module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Kernel.Posterior | {
"line": 155,
"column": 28
} | {
"line": 155,
"column": 62
} | {
"line": 155,
"column": 62
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.deterministic f hf ∥ₖ Kernel.determinist... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountablyGenerated 𝓧\nf : Ω → 𝓧\nhf : Measurable f\n⊢ ⇑(Kernel.copy 𝓧 ∘ₖ Kernel.deterministic f hf) ∘ₘ μ = ⇑(Kerne... | Kernel.parallelComp_self_comp_copy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 15
} | {
"line": 164,
"column": 16
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\nthis : (⇑κ ∘ₘ μ) ⊗ₘ... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nν : Measure 𝓧\ninst✝ : SFinite ν\nh_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν\nthis : (⇑κ ∘ₘ μ) ⊗ₘ κ†μ ≪ ν ⊗ₘ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 501,
"column": 6
} | {
"line": 501,
"column": 43
} | {
"line": 501,
"column": 44
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nt : Set ((n : ℕ) → X n)\nht : t ∈ measurableCylinders X\n⊢ ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S",
"ppTerm": "?m.93",
"... | [
"X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nt : Set ((n : ℕ) → X n)\nht : t ∈ measurableCylinders X\n⊢ ∃ N S, MeasurableSet S ∧ t = cylinder (Iic N) S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | {
"line": 216,
"column": 16
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ... | [
"Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\ninst✝³ : IsFiniteKernel κ\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\ninst✝ : MeasurableSpace.CountableOrCountablyGenerated Ω 𝓧\nh_ac : ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 55
} | {
"line": 507,
"column": 56
} | [
{
"pp": "case compl\nX : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\nthis : ∀ (x₀ : (i : ↥(Iic a)) → X... | [
"case compl\nX : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nt : Set ((n : ℕ) → X n)\nmt : MeasurableSet t\nht : Measurable fun x₀ ↦ (trajFun κ a x₀) t\nthis : ∀ (x₀ : (i : ↥(Iic a)) → X ↑i), IsProb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 40
} | {
"line": 508,
"column": 41
} | [
{
"pp": "case union\nX : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nf : ℕ → Set ((n : ℕ) → X n)\ndisf : Pairwise (Disjoint on f)\nmf : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable... | [
"case union\nX : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nf : ℕ → Set ((n : ℕ) → X n)\ndisf : Pairwise (Disjoint on f)\nmf : ∀ (i : ℕ), MeasurableSet (f i)\nhf : ∀ (i : ℕ), Measurable fun x₀ ↦ (t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Decision.Risk.Basic | {
"line": 117,
"column": 77
} | {
"line": 117,
"column": 88
} | {
"line": 117,
"column": 89
} | [
{
"pp": "Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻... | [
"Θ : Type u_1\n𝓧 : Type u_2\n𝓨 : Type u_4\nmΘ : MeasurableSpace Θ\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nℓ : Θ → 𝓨 → ℝ≥0∞\nhl : Measurable (uncurry ℓ)\nμ : Measure 𝓧\ninst✝¹ : SFinite μ\nπ : Measure Θ\ninst✝ : SFinite π\nhl_pos : μ Set.univ = ∞ → ⨅ y, ∫⁻ (θ : Θ), ℓ θ y ∂π = 0 → ∃ y, ∫⁻ (θ : Θ), ℓ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Binomial | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nm : MeasurableSpace Ω\nP : Measure Ω\nn : ℕ\np : ↑I\nX : Ω → ℕ\nhX : HasLaw X Bin(n, p) P\ns : Set ℕ\nhs : s ⊆ Iio n\n⊢ s.ncard ≤ n",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_2\nm : MeasurableSpace Ω\nP : Measure Ω\nn : ℕ\np : ↑I\nX : Ω → ℕ\nhX : HasLaw X Bin(n, p) P\ns : Set ℕ\nhs : s ⊆ Iio n\n⊢ s.ncard ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Cauchy | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 31
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case pos\nx₀ : ℝ\nγ : ℝ≥0\nh : γ = 0\n⊢ Integrable 0 volume",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"PseudoMetricSpace.toUniformSpace",
"Real.measureSpace",
"Real.normedAddCommGroup",
... | [] | exact integrable_zero _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 76
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\n⊢ p.toOuterMeasure (s ∩ p.support) = p.toOuterMeasure s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"SummationFilter",
"PMF.toOuterMeasure",
... | [] | simp only [toOuterMeasure_apply, PMF.support, Set.indicator_inter_support] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 70
} | {
"line": 257,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s ∩ p.support = t ∩ p.support\n⊢ p.toMeasure s = p.toMeasure t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Mea... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\nht : MeasurableSet t\nh : s ∩ p.support = t ∩ p.support\n⊢ p.toOuterMeasure s = p.toOuterMeasure t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 335,
"column": 4
} | {
"line": 336,
"column": 53
} | {
"line": 336,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\n⊢ p.toMeasure Set.univ = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicato... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\n⊢ ∑' (x : α), p x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gamma | {
"line": 146,
"column": 81
} | {
"line": 149,
"column": 48
} | {
"line": 151,
"column": 0
} | [
{
"pp": "a r : ℝ\nha : 0 < a\nhr : 0 < r\nx : ℝ\n⊢ ↑(cdf (gammaMeasure a r)) x = (∫⁻ (x : ℝ) in Iic x, gammaPDF a r x).toReal",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"instClosedIicTopology",
"Real",
"Measu... | [] | by
have : IsProbabilityMeasure (gammaMeasure a r) := isProbabilityMeasure_gammaMeasure ha hr
simp only [gammaPDF, cdf_eq_real]
simp [gammaMeasure, gammaPDF, measureReal_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑n * p n * (1 / ↑n)) =ᶠ[atTop] p\n⊢ Tendsto p atTop (𝓝 0)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ → ℝ\nr : ℝ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑n * p n * (1 / ↑n)) =ᶠ[atTop] p\n⊢ Tendsto p atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 30
} | {
"line": 64,
"column": 31
} | [
{
"pp": "p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑(n.choose k) * p n ^ k) ~[atTop] fun n ↦ (↑n * p n) ^ k / ↑k.factorial\n⊢ Tendsto (fun n ↦ (↑n * p n) ^ k / ↑k.factorial) atTop (𝓝 (r ^ k / ↑k.factorial))",
"ppTerm": "?m.155",
"assigned": true,
"usedCo... | [
"p : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nthis : (fun n ↦ ↑(n.choose k) * p n ^ k) ~[atTop] fun n ↦ (↑n * p n) ^ k / ↑k.factorial\n⊢ Tendsto (fun n ↦ (↑n * p n) ^ k * (↑k.factorial)⁻¹) atTop (𝓝 (r ^ k * (↑k.factorial)⁻¹))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case refine_1\np : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nhp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2\nhEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹\nthis : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹\n⊢ Tendsto (fun n ↦ ↑n * -p n) a... | [
"case refine_1\np : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nhp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2\nhEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹\nthis : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹\n⊢ Tendsto (fun x ↦ ↑x * p x) atTop (𝓝 r)"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"pp": "case refine_2\np : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nhp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2\nhEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹\nthis : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹\n⊢ Tendsto (fun n ↦ 1 - p n) atT... | [
"case refine_2\np : ℕ → ℝ\nr : ℝ\nk : ℕ\nhr : Tendsto (fun n ↦ ↑n * p n) atTop (𝓝 r)\nhp_lt_half : ∀ᶠ (n : ℕ) in atTop, p n < 1 / 2\nhEq : (fun n ↦ (1 - p n) ^ (n - k)) =ᶠ[atTop] fun n ↦ (1 - p n) ^ n * ((1 - p n) ^ k)⁻¹\nthis : Real.exp (-r) = Real.exp (-r) * (1 ^ k)⁻¹\n⊢ Tendsto (fun n ↦ 1 - p n) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 62
} | {
"line": 103,
"column": 4
} | [
{
"pp": "k : ℕ\nr : ℝ≥0\np : ℕ → ↑unitInterval\nhr : Tendsto (fun n ↦ ↑n * ↑(p n)) atTop (𝓝 ↑r)\nt1 : Tendsto (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) atTop (𝓝 (Po(r) {k}))\n⊢ (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) =ᶠ[atTop] fun n ↦... | [
"k : ℕ\nr : ℝ≥0\np : ℕ → ↑unitInterval\nhr : Tendsto (fun n ↦ ↑n * ↑(p n)) atTop (𝓝 ↑r)\nt1 : Tendsto (fun n ↦ ENNReal.ofReal (↑(n.choose k) * ↑(p n) ^ k * (1 - ↑(p n)) ^ (n - k))) atTop (𝓝 (Po(r) {k}))\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → ENNReal.ofReal (↑(b.choose k) * ↑(p b) ^ k * (1 - ↑(p b)) ^ (b - k)) = Bin(b, p b) {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 76
} | {
"line": 142,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\ng : β → PMF γ\nb : γ\n⊢ ((p.bind f).bind g) b = (p.bind fun a ↦ (f a).bind g) b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"ENNReal.tsum_mul_left",
"HMul.h... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\ng : β → PMF γ\nb : γ\n⊢ ∑' (a : β) (i : α), p i * ((f i) a * (g a) b) = ∑' (a : α) (i : β), p a * ((f a) i * (g i) b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 76
} | {
"line": 148,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nq : PMF β\nf : α → β → PMF γ\nb : γ\n⊢ (p.bind fun a ↦ q.bind (f a)) b = (q.bind fun b ↦ p.bind fun a ↦ f a b) b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.tsum_mul_left",
"HMul.hMul",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nq : PMF β\nf : α → β → PMF γ\nb : γ\n⊢ ∑' (a : α) (i : β), p a * (q i * (f a i) b) = ∑' (a : β) (i : α), p i * (q a * (f i a) b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 146,
"column": 58
} | {
"line": 161,
"column": 13
} | {
"line": 163,
"column": 0
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ charFun Po(ℝ, r) t = cexp (↑↑r * (cexp (↑t * I) - 1))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.inner_apply",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | by
rw [charFun_apply, integral_map .of_discrete (by fun_prop), integral_poissonMeasure r]
simp_rw [Real.inner_apply]
calc ∑' a, (rexp (-r) * r ^ a / a ! : ℝ) * cexp ((a * t : ℝ) * I)
_ = ∑' a, (rexp (-r)) * ((r * cexp (t * I)) ^ a / a !) := by
congr with a
push_cast
rw [mul_pow, ← Complex.exp_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 31
} | {
"line": 241,
"column": 32
} | [
{
"pp": "r : ℝ≥0\nn : ℕ\n⊢ ENNReal.ofReal (poissonPMFReal r n) = (poissonPMF r) n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"ENNReal.ofReal",
"PMF",
"ProbabilityTheory.poissonPMF",
"PMF.instFunLike",
"id",
"Nat",
"ENNReal",
"ProbabilityT... | [
"r : ℝ≥0\nn : ℕ\n⊢ ENNReal.ofReal (poissonPMFReal r n) = ⟨fun n ↦ ENNReal.ofReal (poissonPMFReal r n), ⋯⟩ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 19
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\ng : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ\na : γ\na' : α\nb : β\nh : ¬p a' = 0\nh_1 : ∀ (i : α), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nH : ¬(f a' h) b = 0\n⊢ (f a' h) b = 0",
... | [
"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\ng : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ\na : γ\na' : α\nb : β\nh : ¬p a' = 0\nh_1 : ∀ (i : α), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nH : ¬(f a' h) b = 0\n⊢ (f a' h) b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.TwoValued | {
"line": 46,
"column": 12
} | {
"line": 46,
"column": 70
} | {
"line": 46,
"column": 71
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\n⊢ ∀ᵐ (ω : Ω) ∂μ, 1 - X ω = 0 ∨ 1 - X ω = 1",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtractionMonoid... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\n⊢ ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 29
} | {
"line": 124,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nq : PMF (α → β)\np : PMF α\nb : β\nf : α → β\na : α\n⊢ (q f * if b = f a then p a else 0) = if b = f a then q f * p a else 0",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nq : PMF (α → β)\np : PMF α\nb : β\nf : α → β\na : α\n⊢ (q f * if b = f a then p a else 0) = if b = f a then q f * p a else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 175,
"column": 22
} | {
"line": 175,
"column": 51
} | {
"line": 175,
"column": 52
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\ns : Finset α\nh : ∑ a ∈ s, f a = 1\nh' : ∀ a ∉ s, f a = 0\na : α\n⊢ a ∈ (ofFinset f s h h').support ↔ a ∈ ↑s ∩ Function.support f",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Function.mem_suppor... | [
"α : Type u_1\nf : α → ℝ≥0∞\ns : Finset α\nh : ∑ a ∈ s, f a = 1\nh' : ∀ a ∉ s, f a = 0\na : α\n⊢ ¬f a = 0 → a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 268,
"column": 36
} | {
"line": 268,
"column": 47
} | {
"line": 268,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\n⊢ tsum (s.indicator ⇑p) ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"P... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\n⊢ ∃ x ∈ s, ¬p x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.TwoValued | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 13
} | [
{
"pp": "case inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhμ : IsProbabilityMeasure μ\n⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}",
"ppTerm": "?inr",
"assigned":... | [
"case inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhμ : IsProbabilityMeasure μ\n⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 111,
"column": 21
} | {
"line": 111,
"column": 47
} | {
"line": 111,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhns : μ s ≠ 0\nhnt : μ s ≠ ∞\nhu : IsUniform X s ℙ μ\nt : Set E := toMeasurable μ s\n⊢ μ[|s] = (μ t)⁻¹ • μ.restrict (toMeasurable μ s)",
"ppTerm": "?m.109",
"assign... | [
"E : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhns : μ s ≠ 0\nhnt : μ s ≠ ∞\nhu : IsUniform X s ℙ μ\nt : Set E := toMeasurable μ s\n⊢ μ[|s] = (μ t)⁻¹ • μ.restrict s"
] | restrict_toMeasurable hnt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 66
} | {
"line": 222,
"column": 8
} | [
{
"pp": "α : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"PMF.uniformOfFinset._simp_2",
"Eq.mpr",
"congrArg",
"Finset",
"ENNReal.instCharZero",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAdd... | [
"α : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ ¬s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 266,
"column": 40
} | {
"line": 266,
"column": 51
} | {
"line": 266,
"column": 52
} | [
{
"pp": "α : Type u_1\ns : Finset α\nhs : s.Nonempty\nt : Set α\nx : α\nhx : x ∈ {x ∈ s | x ∈ t}\n⊢ x ∈ s ∧ x ∈ t",
"ppTerm": "?m.188",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Finset α\nhs : s.Nonempty\nt : Set α\nx : α\nhx : x ∈ {x ∈ s | x ∈ t}\n⊢ x ∈ s ∧ x ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 358,
"column": 2
} | {
"line": 359,
"column": 43
} | {
"line": 359,
"column": 44
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nhs : s ≠ 0\na : α\nha : a ∉ s\n⊢ (ofMultiset s hs) a = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"instHDiv",
"congrArg",
"PMF.ofMultiset",
"PMF",
"ENNReal.instCharZero",
"AddMo... | [
"α : Type u_1\ns : Multiset α\nhs : s ≠ 0\na : α\nha : a ∉ s\n⊢ a ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → F\ninst✝ : TopologicalSpace F\nhm : m ≤ mΩ\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ f x.2) (Measure.map (fun ω ↦ (id ω, id ω)) μ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [
"Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → F\ninst✝ : TopologicalSpace F\nhm : m ≤ mΩ\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun x ↦ f x.2) (Measure.map (fun ω ↦ (ω, ω)) μ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | {
"line": 57,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → F\ninst✝ : NormedAddCommGroup F\nhf : Integrable f μ\n⊢ Integrable (fun x ↦ f x.2) (Measure.map (fun ω ↦ (id ω, id ω)) μ)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasurableSpace.prod",
... | [
"Ω : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → F\ninst✝ : NormedAddCommGroup F\nhf : Integrable f μ\n⊢ Integrable (fun x ↦ f x.2) (Measure.map (fun ω ↦ (ω, ω)) μ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 34
} | {
"line": 91,
"column": 35
} | [
{
"pp": "case inr\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : Nonempty Ω\n⊢ IsMarkovKernel (condExpKernel μ m)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
... | [
"case inr\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : Nonempty Ω\n⊢ IsMarkovKernel ((condDistrib id id μ).comap id ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 51
} | {
"line": 57,
"column": 52
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nh : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)\nt : Set Ω\nh_indep : IndepSet t t κ μα\na : α\nh_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞\nh' : IsFiniteMeasure (κ a)\n⊢ (κ a) t = 0 ∨ (κ a) t... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nh : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)\nt : Set Ω\nh_indep : IndepSet t t κ μα\na : α\nh_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞\nh' : IsFiniteMeasure (κ a)\n⊢ (κ a) t = 0 ∨ (κ a) t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nm m0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhm : Indep m m μ\nt : Set Ω\nht : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_2\nm m0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhm : Indep m m μ\nt : Set Ω\nht : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 95,
"column": 4
} | {
"line": 96,
"column": 48
} | {
"line": 97,
"column": 4
} | [
{
"pp": "X : SFinKer\nf₁ : X.carrier → PUnit.{?u.96 + 1} × X.carrier := fun x ↦ (PUnit.unit, x)\nhf₁ : Measurable f₁\nhf₂ : Measurable Prod.snd\n⊢ { carrier := { carrier := PUnit.{u + 1}, str := PUnit.instMeasurableSpace }.carrier × X.carrier,\n str := Prod.instMeasurableSpace } ≅\n X",
"ppTerm": "?... | [
"case refine_1\nX : SFinKer\nf₁ : X.carrier → PUnit.{u + 1} × X.carrier := fun x ↦ (PUnit.unit, x)\nhf₁ : Measurable f₁\nhf₂ : Measurable Prod.snd\n⊢ { hom := Kernel.id.map Prod.snd, property := ⋯ } ≫ { hom := Kernel.id.map f₁, property := ⋯ } =\n 𝟙\n { carrier := { carrier := PUnit.{u + 1}, str := PUnit.i... | refine ⟨⟨Kernel.id.map Prod.snd, inferInstance⟩,
⟨Kernel.id.map f₁, inferInstance⟩, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Invariance | {
"line": 71,
"column": 50
} | {
"line": 71,
"column": 77
} | {
"line": 71,
"column": 78
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\ninst✝ : IsMarkovKernel κ\nπ : Measure α\nh_rev : κ.IsReversible π\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α), (κ x) s ∂π = ∫⁻ (x : α) in s, (κ x) Set.univ ∂π",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"MeasureThe... | [
"α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\ninst✝ : IsMarkovKernel κ\nπ : Measure α\nh_rev : κ.IsReversible π\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α), (κ x) s ∂π = π s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | {
"line": 72,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nc : ℝ≥0∞\nφ : Measure α\nκ : Kernel α α\nhκ : IsIrreducible φ κ\ns : Set α\nhs : MeasurableSet s\nhsp : (c • φ) s > 0\n⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nc : ℝ≥0∞\nφ : Measure α\nκ : Kernel α α\nhκ : IsIrreducible φ κ\ns : Set α\nhs : MeasurableSet s\nhsp : (c • φ) s > 0\n⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | {
"line": 78,
"column": 16
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nφ₁ φ₂ : Measure α\nhφ : φ₁ ≤ φ₂\nκ : Kernel α α\nhκ : IsIrreducible φ₂ κ\ns : Set α\nhs : MeasurableSet s\nhsp : φ₁ s > 0\n⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ProbabilityTh... | [
"α : Type u_1\nmα : MeasurableSpace α\nφ₁ φ₂ : Measure α\nhφ : φ₁ ≤ φ₂\nκ : Kernel α α\nhκ : IsIrreducible φ₂ κ\ns : Set α\nhs : MeasurableSet s\nhsp : φ₁ s > 0\n⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | {
"line": 55,
"column": 90
} | [
{
"pp": "case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nx✝ : π.IsProper\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • ... | [
"case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nx✝ : π.IsProper\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | {
"line": 55,
"column": 90
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x",
"ppTerm... | [
"case refine_2\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 38
} | {
"line": 78,
"column": 38
} | [
{
"pp": "X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nA B : Set X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nμ : Measure X\nhA : MeasurableSet A\nhB : MeasurableSet B\n⊢ ∫⁻ (a : X) in B, (π a) A ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ",
"ppTerm": "?m.54",
"assigned": true,
"usedConst... | [
"X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nA B : Set X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nμ : Measure X\nhA : MeasurableSet A\nhB : MeasurableSet B\n⊢ ∫⁻ (a : X), B.indicator (fun a ↦ (π a) A) a ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ"
] | ← lintegral_indicator (h𝓑𝓧 _ hB) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 59,
"column": 26
} | {
"line": 59,
"column": 67
} | {
"line": 59,
"column": 67
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | [
"Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹ : SigmaFini... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 69,
"column": 28
} | {
"line": 69,
"column": 69
} | {
"line": 69,
"column": 69
} | [
{
"pp": "case pos\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι... | [
"case pos\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹ :... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 102,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | simp only [h, Set.mem_range] at hi
obtain ⟨ω, hω⟩ := hi
specialize hτ_le ω
simp [hω] at hτ_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 102,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | simp only [h, Set.mem_range] at hi
obtain ⟨ω, hω⟩ := hi
specialize hτ_le ω
simp [hω] at hτ_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"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\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)\n⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)",
"ppTerm": "?m.30",
"assigned": false,
... | [
"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\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)\n⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | {
"line": 164,
"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\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')\nω : Ω'\nh_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)\n⊢ AEStrongl... | [
"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\nh_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')\nω : Ω'\nh_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)\n⊢ AEStronglyMeasurable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.Tilted | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 20
} | {
"line": 144,
"column": 21
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp : p = 0\n⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"N... | [
"case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp : p = 0\n⊢ AEStronglyMeasurable X (μ.tilted fun x ↦ t * X x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 72,
"column": 35
} | {
"line": 72,
"column": 46
} | {
"line": 72,
"column": 47
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k % (b + 1) = k",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"i... | [
"k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 21
} | {
"line": 192,
"column": 22
} | [
{
"pp": "case pos\nΩ : 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⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nΩ : 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⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 22
} | {
"line": 211,
"column": 23
} | [
{
"pp": "case pos\nΩ : 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 ... | [
"case pos\nΩ : 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 : κ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 49
} | {
"line": 156,
"column": 50
} | [
{
"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\nM : ℝ≥0\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q M\ns t : T\nh : edist s t = 0\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω ... | [
"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\nM : ℝ≥0\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q M\ns t : T\nh : edist s t = 0\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω : edist (X s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 49
} | {
"line": 172,
"column": 50
} | [
{
"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\nω : Ω\nhω : edist (X s ω) (X t ω) ^ p ... | [
"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\nω : Ω\nhω : edist (X s ω) (X t ω) ^ p = 0 ω\n⊢ edi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 36
} | {
"line": 220,
"column": 37
} | [
{
"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 : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ IsFiniteMeasure (κ ω'... | [
"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 : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ IsFiniteMeasure (κ ω')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 19
} | {
"line": 227,
"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 : 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)\n⊢ (κ ω').real Set.univ ≤ 1",
"ppTerm": ... | [
"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)\n⊢ (κ ω').real Set.univ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 265,
"column": 29
} | {
"line": 265,
"column": 40
} | {
"line": 265,
"column": 41
} | [
{
"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... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 80
} | {
"line": 266,
"column": 81
} | [
{
"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",... | [
"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⊢ ∫ (x : Ω), rexp (-(t * X x)) ∂κ ω' ≤ rexp (↑c * t ^ 2 / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Conditional | {
"line": 885,
"column": 6
} | {
"line": 885,
"column": 17
} | {
"line": 885,
"column": 18
} | [
{
"pp": "case e_f\nΩ : 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 : N... | [
"case e_f\nΩ : 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\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 67
} | {
"line": 306,
"column": 68
} | [
{
"pp": "case refine_3\nΩ : 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\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf id ((κ.map X) ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?refine_3",... | [
"case refine_3\nΩ : 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\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 156,
"column": 70
} | {
"line": 156,
"column": 85
} | {
"line": 157,
"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\np : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\nX : ι → Ω → E\nhX : (fun Y ↦ Locally p �... | [
"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\nX : ι → Ω → E\nhX : (fun Y ↦ Locally p 𝓕 Y P) X\nτ ... | Set.inter_comm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 437,
"column": 10
} | {
"line": 437,
"column": 21
} | {
"line": 437,
"column": 22
} | [
{
"pp": "case hf\nΩ : 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 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ), mgf X ... | [
"case hf\nΩ : 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≥0 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 438,
"column": 10
} | {
"line": 438,
"column": 21
} | {
"line": 438,
"column": 22
} | [
{
"pp": "case hg\nΩ : 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 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ), mgf X ... | [
"case hg\nΩ : 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≥0 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 237,
"column": 2
} | {
"line": 246,
"column": 9
} | {
"line": 247,
"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... | [
"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\nhτ : IsLoc... | suffices (1 / 2) ^ n ≤ P (⋂ k : ℕ, {ω | σ n k ω < min (τ n ω) (T n)}) by
refine (by simp : ¬ (1 / 2 : ℝ≥0∞) ^ n ≤ 0) <| this.trans <| nonpos_iff_eq_zero.2 ?_
rw [measure_eq_zero_iff_ae_notMem]
filter_upwards [(hσ n).tendsto_top] with ω hTop hmem
simp_rw [WithTop.tendsto_nhds_top_iff, eventually_atTop] a... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Probability.Moments.SubGaussian | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 15
} | {
"line": 461,
"column": 16
} | [
{
"pp": "case pos.integrable_exp_mul\nΩ : 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 ... | [
"case pos.integrable_exp_mul\nΩ : 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\n⊢ ∀ (t : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 23
} | [
{
"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... | [
"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.toSubmodule... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 59
} | {
"line": 613,
"column": 70
} | {
"line": 613,
"column": 71
} | [
{
"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 ... | [
"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 ((Kernel.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 89
} | {
"line": 613,
"column": 90
} | [
{
"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 ... | [
"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 ((Kernel.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"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 ... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"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 ... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"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 ... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 13
} | {
"line": 619,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) μ\n⊢ AEStronglyMeasurable X μ",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) μ\n⊢ AEStronglyMeasurable X μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 14
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 13
} | {
"line": 633,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 13
} | {
"line": 637,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\n⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\n⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 44
} | {
"line": 647,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ HasSubgaussianMGF (-X) c μ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Real",
"Pi.instNeg",
"_private.Mathlib.Probabilit... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ Kernel.HasSubgaussianMGF (-X) c (Kernel.const Unit μ) (Measure.dirac ())"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 666,
"column": 8
} | {
"line": 666,
"column": 21
} | {
"line": 666,
"column": 21
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ mgf id (Measure.map X μ) t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.i... | [
"case refine_3\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)"
] | mgf_id_map hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 707,
"column": 2
} | {
"line": 707,
"column": 13
} | {
"line": 707,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 13
} | {
"line": 714,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᵐ[μ] 0",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᵐ[μ] 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 44
} | {
"line": 724,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ HasSubgaussianMGF ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ Kernel.HasSubgaussianMGF (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 27
} | {
"line": 792,
"column": 28
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\nh : μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑(∑ i ∈ Finset.range n, c)))\n⊢ μ.real {ω | ε ≤ ∑ i ∈ Fin... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\nh : μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑(∑ i ∈ Finset.range n, c)))\n⊢ μ.real {ω | ε ≤ ∑ i ∈ Finset.range n,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 265,
"column": 13
} | {
"line": 265,
"column": 67
} | {
"line": 265,
"column": 67
} | [
{
"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... | [
"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 : Ω), X x) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ"
] | integral_truncation_le_integral_of_nonneg hint hnonneg | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 865,
"column": 44
} | {
"line": 865,
"column": 55
} | {
"line": 865,
"column": 56
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nω : Ω\nhab : X ω ∈ Set.Icc a b\n⊢ X ω - ∫ (x : Ω), X x ∂μ ∈ Set.Icc (a - ∫ (x : Ω), X x ∂μ) (b - ∫ (x : Ω), X x ∂μ)",
"ppTerm": "?m.... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nω : Ω\nhab : X ω ∈ Set.Icc a b\n⊢ a ≤ X ω ∧ X ω ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "case mk.mk\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoLinearMap✝¹ : V →ₗ[A] W\nisIntertwining'✝¹ : ∀ (... | [
"case mk.mk\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoLinearMap✝¹ : V →ₗ[A] W\nisIntertwining'✝¹ : ∀ (g : G), toLi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 342,
"column": 35
} | {
"line": 342,
"column": 72
} | {
"line": 342,
"column": 73
} | [
{
"pp": "case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : σ.IntertwiningMap ρ\ninvFu... | [
"case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : σ.IntertwiningMap ρ\ninvFun✝¹ : V → W\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 13
} | {
"line": 367,
"column": 14
} | [
{
"pp": "case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : ρ.IntertwiningMap σ\ninvFu... | [
"case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : ρ.IntertwiningMap σ\ninvFun✝¹ : W → V\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Equiv | {
"line": 173,
"column": 24
} | {
"line": 173,
"column": 49
} | {
"line": 174,
"column": 2
} | [
{
"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... | [] | simp [← x.isIntertwining] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 34,
"column": 27
} | {
"line": 34,
"column": 38
} | {
"line": 34,
"column": 39
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nX Y : Rep k G\nf : H →* G\np : X ⟶ Y\nh : H\n⊢ ↑(Hom.hom p) ∘ₗ (MonoidHom.comp X.ρ f) h = (MonoidHom.comp Y.ρ f) h ∘ₗ ↑(Hom.hom p)",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nX Y : Rep k G\nf : H →* G\np : X ⟶ Y\nh : H\n⊢ (Hom.hom p).toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom p).toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 73
} | {
"line": 70,
"column": 74
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX✝ Y✝ : Rep k G\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : (resFunctor f).map a₁✝ = (resFunctor f).map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX✝ Y✝ : Rep k G\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : (resFunctor f).map a₁✝ = (resFunctor f).map a₂✝\n⊢ (Hom.hom a₁✝).toLinearMap = (Hom.hom a₂✝).toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 75,
"column": 76
} | {
"line": 75,
"column": 87
} | {
"line": 75,
"column": 88
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX Y : Rep k G\nhf : Function.Surjective ⇑f\nf' : res f X ⟶ res f Y\nh : H\n⊢ (Hom.hom f').toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom f').toLinearMap",
"ppTerm": "?m.89",
... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX Y : Rep k G\nhf : Function.Surjective ⇑f\nf' : res f X ⟶ res f Y\nh : H\n⊢ (Hom.hom f').toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom f').toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 50
} | {
"line": 96,
"column": 51
} | [
{
"pp": "case pred\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\nh : (ρ ((fun x ↦ g ^ x) (-↑i))) x = x\n⊢ (ρ ((fun x ↦ g ^ x) ... | [
"case pred\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\nh : (ρ ((fun x ↦ g ^ x) (-↑i))) x = x\n⊢ (ρ g⁻¹) ((ρ (g ^ i)⁻¹) x) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 465,
"column": 8
} | {
"line": 465,
"column": 34
} | {
"line": 465,
"column": 35
} | [
{
"pp": "case hbc\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_... | [
"case hbc\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < c\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 15
} | {
"line": 155,
"column": 16
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx : V\nhx : x ∈ invariants (MonoidHom.comp ρ S.subtype)\nx✝ : ↥S\ns : G\nhs : s ∈ S\n⊢ ((MonoidHom.comp ρ S.s... | [
"k : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx : V\nhx : x ∈ invariants (MonoidHom.comp ρ S.subtype)\nx✝ : ↥S\ns : G\nhs : s ∈ S\n⊢ (ρ s) ((ρ g) x) = (ρ g) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 164,
"column": 58
} | {
"line": 164,
"column": 69
} | {
"line": 164,
"column": 70
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW : Type u_4\ninst✝⁸ : CommRing k\ninst✝⁷ : Group G\ninst✝⁶ : AddCommGroup V✝\ninst✝⁵ : Module k V✝\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\nρ✝ : Representation k G V✝\nσ : Representation k G W\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\... | [
"k : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW : Type u_4\ninst✝⁸ : CommRing k\ninst✝⁷ : Group G\ninst✝⁶ : AddCommGroup V✝\ninst✝⁵ : Module k V✝\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\nρ✝ : Representation k G V✝\nσ : Representation k G W\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Represe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Character | {
"line": 117,
"column": 41
} | {
"line": 119,
"column": 52
} | {
"line": 121,
"column": 0
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Group G\ninst✝⁸ : Field k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : FiniteDimensional k V\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\ninst✝² : FiniteDimensional k W\nρ : Representation k G V\nσ : Representation k G W\nins... | [] | by
simp_rw [mul_comm, ← char_linHom, card_inv_mul_sum_char_eq_finrank,
(invariantsEquivIntertwiningMap ρ σ).finrank_eq] | [anonymous] | Lean.Parser.Term.byTactic |
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