": 71
} | {
"line": 178,
"column": 72
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np✝ q✝ : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp✝ : IsStable 𝓕 p✝\nhq✝ : IsStable 𝓕 q... | [
"ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np✝ q✝ : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp✝ : IsStable 𝓕 p✝\nhq✝ : IsStable 𝓕 q✝\nx✝ : Loca... | ← stoppedProcess_indicator_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 419,
"column": 28
} | {
"line": 419,
"column": 30
} | {
"line": 420,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (X ω + Y ω))) (⇑κ ∘ₘ ν)",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 108,
"column": 22
} | {
"line": 108,
"column": 24
} | {
"line": 109,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation ρ\nc : A[G]\nv : ρ.asModule\nhv : v ∈ σ.toSubmodule.carrier... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 117,
"column": 91
} | {
"line": 117,
"column": 93
} | {
"line": 117,
"column": 94
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid W\ninst✝ : Module A W\nρ : Representation A G W\nσ : Subrepresentation ρ\nv : W\n⊢ v ∈ σ.asSubmodule ↔ v ∈ σ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MonoidAl... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 130,
"column": 60
} | {
"line": 130,
"column": 62
} | {
"line": 131,
"column": 8
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ.toSubmodule.... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 162,
"column": 77
} | {
"line": 162,
"column": 79
} | {
"line": 163,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np q : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\nhq : IsStable 𝓕 q\n⊢ L... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 130,
"column": 6
} | {
"line": 131,
"column": 98
} | {
"line": 133,
"column": 0
} | [
{
"pp": "case single\nA : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ... | [] | exact σ.toSubmodule.smul_mem' ((algebraMap A A) a) <| by
simpa [Representation.ofModule, RestrictScalars.lsmul] using! σ.apply_mem_toSubmodule g hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 124,
"column": 22
} | {
"line": 124,
"column": 24
} | {
"line": 125,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nc : A[G]\nm : M\nhm : m ∈ σ.to... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 135,
"column": 44
} | {
"line": 135,
"column": 46
} | {
"line": 135,
"column": 47
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nM : Type u_4\ninst✝³ : CommSemiring A\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\n⊢ m ∈ σ.asSubmodule' ↔ m ∈ σ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Mono... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 143,
"column": 34
} | {
"line": 143,
"column": 36
} | {
"line": 144,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] M\ng : G\nv : RestrictScalars A A[G] M\nhv : v ∈ { toAddSubmon... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 148,
"column": 88
} | {
"line": 148,
"column": 90
} | {
"line": 148,
"column": 91
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nM : Type u_4\ninst✝³ : CommSemiring A\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] M\nm : M\n⊢ m ∈ ofSubmodule N ↔ m ∈ N",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Re... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 155,
"column": 24
} | {
"line": 155,
"column": 26
} | {
"line": 155,
"column": 27
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] ρ.asModule\na : A\nw : W\nhw : w ∈ N.carrier\n⊢ a • w ∈ N.carr... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 156,
"column": 34
} | {
"line": 156,
"column": 36
} | {
"line": 157,
"column": 4
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] ρ.asModule\ng : G\nw : W\nhw : w ∈ { toAddSubmonoid := N.toAdd... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 175,
"column": 18
} | {
"line": 175,
"column": 20
} | {
"line": 175,
"column": 21
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\n⊢ ∀ {a b : Subrepresentation ρ},\n { toFun := asSubmodule, invFun := ofSubmodu... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 186,
"column": 18
} | {
"line": 186,
"column": 20
} | {
"line": 186,
"column": 21
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\n⊢ ∀ {a b : Submodule A[G] M},\n { toFun := ofSubmodule, invFun := asSubmodule'... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 197,
"column": 17
} | {
"line": 197,
"column": 19
} | {
"line": 198,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁶ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝³ : DenselyOrdered ι\ninst✝² : FirstCountableTopology ι\ninst✝¹ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nins... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 434,
"column": 17
} | {
"line": 434,
"column": 19
} | {
"line": 434,
"column": 20
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 434,
"column": 40
} | {
"line": 434,
"column": 42
} | {
"line": 434,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 434,
"column": 55
} | {
"line": 434,
"column": 57
} | {
"line": 434,
"column": 58
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 213,
"column": 23
} | {
"line": 213,
"column": 25
} | {
"line": 214,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁷ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁶ : TopologicalSpace ι\ninst✝⁵ : OrderTopology ι\n𝓕 : Filtration ι mΩ\nX : ι → Ω → E\np : (ι → Ω → E) → Prop\ninst✝⁴ : Zero E\ninst✝³ : DenselyOrdered ι\ninst✝² : First... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 238,
"column": 12
} | {
"line": 238,
"column": 14
} | {
"line": 238,
"column": 15
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : ConditionallyCompleteLinearOrderBot ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : SecondCountableTopology ι\ninst✝ : IsFiniteMeasure P\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω → WithTop ι... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 87,
"column": 57
} | {
"line": 87,
"column": 59
} | {
"line": 88,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\n⊢ AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"instClosedIicTopology",
"MeasureTheory.AES... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 91,
"column": 90
} | {
"line": 91,
"column": 92
} | {
"line": 92,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nA : ℝ\nx : α\n⊢ |truncation f A x| ≤ |A|",
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"usedConstants": [
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"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Set.Ioc",
"NegZeroClass.toNeg",
"Real.instLE",
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 98,
"column": 60
} | {
"line": 98,
"column": 62
} | {
"line": 98,
"column": 63
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\n⊢ truncation f 0 = 0",
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"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"False",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Set.indicator",
"lt_self_iff_false._simp_1",
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 100,
"column": 95
} | {
"line": 100,
"column": 97
} | {
"line": 101,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nA : ℝ\nx : α\n⊢ |truncation f A x| ≤ |f x|",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Set.Ioc",
"Real.instLE",
"Real",
"le_rfl",
"Real.lat... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 32
} | {
"line": 108,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nA : ℝ\nx : α\nh : |f x| < A\n⊢ truncation f A x = f x",
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"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Set.Ioc",
"NegZeroClass.toNeg",
"Real",
"Preorder.toLT",
"Real.instZer... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 119,
"column": 25
} | {
"line": 119,
"column": 27
} | {
"line": 119,
"column": 28
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nA : ℝ\nh : ∀ (x : α), 0 ≤ f x\nx : α\nhx : 0 < f x\nh'x : f x ≤ A\n⊢ -A < f x",
"ppTerm": "?m.62",
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"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"No... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 76
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nf : α → ℝ\nA : ℝ\nh : ∀ (x : α), 0 ≤ f x\nx : α\nhx : 0 = f x\n⊢ truncation f A x = ((Set.Ioc 0 A).indicator id ∘ f) x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Real",
"Real.instZero",
"congrArg",
"Set.indicato... | [] | simp only [truncation, indicator, hx, id, Function.comp_apply, ite_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.StrongLaw | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 76
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nf : α → ℝ\nA : ℝ\nh : ∀ (x : α), 0 ≤ f x\nx : α\nhx : 0 = f x\n⊢ truncation f A x = ((Set.Ioc 0 A).indicator id ∘ f) x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Real",
"Real.instZero",
"congrArg",
"Set.indicato... | [] | simp only [truncation, indicator, hx, id, Function.comp_apply, ite_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.StrongLaw | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 76
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nf : α → ℝ\nA : ℝ\nh : ∀ (x : α), 0 ≤ f x\nx : α\nhx : 0 = f x\n⊢ truncation f A x = ((Set.Ioc 0 A).indicator id ∘ f) x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Real",
"Real.instZero",
"congrArg",
"Set.indicato... | [] | simp only [truncation, indicator, hx, id, Function.comp_apply, ite_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.StrongLaw | {
"line": 114,
"column": 55
} | {
"line": 114,
"column": 57
} | {
"line": 115,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nA : ℝ\nh : ∀ (x : α), 0 ≤ f x\n⊢ truncation f A = (Set.Ioc 0 A).indicator id ∘ f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Set.Ioc",
"NegZeroCla... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 132,
"column": 79
} | {
"line": 132,
"column": 81
} | {
"line": 133,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : IsFiniteMeasure μ\nhf : AEStronglyMeasurable f μ\nA : ℝ\n⊢ Integrable (truncation f A) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Measu... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 431,
"column": 69
} | {
"line": 431,
"column": 71
} | {
"line": 432,
"column": 8
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 237,
"column": 72
} | {
"line": 237,
"column": 74
} | {
"line": 238,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : ConditionallyCompleteLinearOrderBot ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : SecondCountableTopology ι\ninst✝ : IsFiniteMeasure P\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω → WithTop ι... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 136,
"column": 98
} | {
"line": 136,
"column": 100
} | {
"line": 137,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nhA : 0 ≤ A\nn : ℕ\nhn : n ≠ 0\n⊢ ∫ (x : α), truncation f A x ^ n ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂Measure.map f μ",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real.instIsO... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 440,
"column": 56
} | {
"line": 440,
"column": 58
} | {
"line": 441,
"column": 8
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 167,
"column": 22
} | {
"line": 167,
"column": 24
} | {
"line": 167,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nn : ℕ\nhn : n ≠ 0\nh'f : 0 ≤ f\nM : MeasurableSet (Set.Ioc 0 A)\nM' : MeasurableSet (Set.Ioc A 0)\nhA : A < 0\nthis : ∀ᵐ (x : ℝ) ∂Measure.map f μ, 0 ≤ x\nx : ℝ\nhx : 0 ≤ x\na✝ : A < x\nh''x : x ≤ 0\n⊢ x... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 147,
"column": 74
} | {
"line": 147,
"column": 76
} | {
"line": 148,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nn : ℕ\nhn : n ≠ 0\nh'f : 0 ≤ f\n⊢ ∫ (x : α), truncation f A x ^ n ∂μ = ∫ (y : ℝ) in 0..A, y ^ n ∂Measure.map f μ",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MeasureTheor... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 172,
"column": 82
} | {
"line": 172,
"column": 84
} | {
"line": 173,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nhA : 0 ≤ A\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂Measure.map f μ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 176,
"column": 82
} | {
"line": 176,
"column": 84
} | {
"line": 177,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nh'f : 0 ≤ f\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in 0..A, y ∂Measure.map f μ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 180,
"column": 46
} | {
"line": 180,
"column": 48
} | {
"line": 181,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nh'f : 0 ≤ f\nA : ℝ\n⊢ ∫ (x : α), truncation f A x ∂μ ≤ ∫ (x : α), f x ∂μ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"InnerProductSpace.toNormedSpace",
"Re... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 445,
"column": 55
} | {
"line": 445,
"column": 57
} | {
"line": 446,
"column": 8
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 424,
"column": 14
} | {
"line": 424,
"column": 16
} | {
"line": 425,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\n⊢ ∀ᵐ (ω' : Ω') ∂ν,\n ∀ (t : ℝ), mgf (fun ω ↦ X ω + Y ω) (κ ω') t ≤... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 192,
"column": 75
} | {
"line": 192,
"column": 77
} | {
"line": 193,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\n⊢ Tendsto (fun A ↦ ∫ (x : α), truncation f A x ∂μ) atTop (𝓝 (∫ (x : α), f x ∂μ))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Integrable.aestrongl... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 409,
"column": 75
} | {
"line": 409,
"column": 77
} | {
"line": 410,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\n⊢ HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) κ ν",
"ppTerm":... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 233,
"column": 69
} | {
"line": 233,
"column": 71
} | {
"line": 234,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : ConditionallyCompleteLinearOrderBot ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝¹ : SecondCountableTopology ι\ninst✝ : IsFiniteMeasure P\nτ : ℕ → Ω → WithTop ι\nσ : ℕ → ℕ → Ω → WithTop ι... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 268,
"column": 39
} | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 42
} | [
{
"pp": "x : ℕ → ℕ\nn : ℕ\n⊢ mkStrictMonoAux x n < mkStrictMonoAux x (n + 1)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Process.LocalProperty.0.ProbabilityTheory.mkStrictMonoAux_strictMono._proof_1_4"
],
"usedFVars": [
"x",
"n"
... | [] | by | [anonymous] | by |
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