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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
⊢ ∫⁻ (a : β) in t, (η a) s ∂κ a = ((η ∘ₖ κ) a) s * (κ a) t

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
⊢ ∫⁻ (a : β) in t, (η a) s ∂κ a = ((η ∘ₖ κ) a) s * (κ a) t
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
⊢ s.pi t = s.pi fun i ↦ if i ∈ s then t i else Set.univ

Tactic:
]

Goal state after:
ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
⊢ s.pi t = s.pi fun i ↦ if i ∈ s then t i else Set.univ
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : IsMarkovKernel κ
⊢ μ = μ

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
this : ⇑Kernel.id =ᵐ[μ] ⇑(Kernel.id†μ)
⊢ ⇑(Kernel.id†μ) =ᵐ[μ] ⇑Kernel.id

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
this : ⇑Kernel.id =ᵐ[μ] ⇑(Kernel.id†μ)
⊢ ⇑(Kernel.id†μ) =ᵐ[μ] ⇑Kernel.id
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case ha
ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
this : s.pi t = ⋂ s', (Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ
⊢ Tendsto (fun s' ↦ (infinitePi μ) ((Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ)) atTop
    (𝓝 ((infinitePi μ) (⋂ s', (Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ)))

Tactic:
]

Goal state after:
case ha
ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
this : s.pi t = ⋂ s', (Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ
⊢ Tendsto (fun s' ↦ (infinitePi μ) ((Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ)) atTop
    (𝓝 ((infinitePi μ) (⋂ s', (Subtype.val '' ↑s').pi fun i ↦ if i ∈ s then t i else Set.univ)))
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
⊢ ⇑(Kernel.copy Ω) ∘ₘ μ = ⇑(Kernel.copy Ω) ∘ₘ μ

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.deterministic f hf ∘ₖ (Kernel.deterministic f hf)†μ) =ᵐ[Measure.map f μ] ⇑Kernel.id

Tactic:
refine Kernel.ae_eq_of_compProd_eq ?_

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ Measure.map f μ ⊗ₘ (Kernel.deterministic f hf ∘ₖ (Kernel.deterministic f hf)†μ) = Measure.map f μ ⊗ₘ Kernel.id
Module: Mathlib.Probability.Kernel.Deterministic

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₀ : ((η ∘ₖ κ) a) s = 0
⊢ ∀ᵐ (x : β) ∂κ a, x ∈ t → (η x) s = 0

Tactic:
]

Goal state after:
case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₀ : ((η ∘ₖ κ) a) s = 0
⊢ ∀ᵐ (x : β) ∂κ a, x ∈ t → (η x) s = 0
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ Measure.map f μ ⊗ₘ (Kernel.deterministic f hf ∘ₖ (Kernel.deterministic f hf)†μ) =
    Measure.map f μ ⊗ₘ (Kernel.deterministic f hf ∘ₖ (Kernel.deterministic f hf)†μ)

Tactic:
]

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₀ : (fun a ↦ (η a) s) =ᵐ[κ a] 0
⊢ ∀ᵐ (x : β) ∂κ a, x ∈ t → (η x) s = 0

Tactic:
]

Goal state after:
case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₀ : (fun a ↦ (η a) s) =ᵐ[κ a] 0
⊢ ∀ᵐ (x : β) ∂κ a, x ∈ t → (η x) s = 0
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ
      ⇑(Kernel.id ∥ₖ (Kernel.deterministic f hf)†μ) ∘ₘ ⇑(Kernel.copy 𝓧) ∘ₘ ⇑(Kernel.deterministic f hf) ∘ₘ μ =
    ⇑(Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ
      ⇑(Kernel.id ∥ₖ (Kernel.deterministic f hf)†μ) ∘ₘ ⇑(Kernel.copy 𝓧) ∘ₘ ⇑(Kernel.deterministic f hf) ∘ₘ μ

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
⊢ ∫⁻ (a : β) in t, (η a) s ∂κ a = (κ a) t

Tactic:
]

Goal state after:
case inr
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
⊢ ∫⁻ (a : β) in t, (η a) s ∂κ a = (κ a) t
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
x : (i : ↥(Iic a)) → X ↑i
s✝ : Set ((i : ↥(Iic b)) → X ↑i)
a✝ : MeasurableSet s✝
⊢ ((partialTraj κ a b) x) s✝ = ((partialTraj κ a b) x) s✝

pp: case hf
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
x : (i : ↥(Iic a)) → X ↑i
s✝ : Set ((i : ↥(Iic b)) → X ↑i)
a✝ : MeasurableSet s✝
⊢ Measurable (frestrictLe b)

Tactic:
]

Goal state after:
case hf
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
x : (i : ↥(Iic a)) → X ↑i
s✝ : Set ((i : ↥(Iic b)) → X ↑i)
a✝ : MeasurableSet s✝
⊢ Measurable (frestrictLe b)
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.deterministic f hf ∥ₖ Kernel.id) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ =
    ⇑(Kernel.id ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.deterministic f hf ∥ₖ Kernel.id) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
b : β
⊢ (fun b ↦ (η b) s) b ≤ (fun a ↦ (η b) univ) b

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
b : β
⊢ (fun b ↦ (η b) s) b ≤ (fun a ↦ (η b) univ) b
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
x : (i : ↥(Iic a)) → X ↑i
⊢ (partialTraj κ a b) x = (partialTraj κ a b) x

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
⊢ (κ a) t = (κ a) t

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
⊢ deterministic (frestrictLe₂ ?m.67) ⋯ = deterministic (frestrictLe₂ hab) ⋯

pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a b : ℕ
hab : a ≤ b
⊢ a ≤ b

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.deterministic f hf ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ =
    ⇑(Kernel.deterministic f hf ∥ₖ Kernel.deterministic f hf) ∘ₘ ⇑(Kernel.copy Ω) ∘ₘ μ

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.copy 𝓧 ∘ₖ Kernel.deterministic f hf) ∘ₘ μ = ⇑(Kernel.copy 𝓧 ∘ₖ Kernel.deterministic f hf) ∘ₘ μ

Tactic:
]

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

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
integral_le_kernel : ∫⁻ (b : β) in t, (η b) s ∂κ a ≤ (κ a) t
⊢ (κ a) t - ∫⁻ (a : β) in t, (η a) s ∂κ a ≤ 0

Tactic:
]

Goal state after:
case inr
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
integral_le_kernel : ∫⁻ (b : β) in t, (η b) s ∂κ a ≤ (κ a) t
⊢ (κ a) t - ∫⁻ (a : β) in t, (η a) s ∂κ a ≤ 0
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
inst✝ : MeasurableSpace.CountablyGenerated 𝓧
f : Ω → 𝓧
hf : Measurable f
⊢ ⇑(Kernel.copy 𝓧) ∘ₘ Measure.map f μ = ⇑(Kernel.copy 𝓧) ∘ₘ Measure.map f μ

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_3
inst✝³ : MeasurableSpace γ
κ : Kernel α β
η : Kernel β γ
inst✝² : IsMarkovKernel κ
inst✝¹ : IsMarkovKernel η
inst✝ : IsDeterministic (η ∘ₖ κ)
a : α
s : Set γ
t : Set β
hs : MeasurableSet s
ht : MeasurableSet t
this : ∀ (b : β), (η b) s * t.indicator 1 b = t.indicator (fun b ↦ (η b) s) b
h₁ : ((η ∘ₖ κ) a) s = 1
integral_le_kernel : ∫⁻ (b : β) in t, (η b) s ∂κ a ≤ (κ a) t
⊢ (κ a) t - ∫⁻ (a : β) in t, (η a) s ∂κ a = (κ a) t - ∫⁻ (b : β) in t, (η b) s ∂κ a

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a : ℕ
hf : (fun x ↦ x (a + 1)) = (fun x ↦ x ⟨a + 1, ⋯⟩) ∘ frestrictLe (a + 1)
⊢ κ a = κ a

Tactic:
]

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

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
this : (⇑κ ∘ₘ μ) ⊗ₘ κ†μ ≪ ν ⊗ₘ Kernel.const 𝓧 μ
⊢ ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) b ≪ μ

Tactic:
]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
this : (⇑κ ∘ₘ μ) ⊗ₘ κ†μ ≪ ν ⊗ₘ Kernel.const 𝓧 μ
⊢ ∀ᵐ (b : 𝓧) ∂⇑κ ∘ₘ μ, (κ†μ) b ≪ μ
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case hb
ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
⊢ Tendsto (fun s' ↦ ∏ i ∈ s', (μ ↑i) (t ↑i)) atTop (𝓝 (⨅ s_1, ∏ i ∈ s_1, (μ ↑i) (t ↑i)))

Tactic:
]

Goal state after:
case hb
ι✝ : Type u_1
X✝ : ι✝ → Type u_2
mX✝ : (i : ι✝) → MeasurableSpace (X✝ i)
μ✝ : (i : ι✝) → Measure (X✝ i)
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
s : Set ι
hs : Countable ↑s
t : (i : ι) → Set (X i)
mt : ∀ i ∈ s, MeasurableSet (t i)
s_ne : Nonempty ↑s
⊢ Tendsto (fun s' ↦ ∏ i ∈ s', (μ ↑i) (t ↑i)) atTop (𝓝 (⨅ s_1, ∏ i ∈ s_1, (μ ↑i) (t ↑i)))
Module: Mathlib.Probability.Kernel.IonescuTulcea.Traj

Goal state before:
pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a n : ℕ
η : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)
hη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b
x : (i : ↥(Iic a)) → X ↑i
⊢ ∀ ⦃n_1 : ℕ⦄, n ≤ n_1 → Measure.map (frestrictLe n_1) (η x) = inducedFamily (fun n ↦ (partialTraj κ a n) x) (Iic n_1)

pp: X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a n : ℕ
η : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)
hη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b
x : (i : ↥(Iic a)) → X ↑i
⊢ IsProjectiveMeasureFamily (inducedFamily fun n ↦ (partialTraj κ a n) x)

Tactic:
]

Goal state after:
X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a n : ℕ
η : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)
hη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b
x : (i : ↥(Iic a)) → X ↑i
⊢ ∀ ⦃n_1 : ℕ⦄, n ≤ n_1 → Measure.map (frestrictLe n_1) (η x) = inducedFamily (fun n ↦ (partialTraj κ a n) x) (Iic n_1)

X : ℕ → Type u_1
inst✝¹ : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
a n : ℕ
η : Kernel ((i : ↥(Iic a)) → X ↑i) ((n : ℕ) → X n)
hη : ∀ b ≥ n, η.map (frestrictLe b) = partialTraj κ a b
x : (i : ↥(Iic a)) → X ↑i
⊢ IsProjectiveMeasureFamily (inducedFamily fun n ↦ (partialTraj κ a n) x)
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
this : μ ⊗ₘ κ ≪ μ.prod ν
⊢ Measure.map Prod.swap (μ ⊗ₘ κ) ≪ Measure.map Prod.swap (μ.prod ν)

Tactic:
]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
this : μ ⊗ₘ κ ≪ μ.prod ν
⊢ Measure.map Prod.swap (μ ⊗ₘ κ) ≪ Measure.map Prod.swap (μ.prod ν)
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
⊢ μ ⊗ₘ κ ≪ μ ⊗ₘ Kernel.const Ω ν

Tactic:
]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁴ : IsFiniteMeasure μ
inst✝³ : IsFiniteKernel κ
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
ν : Measure 𝓧
inst✝ : SFinite ν
h_ac : ∀ᵐ (ω : Ω) ∂μ, κ ω ≪ ν
⊢ μ ⊗ₘ κ ≪ μ ⊗ₘ Kernel.const Ω ν
Module: Mathlib.Probability.Kernel.Posterior

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁶ : IsFiniteMeasure μ
inst✝⁵ : IsFiniteKernel κ
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : StandardBorelSpace 𝓧
inst✝¹ : Nonempty 𝓧
inst✝ : IsMarkovKernel κ
this : ⇑κ =ᵐ[μ] ⇑((κ†μ)†(⇑κ ∘ₘ μ))
⊢ ⇑((κ†μ)†(⇑κ ∘ₘ μ)) =ᵐ[μ] ⇑κ

Tactic:
]

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
κ : Kernel Ω 𝓧
μ : Measure Ω
inst✝⁶ : IsFiniteMeasure μ
inst✝⁵ : IsFiniteKernel κ
inst✝⁴ : StandardBorelSpace Ω
inst✝³ : Nonempty Ω
inst✝² : StandardBorelSpace 𝓧
inst✝¹ : Nonempty 𝓧
inst✝ : IsMarkovKernel κ
this : ⇑κ =ᵐ[μ] ⇑((κ†μ)†(⇑κ ∘ₘ μ))
⊢ ⇑((κ†μ)†(⇑κ ∘ₘ μ)) =ᵐ[μ] ⇑κ
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Countable ι
t : (i : ι) → Set (X i)
mt : ∀ (i : ι), MeasurableSet (t i)
⊢ ∏' (x : ι), (μ x) (t x) = ∏' (i : ι), (μ i) (t i)

pp: case hs
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Countable ι
t : (i : ι) → Set (X i)
mt : ∀ (i : ι), MeasurableSet (t i)
⊢ Countable ↑Set.univ

pp: case mt
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Countable ι
t : (i : ι) → Set (X i)
mt : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ i ∈ Set.univ, MeasurableSet (t i)

Tactic:
]

Goal state after:
case hs
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Countable ι
t : (i : ι) → Set (X i)
mt : ∀ (i : ι), MeasurableSet (t i)
⊢ Countable ↑Set.univ

case mt
ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝ : Countable ι
t : (i : ι) → Set (X i)
mt : ∀ (i : ι), MeasurableSet (t i)
⊢ ∀ i ∈ Set.univ, MeasurableSet (t i)
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: ι : Type u_1
X : ι → Type u_2
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
hμ : ∀ (i : ι), IsProbabilityMeasure (μ i)
inst✝¹ : Countable ι
inst✝ : ∀ (i : ι), MeasurableSingletonClass (X i)
f : (i : ι) → X i
⊢ ∏' (i : ι), (μ i) {f i} = ∏' (i : ι), (μ i) {f i}

Tactic:
]

Goal state after: