module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 70
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"PMF.toO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 15
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : Type u_4\nX : S → Ω → E\nY : T → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhXY... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 335,
"column": 4
} | {
"line": 336,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : PMF α\n⊢ p.toMeasure Set.univ = 1",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"Set.univ",
"SummationFilter... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 144,
"column": 58
} | {
"line": 159,
"column": 13
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ charFun Po(ℝ, r) t = cexp (↑↑r * (cexp (↑t * I) - 1))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.inner_apply",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"InnerProductSpace.toNo... | 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": 243,
"column": 2
} | {
"line": 243,
"column": 31
} | [
{
"pp": "r : ℝ≥0\nn : ℕ\n⊢ ENNReal.ofReal (poissonPMFReal r n) = (poissonPMF r) n",
"usedConstants": [
"ENNReal.ofReal",
"PMF",
"ProbabilityTheory.poissonPMF",
"PMF.instFunLike",
"id",
"Nat",
"ENNReal",
"ProbabilityTheory.poissonPMFReal",
"Eq",
"DF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 67,
"column": 35
} | {
"line": 67,
"column": 46
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k % (b + 1) = k",
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"Nat.instMod",
"instHMod",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"ENNReal.tsum_mul_left",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 147,
"column": 4
} | {
"line": 148,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ENNReal.tsum_mul_left",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"Comm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 29
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 174,
"column": 22
} | {
"line": 174,
"column": 51
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Function.mem_support._simp_1",
"congrArg",
"Finset",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 19
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 267,
"column": 36
} | {
"line": 267,
"column": 47
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"Set.indicator",
"PMF.instFunLike",
"Membership.mem",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 30
} | [
{
"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))",
"usedConstants": [
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.PoissonLimitThm | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 62
} | [
{
"pp": "k : ℕ\nr : ℝ≥0\np : ℕ → ℝ≥0\nh : ∀ (n : ℕ), p n ≤ 1\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[atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 110,
"column": 21
} | {
"line": 110,
"column": 47
} | [
{
"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)",
"usedConstants": [
"Eq.mp... | restrict_toMeasurable hnt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 66
} | [
{
"pp": "α : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"usedConstants": [
"PMF.uniformOfFinset._simp_2",
"Eq.mpr",
"congrArg",
"Finset",
"ENNReal.instCharZero",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"id",
"AddMonoidWithOne.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 265,
"column": 40
} | {
"line": 265,
"column": 51
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 43
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nhs : s ≠ 0\na : α\nha : a ∉ s\n⊢ (ofMultiset s hs) a = 0",
"usedConstants": [
"Eq.mpr",
"False",
"instHDiv",
"congrArg",
"PMF.ofMultiset",
"PMF",
"ENNReal.instCharZero",
"AddMonoid.toAddZeroClass",
"PMF.instFunLike"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.WithDensity | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β → ℝ≥0∞\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nhf : Measurable (Function.uncurry f)\na : α\ng : β → ℝ≥0∞\nhg : Measurable g\n⊢ ∫⁻ (a_1 : β), (f a * g) a_1 ∂κ a = ∫⁻ (b : β), f a b * g b ∂κ a",
"usedConstants":... | simp_rw [Pi.mul_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 15
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ninst✝⁶ : Finite ι\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → MeasurableSpace (E i)\ninst✝³ : ∀ (i : ι), CompleteSpace (E i)\ninst✝² : ∀ (i : ι), BorelSpace (E i)\ninst✝¹ : ∀ (i : ι), SecondCountab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 214,
"column": 79
} | {
"line": 215,
"column": 52
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nthis :\n (((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal) a) {x | ... | by
rwa [withDensity_one_sub_rnDerivAux κ η] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 15
} | [
{
"pp": "case refine_2.hX\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ninst✝¹ : Finite ι\nκ : ι → Type u_4\ninst✝ : ∀ (i : ι), Finite (κ i)\nX : (i : ι) → κ i → Ω → ℝ\nhX : HasGaussianLaw (fun ω i j ↦ X i j ω) P\nh : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 15
} | [
{
"pp": "case refine_2.hY\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ninst✝¹ : Finite ι\nκ : ι → Type u_4\ninst✝ : ∀ (i : ι), Finite (κ i)\nX : (i : ι) → κ i → Ω → ℝ\nhX : HasGaussianLaw (fun ω i j ↦ X i j ω) P\nh : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 363,
"column": 6
} | {
"line": 364,
"column": 28
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\ns : Set γ\nhsm : MeasurableSet s\nhs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ\nthis :\n η.wi... | · rw [ne_eq, sub_eq_zero]
exact (hs' x hx).ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Condexp | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | [
{
"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)",
"usedConstants": [
"MeasurableSpace.prod",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | [
{
"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)",
"usedConstants": [
"MeasurableSpace.prod",
"PseudoMetricSpace.toUniformSpace",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 34
} | [
{
"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)",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"ProbabilityTheory.Kernel.comap",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 15
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : Complete... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 22
} | [
{
"pp": "case inl\nΩ : Type u_1\nm mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns : Set Ω\nhs : MeasurableSet s\nh : IsEmpty Ω\nthis : μ = 0\n⊢ (fun ω ↦ ((condExpKernel μ m) ω).real s) =ᶠ[ae μ] μ[s.indicator fun ω ↦ 1 | m ⊓ mΩ]",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 22
} | [
{
"pp": "case inl\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : NormedAddCommGroup F\nf : Ω → F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nhf_int : Integrable f μ\nh : IsEmpty Ω\nthis : μ = 0\n⊢ μ[f | m ⊓ mΩ] =ᶠ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 52
} | [
{
"pp": "case inr\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : NormedAddCommGroup F\nf : Ω → F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nhf_int : Integrable f μ\nh✝ : Nonempty Ω\nhX : Measurable id\nh : μ[f | ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Deterministic | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 26
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nh : IsDeterministic κ\na : α\n⊢ IsZeroOneMeasure (κ a)",
"usedConstants": [
"MeasureTheory.IsZeroOneMeasure.mk",
"MeasureTheory.Measure",
"MeasurableSet",... | refine ⟨fun s hs ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Independence.ZeroOne | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 9
} | [
{
"pp": "Ω : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nt : Set Ω\nh_indep : IndepSet t t μ\n⊢ μ t = 0 ∨ μ t = 1 ∨ μ t = ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 51
} | [
{
"pp": "case h\nα : 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 ∨... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 9
} | [
{
"pp": "Ω : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nt : Set Ω\nh_indep : IndepSet t t μ\n⊢ μ t = 0 ∨ μ t = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Deterministic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_3\ninst✝³ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel β γ\ninst✝² : IsMarkovKernel κ\ninst✝¹ : IsMarkovKernel η\ninst✝ : IsDeterministic (η ∘ₖ κ)\na : α\ns : Set γ\nt : Set β\nhs : MeasurableSet s\nht : M... | rw [comp_apply' _ _ _ (hs.prod ht)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.ZeroOne | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 9
} | [
{
"pp": "Ω : Type u_2\nι : Type u_3\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nβ : Type u_4\np : Set ι → Prop\nf : Filter ι\nns : β → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s μ\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x1 x2 ↦ x1 ≤ x2) ns\nhnsp : ∀ (a : β), p (n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Invariance | {
"line": 71,
"column": 50
} | {
"line": 71,
"column": 77
} | [
{
"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",
"usedConstants": [
"MeasureTheory.lintegral_const",
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 9
} | [
{
"pp": "Ω : Type u_2\nι : Type u_3\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s μ\nt : Set Ω\nht_tail : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 339,
"column": 2
} | {
"line": 340,
"column": 9
} | [
{
"pp": "Ω : Type u_2\nι : Type u_3\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s μ\nt : Set Ω\nht_tail : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ProbabilityTheory.Kernel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ProbabilityTheory.Kernel.instMonoid",
"MeasureTheory... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Conditional | {
"line": 885,
"column": 6
} | {
"line": 885,
"column": 17
} | [
{
"pp": "case e_f.e_a\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 95,
"column": 4
} | {
"line": 96,
"column": 48
} | [
{
"pp": "X : SFinKer\nf₁ : X.carrier → PUnit.{?u.17416 + 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",
"usedCons... | refine ⟨⟨Kernel.id.map Prod.snd, inferInstance⟩,
⟨Kernel.id.map f₁, inferInstance⟩, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Posterior | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteKernel κ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : Nonempty Ω\n⊢ (⇑κ ∘ₘ μ) ⊗ₘ κ†μ = Measure.map Prod.swap (μ ⊗ₘ κ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 153,
"column": 28
} | {
"line": 153,
"column": 66
} | [
{
"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.id ∥ₖ Kernel.deterministic f hf ∘ₖ (Kern... | Kernel.parallelComp_comp_parallelComp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 155,
"column": 28
} | {
"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... | Kernel.parallelComp_self_comp_copy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Posterior | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 59,
"column": 26
} | {
"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... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 69,
"column": 28
} | {
"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... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | [
{
"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 • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | [
{
"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",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 78,
"column": 6
} | {
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory... | ← lintegral_indicator (h𝓑𝓧 _ hB) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 102,
"column": 4
} | {
"line": 105,
"column": 22
} | [
{
"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
} | [
{
"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.Tilted | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 20
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | [
{
"pp": "case h\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\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⊢ A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 21
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 49
} | [
{
"pp": "case h\nT : 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) =ᶠ[ae P] 0\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 211,
"column": 6
} | {
"line": 211,
"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\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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 49
} | [
{
"pp": "case h\nT : 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) =ᶠ[ae P] 0\nω : Ω\nhω : edist (X s ω) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 36
} | [
{
"pp": "case h\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 : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ IsFiniteMeasu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 19
} | [
{
"pp": "case h\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 : IsFiniteMeasure (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ (κ ω').real Set.univ ≤ 1",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 26
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : IsZeroOrMarkovKernel κ\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf (fun x ↦ 0) (κ ω') t ≤ rexp (↑0 * t ^ 2 / 2)",
"usedConstants": [
"MeasureTheory.ae",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 265,
"column": 29
} | {
"line": 265,
"column": 40
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (-X) ω)) (⇑κ ∘ₘ ν)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 80
} | [
{
"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)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 67
} | [
{
"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)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 128,
"column": 10
} | {
"line": 128,
"column": 22
} | [
{
"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... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 130,
"column": 8
} | {
"line": 130,
"column": 68
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ.toSubmodule.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 64
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] M\ng : G\nv : RestrictScalars A A[G] M\nhv : v ∈ { toAddSubmon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 154,
"column": 27
} | {
"line": 154,
"column": 38
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nN : Submodule A[G] ρ.asModule\na : A\nw : W\nhw : w ∈ N.carrier\n⊢ a • w ∈ N.carr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 156,
"column": 70
} | {
"line": 156,
"column": 85
} | [
{
"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 �... | 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
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 438,
"column": 10
} | {
"line": 438,
"column": 21
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 237,
"column": 2
} | {
"line": 246,
"column": 9
} | [
{
"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... | 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
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 59
} | {
"line": 613,
"column": 70
} | [
{
"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 | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 89
} | [
{
"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 | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"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
} | [
{
"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
} | [
{
"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
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 44
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Real",
"Pi.instNeg",
"_private.Mathlib.Probability.Moments.SubGaussian.0.ProbabilityTheory.Ha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Action | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 57
} | [
{
"pp": "case h.a.h.h.h\nG : Type v\ninst✝¹ : Monoid G\nX : Action (Type w) G\nk : Type u\ninst✝ : CommSemiring k\nx1 : X.V\n⊢ ((TensorProduct.AlgebraTensorModule.curry (↑(lid k (linearize k G X))).toLinearMap) 1 ∘ₗ Finsupp.lsingle x1) 1 =\n ((TensorProduct.AlgebraTensorModule.curry\n (((linearize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 666,
"column": 8
} | {
"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)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HM... | mgf_id_map hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Action | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 15
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 693,
"column": 2
} | {
"line": 693,
"column": 13
} | [
{
"pp": "case h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ t ∈ integrableExpSet X μ ↔ t ∈ Set.univ",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"iff_true",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 707,
"column": 2
} | {
"line": 707,
"column": 13
} | [
{
"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))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 13
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᶠ[ae μ] 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.AlgebraRepresentation.Basic | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 59
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 44
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ HasSubgaussianMGF ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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