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.Independence.Conditional | {
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} | {
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} | {
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} | [
{
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Mathlib.Probability.Moments.Tilted | {
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} | {
"line": 75,
"column": 88
} | {
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"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ns : Set Ω\nhs : MeasurableSet s\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (μ.tilted fun x ↦ t * X x) s = ENNReal.ofReal (∫ (a : Ω) in s, rexp (t * X a - cgf X μ t) ∂μ)",
"ppTerm": "?m.42",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 885,
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} | {
"line": 885,
"column": 92
} | {
"line": 886,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 876,
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} | {
"line": 876,
"column": 90
} | {
"line": 877,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.Tilted | {
"line": 83,
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} | {
"line": 83,
"column": 88
} | {
"line": 84,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\ninst✝ : SFinite μ\ns : Set Ω\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (μ.tilted fun x ↦ t * X x) s = ENNReal.ofReal (∫ (a : Ω) in s, rexp (t * X a - cgf X μ t) ∂μ)",
"ppTerm": "?m.42",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 889,
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} | {
"line": 889,
"column": 27
} | {
"line": 889,
"column": 28
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 889,
"column": 39
} | {
"line": 889,
"column": 41
} | {
"line": 889,
"column": 42
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 98,
"column": 94
} | {
"line": 98,
"column": 96
} | {
"line": 99,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : Ω → E\ns : Set Ω\nhs : MeasurableSet s\n⊢ (∫ (x : Ω) in s, g x ∂μ.tilted fun x ↦ t * X x) = ∫ (x : Ω) in s, (rexp (t * X x) / mgf X μ t) • g x ∂μ",
"ppTer... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 888,
"column": 91
} | {
"line": 888,
"column": 93
} | {
"line": 889,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.Tilted | {
"line": 102,
"column": 94
} | {
"line": 102,
"column": 96
} | {
"line": 103,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\ng : Ω → E\ns : Set Ω\n⊢ (∫ (x : Ω) in s, g x ∂μ.tilted fun x ↦ t * X x) = ∫ (x : Ω) in s, (rexp (t * X x) / mgf X μ t) • g x ∂μ",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 893,
"column": 29
} | {
"line": 893,
"column": 31
} | {
"line": 893,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 215,
"column": 79
} | {
"line": 215,
"column": 81
} | {
"line": 216,
"column": 8
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : DiscreteTopology ι\nin... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.Conditional | {
"line": 893,
"column": 43
} | {
"line": 893,
"column": 45
} | {
"line": 893,
"column": 46
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 107,
"column": 92
} | {
"line": 107,
"column": 94
} | {
"line": 108,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : Ω → E\ns : Set Ω\nhs : MeasurableSet s\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (∫ (x : Ω) in s, g x ∂μ.tilted fun x ↦ t * X x) = ∫ (x : Ω) in s, rexp (... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 892,
"column": 56
} | {
"line": 892,
"column": 58
} | {
"line": 893,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 198,
"column": 96
} | {
"line": 198,
"column": 98
} | {
"line": 199,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : CompleteSpace E\nι : Type u_3\ninst✝¹⁰ : LinearOrder ι\ninst✝⁹ : LocallyFiniteOrder ι\ninst✝⁸ : OrderBot ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : DiscreteTopology ι\nin... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 114,
"column": 92
} | {
"line": 114,
"column": 94
} | {
"line": 115,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\ng : Ω → E\ns : Set Ω\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (∫ (x : Ω) in s, g x ∂μ.tilted fun x ↦ t * X x) = ∫ (x : Ω) in s, rexp (t ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 152,
"column": 39
} | {
"line": 152,
"column": 41
} | {
"line": 153,
"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\n⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestronglyMeasu... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 161,
"column": 47
} | {
"line": 161,
"column": 49
} | {
"line": 162,
"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\n⊢ ∀ᵐ (ω' : Ω') ∂ν, AEStronglyMeasurable X (κ ω')",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MeasureTheory.... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 120,
"column": 84
} | {
"line": 120,
"column": 86
} | {
"line": 121,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : Ω → E\n⊢ (∫ (ω : Ω), g ω ∂μ.tilted fun x ↦ t * X x) = ∫ (ω : Ω), (rexp (t * X ω) / mgf X μ t) • g ω ∂μ",
"ppTerm": "?m.39",
"assigned": true,
"use... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 167,
"column": 64
} | {
"line": 167,
"column": 66
} | {
"line": 168,
"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\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants":... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 181,
"column": 70
} | {
"line": 181,
"column": 72
} | {
"line": 181,
"column": 73
} | [
{
"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\np : ℝ≥0\nω' : Ω'\nhi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\nhp : ¬p = 0\n⊢ ↑p ≠ ∞",
"ppTerm": "?m.79",
"ass... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 896,
"column": 84
} | {
"line": 896,
"column": 86
} | {
"line": 897,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.Tilted | {
"line": 124,
"column": 82
} | {
"line": 124,
"column": 84
} | {
"line": 125,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : Ω → E\nht : Integrable (fun ω ↦ rexp (t * X ω)) μ\n⊢ (∫ (ω : Ω), g ω ∂μ.tilted fun x ↦ t * X x) = ∫ (ω : Ω), rexp (t * X ω - cgf X μ t) • g ω ∂μ",
"ppTerm... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 133,
"column": 51
} | {
"line": 133,
"column": 53
} | {
"line": 134,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\n⊢ (∫ (x : Ω), X x ∂μ.tilted fun x ↦ t * X x) = deriv (cgf X μ) t",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 27
} | {
"line": 185,
"column": 28
} | [
{
"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\np : ℝ≥0\nω' : Ω'\nhi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\nhp : ¬p = 0\nhf : ∫⁻ (x : Ω), ENNReal.ofReal (rexp (↑p ... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 903,
"column": 29
} | {
"line": 903,
"column": 31
} | {
"line": 903,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 185,
"column": 72
} | {
"line": 185,
"column": 74
} | {
"line": 185,
"column": 75
} | [
{
"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\np : ℝ≥0\nω' : Ω'\nhi : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\nhp : ¬p = 0\nhf : ∫⁻ (x : Ω), ENNReal.ofReal (rexp (↑p ... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 903,
"column": 43
} | {
"line": 903,
"column": 45
} | {
"line": 903,
"column": 46
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 196,
"column": 57
} | {
"line": 196,
"column": 59
} | {
"line": 197,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : Fintype S\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 141,
"column": 38
} | {
"line": 141,
"column": 40
} | {
"line": 142,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\np : ℝ≥0\n⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"AEMeasurable.aestronglyMea... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 240,
"column": 58
} | {
"line": 240,
"column": 60
} | {
"line": 240,
"column": 61
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : IsProbabilityMe... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 175,
"column": 61
} | {
"line": 175,
"column": 63
} | {
"line": 176,
"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\np : ℝ≥0\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')",
"ppTerm": "?m.26",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 902,
"column": 89
} | {
"line": 902,
"column": 91
} | {
"line": 903,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 195,
"column": 71
} | {
"line": 195,
"column": 73
} | {
"line": 195,
"column": 74
} | [
{
"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\np : ℝ≥0\nhp0 : ¬p = 0\n⊢ ↑p ≠ ∞",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.Tilted | {
"line": 160,
"column": 64
} | {
"line": 160,
"column": 66
} | {
"line": 161,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\n⊢ Var[X; μ.tilted fun x ↦ t * X x] = iteratedDeriv 2 (cgf X μ) t",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.variance_eq_integral",
"... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 237,
"column": 57
} | {
"line": 237,
"column": 59
} | {
"line": 238,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁵ : (s : S) → TopologicalSpace (E s)\ninst✝⁴ : (s : S) → MeasurableSpace (E s)\ninst✝³ : ∀ (s : S), BorelSpace (E s)\ninst✝² : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝¹ : IsProbabilityMe... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 33,
"column": 77
} | {
"line": 33,
"column": 79
} | {
"line": 34,
"column": 4
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\nn : ℕ\n⊢ ∑ a, ↑(p ^ ↑a * (1 - p) ^ (↑(Fin.last n) - ↑a) * ↑(n.choose ↑a)) = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NNReal.instCommSemiring",
"MulOne.toOne"... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 45,
"column": 92
} | {
"line": 45,
"column": 94
} | {
"line": 46,
"column": 2
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\nn : ℕ\ni : Fin (n + 1)\n⊢ (binomial p h n) i = ↑p ^ ↑i * (1 - ↑p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"PMF.binomial",
"ENNReal.ofNNReal",
"Nat... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 50,
"column": 38
} | {
"line": 50,
"column": 40
} | {
"line": 51,
"column": 2
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\nn : ℕ\n⊢ (binomial p h n) 0 = (1 - ↑p) ^ n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PMF.binomial",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"Nat.instOrderedSub",
"ENNReal.ofNNReal",
"Nat.choose",
"HMul.hMul",
... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 55,
"column": 40
} | {
"line": 55,
"column": 42
} | {
"line": 56,
"column": 2
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\nn : ℕ\n⊢ (binomial p h n) (Fin.last n) = ↑p ^ n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"PMF.binomial",
"MulOne.toOne",
"Nat.instOrderedSub",
"ENNReal.ofNNReal",
"Nat.choose",
"H... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 200,
"column": 25
} | {
"line": 200,
"column": 27
} | {
"line": 200,
"column": 28
} | [
{
"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\np : ℝ≥0\nhp0 : ¬p = 0\nh' : ∫⁻ (a : Ω), ‖rexp (↑p * t * X a)‖ₑ ∂⇑κ ∘ₘ ν < ∞\nω : Ω\n⊢ 0 ≤ rexp (t * X ω)",
"ppTerm": "?m.222",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 248,
"column": 41
} | {
"line": 248,
"column": 43
} | {
"line": 249,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nG : Type u_6\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : HasOuterApproxClosed G\nZ : Ω → G\ninst✝ : IsProbabilityMeasure P\nA : Set Ω\nmA : NullMeasurableSet A P\nmZ : AEMeasurable Z P\nh : ∀ (f : G →ᵇ ℝ)... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 60,
"column": 40
} | {
"line": 60,
"column": 42
} | {
"line": 60,
"column": 43
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\nn : ℕ\n⊢ (binomial p h n) (Fin.last n) = ↑p ^ n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"PMF.binomial",
"ENNReal.ofNNReal",
"congrArg",
"PMF",
"CommSemiring.toSemiring",
"PMF.instFunLike",
"instOfNatNat",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 200,
"column": 58
} | {
"line": 200,
"column": 60
} | {
"line": 200,
"column": 61
} | [
{
"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\np : ℝ≥0\nhp0 : ¬p = 0\nh' : ∫⁻ (a : Ω), ‖rexp (↑p * t * X a)‖ₑ ∂⇑κ ∘ₘ ν < ∞\nω : Ω\n⊢ 0 ≤ rexp (↑p * t * X ω)",
"ppTerm": "?m.225... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 39
} | {
"line": 201,
"column": 40
} | [
{
"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\np : ℝ≥0\nhp0 : ¬p = 0\nh' : ∫⁻ (a : Ω), ‖rexp (↑p * t * X a)‖ₑ ∂⇑κ ∘ₘ ν < ∞\nω : Ω\n⊢ 0 ≤ rexp (t * X ω)",
"ppTerm": "?m.229",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 329,
"column": 42
} | {
"line": 329,
"column": 44
} | {
"line": 329,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nF : T → Type u_5\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : (t : T... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 65,
"column": 57
} | {
"line": 65,
"column": 59
} | {
"line": 66,
"column": 2
} | [
{
"pp": "p : ℝ≥0\nh : p ≤ 1\n⊢ binomial p h 1 = map (fun x ↦ bif x then 1 else 0) (bernoulli p h)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"cond",
"Nat.instCanonicallyOrderedAdd",
"PMF.binomial",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"False",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 329,
"column": 56
} | {
"line": 329,
"column": 58
} | {
"line": 329,
"column": 59
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nF : T → Type u_5\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : (t : T... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 34
} | {
"line": 72,
"column": 35
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k % (b + 1) = k",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Nat.instOne",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 73,
"column": 60
} | {
"line": 73,
"column": 62
} | {
"line": 73,
"column": 63
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\n⊢ 1 - ↑x = ENNReal.ofReal (1 - ↑x)",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 342,
"column": 37
} | {
"line": 342,
"column": 39
} | {
"line": 342,
"column": 40
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nF : T → Type u_5\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : (t : T... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 74,
"column": 30
} | {
"line": 74,
"column": 32
} | {
"line": 74,
"column": 33
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\neq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)\n⊢ 1 - ↑x ≥ 0",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 342,
"column": 51
} | {
"line": 342,
"column": 53
} | {
"line": 342,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nF : T → Type u_5\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : (t : T... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 350,
"column": 29
} | {
"line": 350,
"column": 31
} | {
"line": 350,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁸ : (t : T) → TopologicalSpace (F t)\ninst✝⁷ : (t : T) → MeasurableSpace (F t)\ninst✝⁶ : ∀ (t : T), BorelSpace (F t)\ninst✝⁵ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁴ : TopologicalSpace G\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 350,
"column": 43
} | {
"line": 350,
"column": 45
} | {
"line": 350,
"column": 46
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁸ : (t : T) → TopologicalSpace (F t)\ninst✝⁷ : (t : T) → MeasurableSpace (F t)\ninst✝⁶ : ∀ (t : T), BorelSpace (F t)\ninst✝⁵ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁴ : TopologicalSpace G\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 190,
"column": 48
} | {
"line": 190,
"column": 50
} | {
"line": 191,
"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\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 357,
"column": 63
} | {
"line": 357,
"column": 65
} | {
"line": 357,
"column": 66
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁸ : (t : T) → TopologicalSpace (F t)\ninst✝⁷ : (t : T) → MeasurableSpace (F t)\ninst✝⁶ : ∀ (t : T), BorelSpace (F t)\ninst✝⁵ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁴ : TopologicalSpace G\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 357,
"column": 77
} | {
"line": 357,
"column": 79
} | {
"line": 357,
"column": 80
} | [
{
"pp": "Ω : Type u_1\nT : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nF : T → Type u_5\nG : Type u_6\ninst✝⁸ : (t : T) → TopologicalSpace (F t)\ninst✝⁷ : (t : T) → MeasurableSpace (F t)\ninst✝⁶ : ∀ (t : T), BorelSpace (F t)\ninst✝⁵ : ∀ (t : T), HasOuterApproxClosed (F t)\ninst✝⁴ : TopologicalSpace G\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 364,
"column": 68
} | {
"line": 364,
"column": 70
} | {
"line": 364,
"column": 71
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nH : Type u_7\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : TopologicalSpace H\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
"line": 906,
"column": 17
} | {
"line": 906,
"column": 19
} | {
"line": 906,
"column": 20
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 364,
"column": 82
} | {
"line": 364,
"column": 84
} | {
"line": 364,
"column": 85
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nH : Type u_7\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : TopologicalSpace H\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 211,
"column": 23
} | {
"line": 211,
"column": 25
} | {
"line": 211,
"column": 26
} | [
{
"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'\nh : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\nh0 : ... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 371,
"column": 63
} | {
"line": 371,
"column": 65
} | {
"line": 371,
"column": 66
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nH : Type u_7\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : TopologicalSpace H\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 371,
"column": 77
} | {
"line": 371,
"column": 79
} | {
"line": 371,
"column": 80
} | [
{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\nH : Type u_7\ninst✝⁸ : (s : S) → TopologicalSpace (E s)\ninst✝⁷ : (s : S) → MeasurableSpace (E s)\ninst✝⁶ : ∀ (s : S), BorelSpace (E s)\ninst✝⁵ : ∀ (s : S), HasOuterApproxClosed (E s)\ninst✝⁴ : TopologicalSpace H\ninst... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
"line": 385,
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} | {
"line": 385,
"column": 52
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{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁴ : (s : S) → TopologicalSpace (E s)\ninst✝³ : (s : S) → MeasurableSpace (E s)\ninst✝² : ∀ (s : S), BorelSpace (E s)\ninst✝¹ : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝ : IsProbabilityMea... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.BoundedContinuousFunction | {
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} | {
"line": 397,
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} | {
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{
"pp": "Ω : Type u_1\nS : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : S → Type u_4\ninst✝⁴ : (s : S) → TopologicalSpace (E s)\ninst✝³ : (s : S) → MeasurableSpace (E s)\ninst✝² : ∀ (s : S), BorelSpace (E s)\ninst✝¹ : ∀ (s : S), HasOuterApproxClosed (E s)\nX : (s : S) → Ω → E s\ninst✝ : IsProbabilityMea... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
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{
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Mathlib.Probability.ProbabilityMassFunction.Binomial | {
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"line": 71,
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} | {
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{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = (binomial x h b) (Fin.ofNat (b + 1) k)",
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... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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"line": 215,
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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\nω' : Ω'\nh : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\n⊢ log... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
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"line": 906,
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} | {
"line": 906,
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{
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Mathlib.Probability.Moments.SubGaussian | {
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"line": 205,
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} | {
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{
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"M... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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{
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Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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{
"pp": "α : Type u_1\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSingletonClass α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : PMF α\nf : α → E\nhf : Integrable f p.toMeasure\n⊢ ∑' (a : ↑p.support), (p.toMeasure {↑a}).toReal • f ↑a = ∑' (a : ↑p.suppor... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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{
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Mathlib.Probability.Process.Kolmogorov | {
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{
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"usedConstants... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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{
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"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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{
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"ppTerm": "?m.21",... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
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} | {
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{
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"as... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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} | {
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{
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"... | [] | by | [anonymous] | by |
Mathlib.Probability.ProbabilityMassFunction.Integrals | {
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} | {
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{
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Mathlib.Probability.Independence.Conditional | {
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} | {
"line": 870,
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} | {
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{
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Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 115,
"column": 71
} | {
"line": 115,
"column": 72
} | [
{
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Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 218,
"column": 42
} | {
"line": 219,
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} | [
{
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 112,
"column": 64
} | {
"line": 113,
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} | [
{
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"ppTerm": "?m.16",... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 130,
"column": 30
} | {
"line": 131,
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} | [
{
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"pp... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
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} | {
"line": 938,
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} | {
"line": 938,
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{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh_indep : iCondIndepFun m' hm' f μ\nhf : ∀ (i : ι), Measurable (f i)\ni✝ j✝ k l : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 225,
"column": 38
} | {
"line": 226,
"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\nω' : Ω'\nh : IsFiniteMeasure (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nthis : (κ ω').real Set.univ ≤ 1\n⊢ (κ ω') S... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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"column": 49
} | {
"line": 140,
"column": 51
} | {
"line": 141,
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} | [
{
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Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 223,
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.Probability.Independence.Conditional | {
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} | {
"line": 939,
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} | {
"line": 939,
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{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh_indep : iCondIndepFun m' hm' f μ\nhf : ∀ (i : ι), Measurable (f i)\ni j k l : ι\... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 139,
"column": 26
} | {
"line": 140,
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} | [
{
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Mathlib.Probability.Independence.Conditional | {
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} | {
"line": 934,
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} | {
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{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh_indep : iCondIndepFun m' hm' f μ\nhf : ∀ (i : ι), Measurable (f i)\ni j k l : ι\... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 154,
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} | {
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{
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Mathlib.Probability.Process.Kolmogorov | {
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} | {
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} | {
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{
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"assign... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
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} | {
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{
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"ppTerm": "?m... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
"line": 170,
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} | {
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} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q 0\ns t : T\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\n⊢ ∀ᵐ (ω : Ω) ∂P, edist (X s ω) (X t ω)... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
"line": 176,
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} | {
"line": 176,
"column": 13
} | {
"line": 176,
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} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q 0\ns t : T\n⊢ 0 * edist s t ^ q = 0",
"ppTerm": "?m.146",
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"usedConstants... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.Kolmogorov | {
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} | {
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} | {
"line": 170,
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} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q 0\ns t : T\n⊢ ∀ᵐ (ω : Ω) ∂P, edist (X s ω) (X t ω) = 0",
"ppTerm": "?m.21",
"assigned": true,
... | [] | by | [anonymous] | by |
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