mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case iUnion.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
this : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
f : ℕ → Set γ
hf_disj : Pairwise (Function.onFun Disjoint f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), ∫ (x : γ) in f i, κ.density ν a x s ∂ν a = (κ a).real (f i ×ˢ s)
⊢ ∀ (i : ℕ), MeasurableSet (f i ×ˢ s)

Tactic:
·

Goal state after:
case iUnion.h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
this : IsFiniteKernel κ
hgen : mγ = generateFrom {s | ∃ n, MeasurableSet s}
hpi : IsPiSystem {s | ∃ n, MeasurableSet s}
f : ℕ → Set γ
hf_disj : Pairwise (Function.onFun Disjoint f)
hf : ∀ (i : ℕ), MeasurableSet (f i)
h_eq : ∀ (i : ℕ), ∫ (x : γ) in f i, κ.density ν a x s ∂ν a = (κ a).real (f i ×ˢ s)
⊢ ∀ (i : ℕ), MeasurableSet (f i ×ˢ s)
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
x : ((i : ↥(Iic a)) → X ↑i) × ((i : ↥(Ioc a b)) → X ↑i)
x✝ :
  (∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩) ∧
    ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
i : ι
hi : i ≤ b
h1 : ∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
h2 : ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
hi3 : i ≤ a
⊢ x.1 ⟨i, ⋯⟩ ∈ s ⟨i, ⋯⟩

pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
x : ((i : ↥(Iic a)) → X ↑i) × ((i : ↥(Ioc a b)) → X ↑i)
x✝ :
  (∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩) ∧
    ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
i : ι
hi : i ≤ b
h1 : ∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
h2 : ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
hi3 : ¬i ≤ a
⊢ x.2 ⟨i, ⋯⟩ ∈ s ⟨i, ⋯⟩

Tactic:
·

Goal state after:
case pos
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
x : ((i : ↥(Iic a)) → X ↑i) × ((i : ↥(Ioc a b)) → X ↑i)
x✝ :
  (∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩) ∧
    ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
i : ι
hi : i ≤ b
h1 : ∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
h2 : ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
hi3 : i ≤ a
⊢ x.1 ⟨i, ⋯⟩ ∈ s ⟨i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.Maps

Goal state before:
pp: case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
x : ((i : ↥(Iic a)) → X ↑i) × ((i : ↥(Ioc a b)) → X ↑i)
x✝ :
  (∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩) ∧
    ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
i : ι
hi : i ≤ b
h1 : ∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
h2 : ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
hi3 : ¬i ≤ a
⊢ x.2 ⟨i, ⋯⟩ ∈ s ⟨i, ⋯⟩

Tactic:
·

Goal state after:
case neg
ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : DecidableLE ι
X : ι → Type u_2
inst✝ : LocallyFiniteOrderBot ι
a b : ι
hab : a ≤ b
s : (i : ↥(Iic b)) → Set (X ↑i)
x : ((i : ↥(Iic a)) → X ↑i) × ((i : ↥(Ioc a b)) → X ↑i)
x✝ :
  (∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩) ∧
    ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
i : ι
hi : i ≤ b
h1 : ∀ (a_1 : ι) (b_1 : a_1 ≤ a), x.1 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
h2 : ∀ (a_1 : ι) (b_1 : a < a_1 ∧ a_1 ≤ b), x.2 ⟨a_1, ⋯⟩ ∈ s ⟨a_1, ⋯⟩
hi3 : ¬i ≤ a
⊢ x.2 ⟨i, ⋯⟩ ∈ s ⟨i, ⋯⟩
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab : a ≤ a
⊢ partialTraj κ a a =
    Nat.leRec Kernel.id
      (fun k x κ_k ↦ (Kernel.id ×ₖ (κ k).map ⇑(piSingleton k) ∘ₖ κ_k).map (_root_.IicProdIoc k (k + 1))) hab

pp: case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab✝ : a ≤ b
hab : a < b
⊢ partialTraj κ a b =
    Nat.leRec Kernel.id
      (fun k x κ_k ↦ (Kernel.id ×ₖ (κ k).map ⇑(piSingleton k) ∘ₖ κ_k).map (_root_.IicProdIoc k (k + 1))) hab✝

Tactic:
·

Goal state after:
case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab : a ≤ a
⊢ partialTraj κ a a =
    Nat.leRec Kernel.id
      (fun k x κ_k ↦ (Kernel.id ×ₖ (κ k).map ⇑(piSingleton k) ∘ₖ κ_k).map (_root_.IicProdIoc k (k + 1))) hab
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab✝ : a ≤ b
hab : a < b
⊢ partialTraj κ a b =
    Nat.leRec Kernel.id
      (fun k x κ_k ↦ (Kernel.id ×ₖ (κ k).map ⇑(piSingleton k) ∘ₖ κ_k).map (_root_.IicProdIoc k (k + 1))) hab✝

Tactic:
·

Goal state after:
case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab✝ : a ≤ b
hab : a < b
⊢ partialTraj κ a b =
    Nat.leRec Kernel.id
      (fun k x κ_k ↦ (Kernel.id ×ₖ (κ k).map ⇑(piSingleton k) ∘ₖ κ_k).map (_root_.IicProdIoc k (k + 1))) hab✝
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ENNReal.ofReal (∫ (x : γ) in A, κ.density ν a x s ∂ν a) = (κ a) (A ×ˢ s)

pp: case hfi
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ Integrable (fun x ↦ κ.density ν a x s) ((ν a).restrict A)

pp: case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᵐ[(ν a).restrict A] fun x ↦ κ.density ν a x s

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ ENNReal.ofReal (∫ (x : γ) in A, κ.density ν a x s ∂ν a) = (κ a) (A ×ˢ s)
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ Integrable ((Y ⁻¹' s).indicator fun ω ↦ 1) μ

pp: case refine_2
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → μ s_1 < ∞ → IntegrableOn (fun a ↦ ((condDistrib Y X μ) (X a)).real s) s_1 μ

pp: case refine_3
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      μ s_1 < ∞ →
        ∫ (x : α) in s_1, ((condDistrib Y X μ) (X x)).real s ∂μ = ∫ (x : α) in s_1, (Y ⁻¹' s).indicator (fun ω ↦ 1) x ∂μ

pp: case refine_4
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun a ↦ ((condDistrib Y X μ) (X a)).real s) μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ Integrable ((Y ⁻¹' s).indicator fun ω ↦ 1) μ
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a b : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
hab : a ≤ b
⊢ ¬b + 1 ≤ a

Tactic:
by lia

Goal state after:
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → μ s_1 < ∞ → IntegrableOn (fun a ↦ ((condDistrib Y X μ) (X a)).real s) s_1 μ

pp: case refine_3
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      μ s_1 < ∞ →
        ∫ (x : α) in s_1, ((condDistrib Y X μ) (X x)).real s ∂μ = ∫ (x : α) in s_1, (Y ⁻¹' s).indicator (fun ω ↦ 1) x ∂μ

pp: case refine_4
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun a ↦ ((condDistrib Y X μ) (X a)).real s) μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α), MeasurableSet s_1 → μ s_1 < ∞ → IntegrableOn (fun a ↦ ((condDistrib Y X μ) (X a)).real s) s_1 μ
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case hfi
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ Integrable (fun x ↦ κ.density ν a x s) ((ν a).restrict A)

pp: case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᵐ[(ν a).restrict A] fun x ↦ κ.density ν a x s

Tactic:
·

Goal state after:
case hfi
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ Integrable (fun x ↦ κ.density ν a x s) ((ν a).restrict A)
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᵐ[(ν a).restrict A] fun x ↦ κ.density ν a x s

Tactic:
·

Goal state after:
case f_nn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
s : Set β
hs : MeasurableSet s
A : Set γ
hA : MeasurableSet A
this : IsFiniteKernel κ
⊢ 0 ≤ᵐ[(ν a).restrict A] fun x ↦ κ.density ν a x s
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      μ s_1 < ∞ →
        ∫ (x : α) in s_1, ((condDistrib Y X μ) (X x)).real s ∂μ = ∫ (x : α) in s_1, (Y ⁻¹' s).indicator (fun ω ↦ 1) x ∂μ

pp: case refine_4
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun a ↦ ((condDistrib Y X μ) (X a)).real s) μ

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      μ s_1 < ∞ →
        ∫ (x : α) in s_1, ((condDistrib Y X μ) (X x)).real s ∂μ = ∫ (x : α) in s_1, (Y ⁻¹' s).indicator (fun ω ↦ 1) x ∂μ
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ∞} ∈ 𝓝 ((κ a) univ)

pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)
⊢ Tendsto (fun m ↦ (κ a).real (univ ×ˢ seq m)) atTop (𝓝 ((κ a).real univ))

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ {a | a ≠ ∞} ∈ 𝓝 ((κ a) univ)
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case refine_1.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1

pp: case refine_1.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ ∈ Iio ((κ a) univ + 1)

Tactic:
·

Goal state after:
case refine_1.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
x : ℝ≥0∞
hx : x ∈ Iio ((κ a) univ + 1)
⊢ x < (κ a) univ + 1
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case refine_1.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ ∈ Iio ((κ a) univ + 1)

Tactic:
·

Goal state after:
case refine_1.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
a : α
seq : ℕ → Set β
hseq : Monotone seq
hseq_iUnion : ⋃ i, seq i = univ
hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)
this : IsFiniteKernel κ
⊢ (κ a) univ ∈ Iio ((κ a) univ + 1)
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a✝ b✝ c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
a b : ℕ
hab : a ≤ b
⊢ IsSFiniteKernel (partialTraj κ a b)

pp: case inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a✝ b✝ c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
a b : ℕ
hba : b ≤ a
⊢ IsSFiniteKernel (partialTraj κ a b)

Tactic:
·

Goal state after:
case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a✝ b✝ c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
a b : ℕ
hab : a ≤ b
⊢ IsSFiniteKernel (partialTraj κ a b)
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case refine_4
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun a ↦ ((condDistrib Y X μ) (X a)).real s) μ

Tactic:
·

Goal state after:
case refine_4
α : Type u_1
β : Type u_2
Ω : Type u_3
inst✝³ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
s : Set Ω
hX : Measurable X
hY : Measurable Y
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun a ↦ ((condDistrib Y X μ) (X a)).real s) μ
Module: Mathlib.Probability.Kernel.CondDistrib

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hX : Measurable X
hY : AEMeasurable Y μ
hf_int : Integrable f (Measure.map (fun a ↦ (X a, Y a)) μ)
hf_int' : Integrable (fun a ↦ f (X a, Y a)) μ
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s < ∞
⊢ IntegrableOn (fun a ↦ ∫ (y : Ω), f (X a, y) ∂(condDistrib Y X μ) (X a)) s μ

pp: case refine_2
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hX : Measurable X
hY : AEMeasurable Y μ
hf_int : Integrable f (Measure.map (fun a ↦ (X a, Y a)) μ)
hf_int' : Integrable (fun a ↦ f (X a, Y a)) μ
⊢ ∀ (s : Set α),
    MeasurableSet s →
      μ s < ∞ → ∫ (x : α) in s, ∫ (y : Ω), f (X x, y) ∂(condDistrib Y X μ) (X x) ∂μ = ∫ (x : α) in s, f (X x, Y x) ∂μ

pp: case refine_3
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hX : Measurable X
hY : AEMeasurable Y μ
hf_int : Integrable f (Measure.map (fun a ↦ (X a, Y a)) μ)
hf_int' : Integrable (fun a ↦ f (X a, Y a)) μ
⊢ AEStronglyMeasurable (fun a ↦ ∫ (y : Ω), f (X a, y) ∂(condDistrib Y X μ) (X a)) μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup F
mα : MeasurableSpace α
μ : Measure α
inst✝² : IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hX : Measurable X
hY : AEMeasurable Y μ
hf_int : Integrable f (Measure.map (fun a ↦ (X a, Y a)) μ)
hf_int' : Integrable (fun a ↦ f (X a, Y a)) μ
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s < ∞
⊢ IntegrableOn (fun a ↦ ∫ (y : Ω), f (X a, y) ∂(condDistrib Y X μ) (X a)) s μ