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

Goal state before:
pp: X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ ∫⁻ (a_1 : X (a + 1)),
      f
        (updateFinset x₀ (Iic (a + 1))
          (_root_.IicProdIoc a (a + 1) (frestrictLe a x₀, (piSingleton a) a_1))) ∂(κ a) (frestrictLe a x₀) =
    ∫⁻ (x : X (a + 1)), f (update x₀ (a + 1) x) ∂(κ a) (frestrictLe a x₀)

pp: case hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ Measurable ⇑(piSingleton a)

pp: case hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ Measurable fun b ↦ f (updateFinset x₀ (Iic (a + 1)) (_root_.IicProdIoc a (a + 1) (frestrictLe a x₀, b)))

pp: case hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ Measurable fun a_1 ↦ f (updateFinset x₀ (Iic (a + 1)) (_root_.IicProdIoc a (a + 1) a_1))

pp: case hf
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ Measurable (_root_.IicProdIoc a (a + 1))

pp: case hg
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ Measurable fun z ↦ f (updateFinset x₀ (Iic (a + 1)) z)

Tactic:
·

Goal state after:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
x₀ : (n : ℕ) → X n
⊢ ∫⁻ (a_1 : X (a + 1)),
      f
        (updateFinset x₀ (Iic (a + 1))
          (_root_.IicProdIoc a (a + 1) (frestrictLe a x₀, (piSingleton a) a_1))) ∂(κ a) (frestrictLe a x₀) =
    ∫⁻ (x : X (a + 1)), f (update x₀ (a + 1) x) ∂(κ a) (frestrictLe a x₀)
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (η.withDensity (κ.rnDeriv η)) a = 0
⊢ (η.withDensity (κ.rnDeriv η)) a + (κ.singularPart η) a ⟂ₘ η a

pp: case refine_2
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (η.withDensity (κ.rnDeriv η)) a + (κ.singularPart η) a ⟂ₘ η a
⊢ (η.withDensity (κ.rnDeriv η)) a = 0

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : (η.withDensity (κ.rnDeriv η)) a = 0
⊢ (η.withDensity (κ.rnDeriv η)) a + (κ.singularPart η) a ⟂ₘ η a
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ ∫⁻ (x : γ), κ.rnDeriv η a x ∂η a ≤ κ.bound

Tactic:
·

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
hκ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ Measurable (Function.uncurry (κ.rnDeriv η))
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inl.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))
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab hbc : a ≤ a
⊢ lmarginalPartialTraj κ a a (lmarginalPartialTraj κ a a f) x₀ = lmarginalPartialTraj κ a a f x₀

pp: case inl.inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a ≤ a
hbc✝ : a ≤ c
hbc : a < c
⊢ lmarginalPartialTraj κ a a (lmarginalPartialTraj κ a c f) x₀ = lmarginalPartialTraj κ a c f x₀

pp: case inr.inl
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
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b ≤ b
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b b f) x₀ = lmarginalPartialTraj κ a b f x₀

pp: case inr.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))
hab✝ : a ≤ b
hbc✝ : b ≤ c
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b < c
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b c f) x₀ = lmarginalPartialTraj κ a c f x₀

Tactic:
·

Goal state after:
case inl.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))
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab hbc : a ≤ a
⊢ lmarginalPartialTraj κ a a (lmarginalPartialTraj κ a a f) x₀ = lmarginalPartialTraj κ a a f x₀
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inl.inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a ≤ a
hbc✝ : a ≤ c
hbc : a < c
⊢ lmarginalPartialTraj κ a a (lmarginalPartialTraj κ a c f) x₀ = lmarginalPartialTraj κ a c f x₀

pp: case inr.inl
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
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b ≤ b
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b b f) x₀ = lmarginalPartialTraj κ a b f x₀

pp: case inr.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))
hab✝ : a ≤ b
hbc✝ : b ≤ c
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b < c
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b c f) x₀ = lmarginalPartialTraj κ a c f x₀

Tactic:
·

Goal state after:
case inl.inr
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
a c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a ≤ a
hbc✝ : a ≤ c
hbc : a < c
⊢ lmarginalPartialTraj κ a a (lmarginalPartialTraj κ a c f) x₀ = lmarginalPartialTraj κ a c f x₀
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inr.inl
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
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b ≤ b
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b b f) x₀ = lmarginalPartialTraj κ a b f x₀

pp: case inr.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))
hab✝ : a ≤ b
hbc✝ : b ≤ c
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b < c
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b c f) x₀ = lmarginalPartialTraj κ a c f x₀

Tactic:
·

Goal state after:
case inr.inl
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
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b ≤ b
⊢ lmarginalPartialTraj κ a b (lmarginalPartialTraj κ b b f) x₀ = lmarginalPartialTraj κ a b f x₀
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case inr.convert_2
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)
A : ℕ → Set ((i : ι) → X i)
A_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X
A_anti : Antitone A
A_inter : ⋂ n, A n = ∅
this : ∀ (i : ι), Nonempty (X i)
s : ℕ → Finset ι
S : (n : ℕ) → Set ((i : ↥(s n)) → X ↑i)
mS : ∀ (n : ℕ), MeasurableSet (S n)
A_eq : ∀ (n : ℕ), A n = cylinder (s n) (S n)
u : Set ι := ⋯
t : ℕ → Finset ↑u := ⋯
f : ((i : ↑u) → X ↑i) → (i : ι) → X i := ⋯
aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := ⋯
g : (n : ℕ) → ((a : ↥(t n)) → X ↑((aux n) a)) ≃ ((b : ↥(s n)) → X ↑b) := ⋯
r_comp_f : ∀ (n : ℕ), (s n).restrict ∘ f = ⇑(g n) ∘ fun x i ↦ x ↑i
B : ℕ → Set ((i : ↑u) → X ↑i) := ⋯
T : (n : ℕ) → Set ((a : ↥(t n)) → X ↑((aux n) a)) := ⋯
B_eq : ∀ (n : ℕ), B n = cylinder (t n) (T n)
mT : ∀ (n : ℕ), MeasurableSet (T n)
B_mem : ∀ (n : ℕ), B n ∈ measurableCylinders fun i ↦ X ↑i
mB : ∀ (n : ℕ), MeasurableSet (B n)
B_anti : Antitone B
B_inter : ⋂ n, B n = ∅
u_inf : Infinite ↑u
count_u : Countable ↑u
φ : ℕ ≃ ↑u
⊢ IsFiniteMeasure (Measure.map (⇑(piCongrLeft (fun i ↦ X ↑i) φ)) (infinitePiNat fun n ↦ μ ↑(φ n)))

Tactic:
·

Goal state after:
case inr.convert_2
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)
A : ℕ → Set ((i : ι) → X i)
A_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X
A_anti : Antitone A
A_inter : ⋂ n, A n = ∅
this : ∀ (i : ι), Nonempty (X i)
s : ℕ → Finset ι
S : (n : ℕ) → Set ((i : ↥(s n)) → X ↑i)
mS : ∀ (n : ℕ), MeasurableSet (S n)
A_eq : ∀ (n : ℕ), A n = cylinder (s n) (S n)
u : Set ι := ⋯
t : ℕ → Finset ↑u := ⋯
f : ((i : ↑u) → X ↑i) → (i : ι) → X i := ⋯
aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) := ⋯
g : (n : ℕ) → ((a : ↥(t n)) → X ↑((aux n) a)) ≃ ((b : ↥(s n)) → X ↑b) := ⋯
r_comp_f : ∀ (n : ℕ), (s n).restrict ∘ f = ⇑(g n) ∘ fun x i ↦ x ↑i
B : ℕ → Set ((i : ↑u) → X ↑i) := ⋯
T : (n : ℕ) → Set ((a : ↥(t n)) → X ↑((aux n) a)) := ⋯
B_eq : ∀ (n : ℕ), B n = cylinder (t n) (T n)
mT : ∀ (n : ℕ), MeasurableSet (T n)
B_mem : ∀ (n : ℕ), B n ∈ measurableCylinders fun i ↦ X ↑i
mB : ∀ (n : ℕ), MeasurableSet (B n)
B_anti : Antitone B
B_inter : ⋂ n, B n = ∅
u_inf : Infinite ↑u
count_u : Countable ↑u
φ : ℕ ≃ ↑u
⊢ IsFiniteMeasure (Measure.map (⇑(piCongrLeft (fun i ↦ X ↑i) φ)) (infinitePiNat fun n ↦ μ ↑(φ n)))
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ ∫⁻ (b : γ) in s, κ.rnDeriv η a b ∂η a ≤ (κ a) s

pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

Tactic:
·

Goal state after:
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ ∫⁻ (b : γ) in s, κ.rnDeriv η a b ∂η a ≤ (κ a) s
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

Tactic:
·

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
s : Set γ
hs : MeasurableSet s
x✝ : s.Nonempty
⊢ Measurable (Function.uncurry (κ.rnDeriv η))
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inr.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))
hab✝ : a ≤ b
hbc✝ : b ≤ c
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b < c
⊢ ∫⁻ (z : (i : ↥(Iic b)) → X ↑i),
      ∫⁻ (z : (i : ↥(Iic c)) → X ↑i),
        f (updateFinset x₀ (Iic c) z) ∂(partialTraj κ b c) z ∂(partialTraj κ a b) ((Iic a).restrict x₀) =
    ∫⁻ (z : (i : ↥(Iic c)) → X ↑i), f (updateFinset x₀ (Iic c) z) ∂(partialTraj κ a c) ((Iic a).restrict x₀)

Tactic:
rw [← lintegral_comp, partialTraj_comp_partialTraj hab.le hbc.le]

Goal state after:
case inr.inr.hg
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))
hab✝ : a ≤ b
hbc✝ : b ≤ c
f : ((n : ℕ) → X n) → ℝ≥0∞
hf : Measurable f
x₀ : (n : ℕ) → X n
hab : a < b
hbc : b < c
⊢ Measurable fun z ↦ f (updateFinset x₀ (Iic c) z)
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
pp: case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ Measurable (Function.uncurry (κ.rnDeriv η))

pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ (η a).withDensity (κ.rnDeriv η a) = κ a

Tactic:
·

Goal state after:
case hf
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
h : κ a ≪ η a
⊢ Measurable (Function.uncurry (κ.rnDeriv η))
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case hC_gen
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)
I : Finset ι
s : Set ((i : ↥I) → X ↑i)
hs : MeasurableSet s
⊢ MeasurableSpace.pi = MeasurableSpace.generateFrom (measurableCylinders X)

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)
I : Finset ι
s : Set ((i : ↥I) → X ↑i)
hs : MeasurableSet s
⊢ I.restrict ⁻¹' s ∈ measurableCylinders X

Tactic:
·

Goal state after:
case hC_gen
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)
I : Finset ι
s : Set ((i : ↥I) → X ↑i)
hs : MeasurableSet s
⊢ MeasurableSpace.pi = MeasurableSpace.generateFrom (measurableCylinders X)
Module: Mathlib.Probability.ProductMeasure

Goal state before:
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)
I : Finset ι
s : Set ((i : ↥I) → X ↑i)
hs : MeasurableSet s
⊢ I.restrict ⁻¹' s ∈ measurableCylinders X

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)
I : Finset ι
s : Set ((i : ↥I) → X ↑i)
hs : MeasurableSet s
⊢ I.restrict ⁻¹' s ∈ measurableCylinders X
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case refine_1
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
x : (n : ℕ) → X n
⊢ IsProbabilityMeasure (((partialTraj κ b c).map (restrict₂ ⋯)) (frestrictLe b x))

pp: case refine_2
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
x : (n : ℕ) → X n
y : (i : ↥(Ioc b c)) → X ↑i
i : ℕ
hi : i ∈ ↑(Iic a)
⊢ Function.updateFinset x (Ioc b c) y i = x i

Tactic:
·

Goal state after:
case refine_1
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
x : (n : ℕ) → X n
⊢ IsProbabilityMeasure (((partialTraj κ b c).map (restrict₂ ⋯)) (frestrictLe b x))
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case refine_2
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
x : (n : ℕ) → X n
y : (i : ↥(Ioc b c)) → X ↑i
i : ℕ
hi : i ∈ ↑(Iic a)
⊢ Function.updateFinset x (Ioc b c) y i = x i

Tactic:
·

Goal state after:
case refine_2
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
c : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hab : a ≤ b
x : (n : ℕ) → X n
y : (i : ↥(Ioc b c)) → X ↑i
i : ℕ
hi : i ∈ ↑(Iic a)
⊢ Function.updateFinset x (Ioc b c) y i = x i
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 c : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
d : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hac : a ≤ c
had : a ≤ d
this : ∀ {c d : ℕ}, a ≤ c → a ≤ d → c ≤ d → lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f
hcd : ¬c ≤ d
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

pp: 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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

Tactic:
·

Goal state after:
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
d : ℕ
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
hac : a ≤ c
had : a ≤ d
this : ∀ {c d : ℕ}, a ≤ c → a ≤ d → c ≤ d → lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f
hcd : ¬c ≤ d
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hbc : b ≤ c
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hcb : c ≤ b
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hbc : b ≤ c
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : ¬SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : ¬SFinite μ
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ η : Kernel α β
inst✝² : IsFiniteKernel κ
inst✝¹ : IsFiniteKernel η
inst✝ : MeasurableSpace.CountableOrCountablyGenerated α β
μ ν : Measure α
hκη : ∀ᵐ (a : α) ∂μ, κ a ⟂ₘ η a
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
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 c✝ : ℕ
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hcb : c ≤ b
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f

Tactic:
·

Goal state after:
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))
inst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)
f : ((n : ℕ) → X n) → ℝ≥0∞
mf : Measurable f
hf : DependsOn f ↑(Iic a)
c d : ℕ
hac : a ≤ c
had : a ≤ d
hcd : c ≤ d
hcb : c ≤ b
⊢ lmarginalPartialTraj κ b c f = lmarginalPartialTraj κ b d f