mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[fun x ↦ L₁ (X i x), fun x ↦ L₂ (X j x); P] = 0
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
ω : Ω
⊢ (L fun x ↦ X x ω) = ∑ i, (L ∘SL single ℝ E i) (X i ω)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
⊢ HasIndepIncrements B P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
⊢ Iic ↑↑i ⊆ Iic ↑↑j

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
law : ∀ (t : ℝ≥0), HasLaw (X t) (gaussianReal 0 t) P
incr : HasIndepIncrements X P
⊢ HasLaw (X 0) (gaussianReal 0 0) P

Tactic:
exact law 0

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
law : ∀ (t : ℝ≥0), HasLaw (X t) (gaussianReal 0 t) P
incr : HasIndepIncrements X P
⊢ ∀ᵐ (ω : Ω) ∂P, X 0 ω = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
law : ∀ (t : ℝ≥0), HasLaw (X t) (gaussianReal 0 t) P
incr : HasIndepIncrements X P
h0 : ∀ᵐ (ω : Ω) ∂P, X 0 ω = 0
s t : ℝ≥0
hst : s ≤ t
h1 : X s ⟂ᵢ[P] X t - X s
this : IsProbabilityMeasure P
⊢ X t = X t - X s + X s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (s ×ˢ Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
⊢ ∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
⊢ (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) = fun b ↦
    (ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x)).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∫⁻ (x : α) in s, preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
⊢ Integrable (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) (ν a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
inst✝ : IsFiniteMeasure ρ
⊢ ∫⁻ (x : α), preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) univ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (a : α) ∂ρ.fst, Monotone fun r ↦ preCDF ρ r a

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
⊢ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = (κ a).real (s ×ˢ Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
⊢ ∫ (b : β), ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = (κ a).real (univ ×ˢ Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
q : ℚ
⊢ Measurable fun t ↦ f (a, t) q

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
⊢ ∀ᵐ (a : α) ∂ρ.fst, ∀ (r : ℚ), preCDF ρ r a ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[fun x ↦ L₁ (X i x), fun x ↦ L₂ (X j x); P] = 0
this✝¹ : IsProbabilityMeasure P
this✝ : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this : ∀ (ω : Ω), (L fun x ↦ X x ω) = ∑ i, (L ∘SL single ℝ E i) (X i ω)
i : ι
⊢ i ∈ univ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
⊢ ∀ᵐ (c : β) ∂ν a, Monotone (f (a, c))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → ℚ
hseq : Antitone seq
hseq_tendsto : Tendsto seq atTop atBot
⊢ ∀ᵐ (c : β) ∂ν a, Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ∫ (x : α) in s, (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd ↑r).real s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
r : ℚ
inst✝ : IsFiniteMeasure ρ
⊢ ∫ (x : α), (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd ↑r).real univ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[⇑L₁ ∘ X i, ⇑L₂ ∘ X j; P] = 0
⊢ iIndepFun X P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
⊢ AEStronglyMeasurable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst

Tactic:
exact measurable_preCDF.ennreal_toReal.aestronglyMeasurable

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
law : ∀ (t : ℝ≥0), HasLaw (X t) (gaussianReal 0 t) P
incr : HasIndepIncrements X P
⊢ IsPreBrownianReal X P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → InnerProductSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (x : E i) (y : E j), cov[fun ω ↦ ⟪x, X i ω⟫, fun ω ↦ ⟪y, X j ω⟫; P] = 0
i j : ι
hij : i ≠ j
L₁ : StrongDual ℝ (E i)
L₂ : StrongDual ℝ (E j)
⊢ cov[⇑L₁ ∘ X i, ⇑L₂ ∘ X j; P] = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
t : Set α
x✝¹ : MeasurableSet t
x✝ : ρ.fst t ≠ ∞
⊢ ∀ᵐ (a : α) ∂ρ.fst, ENNReal.ofReal (preCDF ρ x a).toReal ≤ 1

Tactic:
filter_upwards [preCDF_le_one ρ] with a ha using ENNReal.ofReal_toReal_le.trans (ha _)

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
n : ℕ
t : Fin (n + 1) → ℝ≥0
ht : Monotone t
⊢ ∀ (i : Fin n), Measurable fun x ↦ -x

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
⊢ Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
⊢ Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst

Tactic:
by
  refine integrable_of_forall_fin_meas_le _ (measure_lt_top ρ.fst univ) ?_ fun t _ _ ↦ ?_
  · exact measurable_preCDF.ennreal_toReal.aestronglyMeasurable
  · simp_rw [← ofReal_norm, Real.norm_of_nonneg ENNReal.toReal_nonneg]
    rw [← lintegral_one]
    refine (setLIntegral_le_lintegral _ _).trans (lintegral_mono_ae ?_)
    filter_upwards [preCDF_le_one ρ] with a ha using ENNReal.ofReal_toReal_le.trans (ha _)

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
r r' : ℚ
hrr' : r ≤ r'
⊢ ∀ᵐ (c : α) ∂(Kernel.const Unit ρ.fst) a, (preCDF ρ r (a, c).2).toReal ≤ (preCDF ρ r' (a, c).2).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝¹ : Finite ι
κ : ι → Type u_4
inst✝ : ∀ (i : ι), Finite (κ i)
X : (i : ι) → κ i → Ω → ℝ
hX : HasGaussianLaw (fun ω i j ↦ X i j ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0
this : IsProbabilityMeasure P
⊢ (fun i ω j ↦ X i j ω) = fun i ↦ ofLp ∘ toLp 2 ∘ fun ω j ↦ X i j ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
⊢ IsPreBrownianReal (-B) P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x✝ : Unit
q : ℚ
⊢ ∀ᵐ (c : α) ∂(Kernel.const Unit ρ.fst) x✝, 0 ≤ (preCDF ρ q (x✝, c).2).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → ℚ
hseq : Monotone seq
hseq_tendsto : Tendsto seq atTop atTop
⊢ ∀ᵐ (c : β) ∂ν a, Tendsto (fun m ↦ f (a, c) (seq m)) atTop (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
q : ℚ
⊢ ∀ᵐ (c : α) ∂(Kernel.const Unit ρ.fst) a, (preCDF ρ q (a, c).2).toReal ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n ↦ f (a, t) ↑n) atTop (𝓝 1)
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (f (a, t)) atTop (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (f (a, t)) atTop (𝓝 1)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : ℕ → ℚ
x✝ : Antitone s
hs_tendsto : Tendsto s atTop atBot
⊢ Tendsto (fun m ↦ ∫ (c : α), (preCDF ρ (s m) (a, c).2).toReal ∂(Kernel.const Unit ρ.fst) a) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝¹ : Finite ι
κ : ι → Type u_4
inst✝ : ∀ (i : ι), Finite (κ i)
X : (i : ι) → κ i → Ω → ℝ
hX : HasGaussianLaw (fun ω i j ↦ X i j ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (k : κ i) (l : κ j), cov[X i k, X j l; P] = 0
this✝² : IsProbabilityMeasure P
this✝¹ : (fun i ω j ↦ X i j ω) = fun i ↦ ofLp ∘ toLp 2 ∘ fun ω j ↦ X i j ω
this✝ : (i : ι) → Fintype (κ i) := fun i ↦ Fintype.ofFinite (κ i)
this : Fintype ι := Fintype.ofFinite ι
⊢ ∀ (i : ι), Measurable ofLp

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun q ↦ f (a, t) (-q)) atTop (𝓝 0)
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (f (a, t)) atBot (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : ℕ → ℚ
x✝ : Monotone s
hs_tendsto : Tendsto s atTop atTop
⊢ Tendsto (fun m ↦ ∫ (c : α), (preCDF ρ (s m) (a, c).2).toReal ∂(Kernel.const Unit ρ.fst) a) atTop
    (𝓝 (((Kernel.const Unit ρ.fst) a).real univ))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : Set α
hs : MeasurableSet s
q : ℚ
⊢ ∫ (c : α) in s, (preCDF ρ q (a, c).2).toReal ∂(Kernel.const Unit ρ.fst) a =
    ((Kernel.const Unit ρ) a).real (s ×ˢ Iic ↑q)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ condCDF ρ a = stieltjesOfMeasurableRat (fun p r ↦ (preCDF ρ r p.2).toReal) ⋯ ((), a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
⊢ IsCondKernelCDF (fun p ↦ condCDF ρ p.2) (Kernel.const Unit ρ) (Kernel.const Unit ρ.fst)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
⊢ (fun a ↦ ↑(condCDF ρ a) ↑r) =ᵐ[ρ.fst] fun a ↦ (preCDF ρ r a).toReal

Tactic:
by

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
c : ℝ≥0
hc : c ≠ 0
⊢ IsPreBrownianReal (fun t ω ↦ (√↑c)⁻¹ * B (c * t) ω) P

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
this : ∀ᵐ (t : β) ∂ν a, Tendsto (fun n ↦ f (a, t) (-↑n)) atTop (𝓝 0)
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (fun q ↦ f (a, t) (-q)) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
⊢ (fun a ↦ ENNReal.ofReal (↑(condCDF ρ a) ↑r)) =ᵐ[ρ.fst] preCDF ρ r

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ᵐ (t : β) ∂ν a, Tendsto (f (a, t)) atBot (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
x : ℝ
⊢ (condCDF ρ a).measure (Iic x) = ENNReal.ofReal (↑(condCDF ρ a) x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
a : α
⊢ (condCDF ρ a).measure univ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
a : α
⊢ ∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), BddBelow (range fun r ↦ f (a, t) ↑r)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
⊢ ∀ s ∈ range Iic, Measurable fun a ↦ (condCDF ρ a).measure s

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ (ν a) univ ^ (ENNReal.toReal 1)⁻¹ * ENNReal.ofReal 1 ≤ (ν a) univ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsProbabilityMeasure μ
x : ℝ
⊢ ENNReal.ofReal (↑(cdf μ) x) = μ (Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsProbabilityMeasure μ
x : ℝ
⊢ ↑(cdf μ) x = μ.real (Iic x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ℝ
⊢ IsProbabilityMeasure (cdf μ).measure

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsProbabilityMeasure μ
⊢ (cdf μ).measure = μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: f : StieltjesFunction ℝ
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ f.measure univ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: f : StieltjesFunction ℝ
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ cdf f.measure = f

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ↑I
inst✝ : IsProbabilityMeasure μ
x : ↑I
⊢ AEMeasurable Subtype.val μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
t : β
hbdd_below : ∀ (q : ℚ), BddBelow (range fun r ↦ f (a, t) ↑r)
h_nonneg : ∀ (q : ℚ), 0 ≤ f (a, t) q
h_le_one : ∀ (q : ℚ), f (a, t) q ≤ 1
⊢ q + 1 ∈ Ioi q

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ : Measure ↑I
inst✝ : IsProbabilityMeasure μ
x : ↑I
⊢ ↑(cdf (Measure.map Subtype.val μ)) ↑x = μ.real (Icc 0 x)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ ν : Measure ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
h : cdf μ = cdf ν
⊢ μ = ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.CDF

Goal state before:
pp: μ ν : Measure ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsProbabilityMeasure ν
h : μ = ν
⊢ cdf μ = cdf ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
⊢ Integrable (fun t ↦ ⨅ r, f (a, t) ↑r) (ν a)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
r r' : { r' // t < r' }
hrr' : r ≤ r'
⊢ Iic ↑↑r ⊆ Iic ↑↑r'

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
t : ℚ
⊢ ⨅ r, ρ (s ×ˢ Iic ↑↑r) = ρ (s ×ˢ Iic ↑t)

Tactic:
by

Goal state after: