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.Process.LocalProperty | {
"line": 271,
"column": 9
} | {
"line": 271,
"column": 11
} | {
"line": 271,
"column": 12
} | [
{
"pp": "x : ℕ → ℕ\n⊢ x 0 ≤ mkStrictMonoAux x 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instReflLe",
"Std.le_refl._simp_1",
"instOfNatNat",
"LE.le",
"instLENat",
"Nat.instPreorder",
"Nat",
"of_eq_true",
"OfNat.ofNat"
],
... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 272,
"column": 13
} | {
"line": 272,
"column": 15
} | {
"line": 272,
"column": 16
} | [
{
"pp": "x : ℕ → ℕ\nn : ℕ\n⊢ x (n + 1) ≤ mkStrictMonoAux x (n + 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Process.LocalProperty.0.ProbabilityTheory.le_mkStrictMonoAux._proof_1_6"
],
"usedFVars": [
"x",
"n"
],
"usedGoa... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 226,
"column": 77
} | {
"line": 226,
"column": 79
} | {
"line": 227,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ = ∑ j ∈ range K, ∑ i ∈ Ico j N, ∫ (x : ℝ) in ↑i..↑(... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 278,
"column": 69
} | {
"line": 278,
"column": 71
} | {
"line": 279,
"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.Moments.SubGaussian | {
"line": 464,
"column": 24
} | {
"line": 464,
"column": 26
} | {
"line": 464,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\nη : Kernel Ω Ω''\nh : HasSubgaussianMGF Y cY η (⇑κ ∘ₘ ν)\nhν : SFinite ν\nhκ : IsSFiniteKernel κ\nh2 : ∀ᵐ (ω' : Ω) ∂Measure.... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 455,
"column": 57
} | {
"line": 455,
"column": 59
} | {
"line": 456,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\nη : Kernel Ω Ω''\nh : HasSubgaussianMGF Y cY η (⇑κ ∘ₘ ν)\n⊢ HasSubgaussianMGF Y cY (prodMkLeft Ω' η) (ν ⊗ₘ κ)",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 297,
"column": 34
} | {
"line": 297,
"column": 36
} | {
"line": 297,
"column": 37
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 231,
"column": 91
} | {
"line": 231,
"column": 93
} | {
"line": 232,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ j ∈ range K, ∑ i ∈ Ico j N, ∫ (x : ℝ) in ↑i..↑(i + 1), 1 ∂ρ =\n ∑ i ∈ range N, ∑ j ∈ ra... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 235,
"column": 72
} | {
"line": 235,
"column": 74
} | {
"line": 236,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ i ∈ range N, ∑ j ∈ range (min (i + 1) K), ∫ (x : ℝ) in ↑i..↑(i + 1), 1 ∂ρ ≤\n ∑ i ∈ ran... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 244,
"column": 42
} | {
"line": 244,
"column": 44
} | {
"line": 245,
"column": 10
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\ni : ℕ\na✝ : i ∈ range N\n⊢ ↑i ≤ ↑(i + 1)",
"ppTerm": "?m.522",
"assigned": true,
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 300,
"column": 71
} | {
"line": 300,
"column": 73
} | {
"line": 300,
"column": 74
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 300,
"column": 82
} | {
"line": 300,
"column": 84
} | {
"line": 300,
"column": 85
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 479,
"column": 52
} | {
"line": 479,
"column": 54
} | {
"line": 480,
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 292,
"column": 74
} | {
"line": 292,
"column": 76
} | {
"line": 293,
"column": 2
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 242,
"column": 60
} | {
"line": 242,
"column": 62
} | {
"line": 243,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ i ∈ range N, (↑i + 1) * ∫ (x : ℝ) in ↑i..↑(i + 1), 1 ∂ρ ≤ ∑ i ∈ range N, ∫ (x : ℝ) in ↑i..... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 254,
"column": 35
} | {
"line": 254,
"column": 37
} | {
"line": 255,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ i ∈ range N, ∫ (x : ℝ) in ↑i..↑(i + 1), x + 1 ∂ρ = ∫ (x : ℝ) in 0..↑N, x + 1 ∂ρ",
"ppT... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 470,
"column": 70
} | {
"line": 470,
"column": 72
} | {
"line": 471,
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 258,
"column": 51
} | {
"line": 258,
"column": 53
} | {
"line": 259,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∫ (x : ℝ) in 0..↑N, x + 1 ∂ρ = ∫ (x : ℝ) in 0..↑N, x ∂ρ + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ",
"pp... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 262,
"column": 51
} | {
"line": 262,
"column": 53
} | {
"line": 263,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∫ (x : ℝ) in 0..↑N, x ∂ρ + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ = (∫ (a : Ω), truncation X (↑N) a) + ∫ (... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 264,
"column": 38
} | {
"line": 264,
"column": 40
} | {
"line": 265,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ (∫ (a : Ω), truncation X (↑N) a) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + ∫ (x : ℝ) i... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 266,
"column": 22
} | {
"line": 266,
"column": 24
} | {
"line": 267,
"column": 8
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ (∫ (a : Ω), X a) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1",
"ppTerm": "?m.384",... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 273,
"column": 97
} | {
"line": 273,
"column": 99
} | {
"line": 274,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\nA : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1\n⊢ ∀ (a b : ℝ), ℙ {ω | X ω... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 280,
"column": 80
} | {
"line": 280,
"column": 82
} | {
"line": 280,
"column": 83
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\nA : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1\nB : ∀ (a b : ℝ), ℙ {ω | X... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 281,
"column": 80
} | {
"line": 281,
"column": 82
} | {
"line": 282,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\nA : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1\nB : ∀ (a b : ℝ), ℙ {ω | X... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 283,
"column": 73
} | {
"line": 283,
"column": 75
} | {
"line": 284,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\nA : ∑ j ∈ range K, ∫ (x : ℝ) in ↑j..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + 1\nB : ∀ (a b : ℝ), ℙ {ω | X... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 220,
"column": 82
} | {
"line": 220,
"column": 84
} | {
"line": 221,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\n⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 298,
"column": 83
} | {
"line": 298,
"column": 85
} | {
"line": 299,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK i : ℕ\nx✝ : i ∈ range K\n⊢ {ω | X ω ∈ Set.Ioi ↑i} = ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N}",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"Set.Subset.antisym... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 501,
"column": 70
} | {
"line": 501,
"column": 72
} | {
"line": 502,
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 296,
"column": 60
} | {
"line": 296,
"column": 62
} | {
"line": 297,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\n⊢ Tendsto (fun N ↦ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))",
"ppTerm": "?m.142",
"assigned": true,
"usedC... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 506,
"column": 44
} | {
"line": 506,
"column": 46
} | {
"line": 506,
"column": 47
} | [
{
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 290,
"column": 53
} | {
"line": 290,
"column": 55
} | {
"line": 291,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\n⊢ ∑' (j : ℕ), ℙ {ω | X ω ∈ Set.Ioi ↑j} < ∞",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Iff.mpr",
"Real.instIsOrde... | [] | by | [anonymous] | by |
Mathlib.Probability.Process.LocalProperty | {
"line": 307,
"column": 65
} | {
"line": 307,
"column": 67
} | {
"line": 308,
"column": 2
} | [
{
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Mathlib.Probability.Moments.SubGaussian | {
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} | {
"line": 507,
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{
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Mathlib.Probability.StrongLaw | {
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{
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"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Action | {
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"line": 45,
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} | {
"line": 45,
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
"line": 46,
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.Probability.StrongLaw | {
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} | {
"line": 323,
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{
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Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
"line": 87,
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | {
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{
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Mathlib.Probability.StrongLaw | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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"F... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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{
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"F... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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{
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"F... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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} | {
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{
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Mathlib.Probability.StrongLaw | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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"ppTerm": "?m.124",
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"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
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} | {
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"column": 78
} | {
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} | [
{
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Mathlib.RepresentationTheory.Action | {
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"column": 26
} | {
"line": 166,
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} | {
"line": 167,
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} | [
{
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"ppTerm": "?m.124",
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"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
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{
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Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | {
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{
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"usedConstants": [
"NonAssocSemiring... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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} | {
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{
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"NonAss... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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} | {
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{
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"NonAss... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
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{
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Mathlib.Probability.StrongLaw | {
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{
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Mathlib.RepresentationTheory.Action | {
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} | {
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} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nk : Type u\ninst✝ : CommSemiring k\nX : Action (Type u) G\n⊢ ↑(lid k (linearize k G X)).symm =\n ((rTensor (linearize k G X) (η k G)).comp (δ (𝟙_ (Action (Type u) G)) X)).comp (linearizeMap (λ_ X).inv)",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
"line": 213,
"column": 35
} | {
"line": 213,
"column": 37
} | {
"line": 214,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nk : Type u\ninst✝ : CommSemiring k\nX : Action (Type u) G\n⊢ ↑(rid k (linearize k G X)).symm =\n ((lTensor (linearize k G X) (η k G)).comp (δ X (𝟙_ (Action (Type u) G)))).comp (linearizeMap (ρ_ X).inv)",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 347,
"column": 43
} | {
"line": 347,
"column": 45
} | {
"line": 347,
"column": 46
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\nIk... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
"line": 217,
"column": 53
} | {
"line": 217,
"column": 55
} | {
"line": 218,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nX Y : Action (Type w) G\nk : Type u\ninst✝ : CommSemiring k\n⊢ (μ X Y).comp (δ X Y) = IntertwiningMap.id (linearize k G (X ⊗ Y))",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Monoid",
"NonAssocSemiring.toAd... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
"line": 221,
"column": 53
} | {
"line": 221,
"column": 55
} | {
"line": 222,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝¹ : Monoid G\nX Y : Action (Type w) G\nk : Type u\ninst✝ : CommSemiring k\n⊢ (δ X Y).comp (μ X Y) = IntertwiningMap.id ((linearize k G X).tprod (linearize k G Y))",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Monoid",
"NonAssoc... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 511,
"column": 37
} | {
"line": 511,
"column": 39
} | {
"line": 512,
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 349,
"column": 58
} | {
"line": 349,
"column": 60
} | {
"line": 349,
"column": 61
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\nIk... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 493,
"column": 69
} | {
"line": 493,
"column": 71
} | {
"line": 494,
"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\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\ninst✝¹ : SFinite ν\nη : Kernel (Ω' × Ω) Ω''\ninst✝ : IsZeroOrMarkovKernel η\nhX : HasSubgaussianMGF X c ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
"line": 230,
"column": 59
} | {
"line": 230,
"column": 61
} | {
"line": 231,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Monoid G\ninst✝ : Semiring k\nX : Type w\ng : G\n⊢ (linearize k G (Action.trivial G X)) g = LinearMap.id",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"R... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 353,
"column": 35
} | {
"line": 353,
"column": 37
} | {
"line": 353,
"column": 38
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\nIk... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 350,
"column": 29
} | {
"line": 350,
"column": 31
} | {
"line": 351,
"column": 14
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\nIk... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Action | {
"line": 240,
"column": 25
} | {
"line": 240,
"column": 27
} | {
"line": 240,
"column": 28
} | [
{
"pp": "k : Type u\nG : Type v\nV : Type u'\nW : Type v'\ninst✝⁵ : Monoid G\ninst✝⁴ : Semiring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AddCommGroup W\ninst✝ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nX✝ Y Z : Action (Type w) G\nX : Type w\ng : G\n⊢ ↑(LinearEquiv.refl k ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 354,
"column": 23
} | {
"line": 354,
"column": 25
} | {
"line": 354,
"column": 26
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\nk : ℕ\na✝ : k ∈ range K\nIk... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.AlgebraRepresentation.Basic | {
"line": 29,
"column": 58
} | {
"line": 29,
"column": 60
} | {
"line": 30,
"column": 2
} | [
{
"pp": "A : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝⁹ : Field k\ninst✝⁸ : Ring A\ninst✝⁷ : Algebra k A\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module k V\ninst✝⁴ : Module A V\ninst✝³ : IsScalarTower k A V\ninst✝² : IsSimpleModule A V\ninst✝¹ : FiniteDimensional k V\ninst✝ : IsAlgClosed k\n⊢ Function.Bijective ⇑(... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.AlgebraRepresentation.Basic | {
"line": 44,
"column": 24
} | {
"line": 44,
"column": 26
} | {
"line": 45,
"column": 6
} | [
{
"pp": "A : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : Ring A\ninst✝⁸ : Algebra k A\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module A V\ninst✝⁴ : IsScalarTower k A V\ninst✝³ : IsSimpleModule A V\ninst✝² : FiniteDimensional k V\ninst✝¹ : IsAlgClosed k\ninst✝ : IsMulCommutat... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 340,
"column": 58
} | {
"line": 340,
"column": 60
} | {
"line": 341,
"column": 6
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\n⊢ ∑ k ∈ range K, 2 / (↑k + ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.AlgebraRepresentation.Basic | {
"line": 40,
"column": 30
} | {
"line": 40,
"column": 32
} | {
"line": 41,
"column": 2
} | [
{
"pp": "A : Type u_1\nV : Type u_2\nk : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : Ring A\ninst✝⁸ : Algebra k A\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module A V\ninst✝⁴ : IsScalarTower k A V\ninst✝³ : IsSimpleModule A V\ninst✝² : FiniteDimensional k V\ninst✝¹ : IsAlgClosed k\ninst✝ : IsMulCommutat... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 39,
"column": 52
} | {
"line": 39,
"column": 54
} | {
"line": 39,
"column": 55
} | [
{
"pp": "k : Type u\ninst✝⁸ : Semiring k\nG : Type v\ninst✝⁷ : Monoid G\nV : Type v'\ninst✝⁶ : AddCommMonoid V\ninst✝⁵ : Module k V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\ninst✝³ : Module k W\nH : Type w\ninst✝² : Subsingleton H\ninst✝¹ : MulOneClass H\ninst✝ : MulAction G H\ng : G\n⊢ ↑(MonoidAlgebra.uniqueLine... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 53,
"column": 73
} | {
"line": 53,
"column": 75
} | {
"line": 53,
"column": 76
} | [
{
"pp": "k : Type u\ninst✝⁸ : Semiring k\nG : Type v\ninst✝⁷ : Monoid G\nV : Type v'\ninst✝⁶ : AddCommMonoid V\ninst✝⁵ : Module k V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\ninst✝³ : Module k W\nH : Type w\ninst✝² : Subsingleton H\ninst✝¹ : MulOneClass H\ninst✝ : MulAction G H\ng : G\n⊢ ↑(MonoidAlgebra.mapDomainL... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 57,
"column": 73
} | {
"line": 57,
"column": 75
} | {
"line": 58,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Semiring k\nG : Type v\ninst✝ : Monoid G\nf : Fin 1 → G\nr : k\n⊢ (diagonalOneEquivLeftRegular k G) (MonoidAlgebra.single f r) = MonoidAlgebra.single (f 0) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"Semiring.toModule",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 62,
"column": 87
} | {
"line": 62,
"column": 89
} | {
"line": 63,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Semiring k\nG : Type v\ninst✝ : Monoid G\ng : G\nr : k\n⊢ (diagonalOneEquivLeftRegular k G).symm (MonoidAlgebra.single g r) = MonoidAlgebra.single (uniqueElim g) r",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Representation.Equiv.symm",
"Semiri... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 80,
"column": 23
} | {
"line": 80,
"column": 25
} | {
"line": 80,
"column": 26
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 84,
"column": 69
} | {
"line": 84,
"column": 71
} | {
"line": 85,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝³ : Monoid G\nV : Type v'\ninst✝² : AddCommMonoid V\nk : Type u\ninst✝¹ : CommSemiring k\ninst✝ : Module k V\nσ : Representation k G V\nα : Type w'\ni : α\ng : G\nr : k\nf : α → V\n⊢ (σ.freeLift f) (single i (MonoidAlgebra.single g r)) = r • (σ g) (f i)",
"ppTerm": "?m.40",
"as... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 564,
"column": 83
} | {
"line": 564,
"column": 85
} | {
"line": 565,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nhm : m ≤ mΩ\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nX : Ω → ℝ\nc : ℝ≥0\nh : HasCondSubgaussianMGF m hm X c μ\nt : ℝ\n⊢ ∀ᵐ (ω' : Ω) ∂μ.trim hm, μ[fun ω ↦ rexp (t * X ω) | m] ω' ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.65",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 96,
"column": 16
} | {
"line": 96,
"column": 18
} | {
"line": 96,
"column": 19
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 97,
"column": 17
} | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 20
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 19
} | {
"line": 613,
"column": 21
} | {
"line": 613,
"column": 22
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : HasSubgaussianMGF X c μ\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) μ\nh2 : ∀ (t : ℝ), mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 103,
"column": 52
} | {
"line": 103,
"column": 54
} | {
"line": 104,
"column": 4
} | [
{
"pp": "k✝ : Type u\ninst✝¹² : Semiring k✝\nG : Type v\ninst✝¹¹ : Monoid G\nV : Type v'\ninst✝¹⁰ : AddCommMonoid V\ninst✝⁹ : Module k✝ V\nW : Type w'\ninst✝⁸ : AddCommMonoid W\ninst✝⁷ : Module k✝ W\nH : Type w\ninst✝⁶ : Subsingleton H\ninst✝⁵ : MulOneClass H\ninst✝⁴ : MulAction G H\nk : Type u\ninst✝³ : CommSe... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 107,
"column": 84
} | {
"line": 107,
"column": 86
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\nf : α →₀ V\nw : W\n⊢ (σ.fin... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 29
} | {
"line": 613,
"column": 31
} | {
"line": 613,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : HasSubgaussianMGF X c μ\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) μ\nh2 : ∀ (t : ℝ), mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ((Kernel.const Unit μ) ω') t ≤ rexp (↑c *... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 112,
"column": 63
} | {
"line": 112,
"column": 65
} | {
"line": 113,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\nf : α →₀ V\nw : W\ni : α\n⊢... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 117,
"column": 92
} | {
"line": 117,
"column": 94
} | {
"line": 118,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\ni : α\nv : V\nw : W\n⊢ (σ.f... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 56
} | {
"line": 613,
"column": 58
} | {
"line": 613,
"column": 59
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 124,
"column": 53
} | {
"line": 124,
"column": 55
} | {
"line": 125,
"column": 4
} | [
{
"pp": "k✝ : Type u\ninst✝¹² : Semiring k✝\nG : Type v\ninst✝¹¹ : Monoid G\nV : Type v'\ninst✝¹⁰ : AddCommMonoid V\ninst✝⁹ : Module k✝ V\nW : Type w'\ninst✝⁸ : AddCommMonoid W\ninst✝⁷ : Module k✝ W\nH : Type w\ninst✝⁶ : Subsingleton H\ninst✝⁵ : MulOneClass H\ninst✝⁴ : MulAction G H\nk : Type u\ninst✝³ : CommSe... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Equiv | {
"line": 128,
"column": 85
} | {
"line": 128,
"column": 87
} | {
"line": 129,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝⁶ : Monoid G\nV : Type v'\ninst✝⁵ : AddCommMonoid V\nW : Type w'\ninst✝⁴ : AddCommMonoid W\nk : Type u\ninst✝³ : CommSemiring k\ninst✝² : Module k V\ninst✝¹ : Module k W\nσ : Representation k G V\nρ : Representation k G W\nα : Type w'\ninst✝ : DecidableEq α\nv : V\nf : α →₀ W\n⊢ (σ.fin... | [] | by | [anonymous] | by |
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