mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ MeasurableSet {a | κ a ≪ η a}

Tactic:
simp_rw [← singularPart_eq_zero_iff_absolutelyContinuous,
    singularPart_eq_zero_iff_measure_eq_zero]

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
⊢ MeasurableSet {a | (κ a) (κ.mutuallySingularSetSlice η a) = 0}
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 : Kernel α γ
inst✝ : IsFiniteKernel κ
a✝ : α
⊢ κ a✝ ≪ κ a✝

Tactic:
]

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ η ξ : Kernel α γ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel η
h : κ = η.withDensity f + ξ
hf : Measurable (Function.uncurry f)
a : α
hξ : ξ a ⟂ₘ η a
⊢ ξ a + (η a).withDensity (f a) = ξ a + (η a).withDensity (f a)

Tactic:
]

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

Goal state before:
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

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))

Tactic:
]

Goal state after:
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

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.RadonNikodym

Goal state before:
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
⊢ ∫⁻ (a_1 : γ), (∂κ a/∂η a) a_1 ∂η a ≤ κ.bound

Tactic:
]

Goal state after:
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
⊢ ∫⁻ (a_1 : γ), (∂κ a/∂η a) a_1 ∂η a ≤ κ.bound
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 ν : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
x : γ
h1 : (κ.singularPart ν).rnDeriv ν a x = (∂(κ.singularPart ν) a/∂ν a) x
h2 : (∂(κ.singularPart ν) a/∂ν a) x = 0 x
⊢ 0 x = 0 x

Tactic:
]

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case e'_5
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 := ⋃ n, ↑(s n)
t : ℕ → Finset ↑u := fun n ↦ (s n).preimage Subtype.val ⋯
f : ((i : ↑u) → X ↑i) → (i : ι) → X i := fun x i ↦ if hi : i ∈ u then x ⟨i, hi⟩ else Classical.ofNonempty
aux : (n : ℕ) → ↥(t n) ≃ ↥(s n) :=
  fun n ↦ { toFun := fun i ↦ ⟨↑↑i, ⋯⟩, invFun := fun i ↦ ⟨⟨↑i, ⋯⟩, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }
g : (n : ℕ) → ((a : ↥(t n)) → X ↑((aux n) a)) ≃ ((b : ↥(s n)) → X ↑b) :=
  fun n ↦ Equiv.piCongrLeft (fun i ↦ X ↑i) (aux n)
r_comp_f : ∀ (n : ℕ), (s n).restrict ∘ f = ⇑(g n) ∘ fun x i ↦ x ↑i
B : ℕ → Set ((i : ↑u) → X ↑i) := fun n ↦ f ⁻¹' A n
T : (n : ℕ) → Set ((a : ↥(t n)) → X ↑((aux n) a)) := fun n ↦ ⇑(g n) ⁻¹' S n
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
⊢ 0 = 0

Tactic:
]

Goal state after:
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 f x₀ = lmarginalPartialTraj κ a a f x₀

Tactic:
]

Goal state after:
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 c f x₀ = lmarginalPartialTraj κ a c f x₀

Tactic:
]

Goal state after:
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 f x₀ = lmarginalPartialTraj κ a b f x₀

Tactic:
]

Goal state after:
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
⊢ ((η a).withDensity (∂κ a/∂η a)) s ≤ (κ a) s

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
⊢ ((η a).withDensity (∂κ a/∂η a)) s ≤ (κ a) s
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
h : κ a ≪ η a
s : Set γ
hs : MeasurableSet s
⊢ (κ a) s = (κ a) s

Tactic:
]

Goal state after:
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
h : κ a ≪ η a
⊢ (κ a) univ = (κ a) univ

Tactic:
]

Goal state after:
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

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
⊢ ∫⁻ (c_1 : (i : ↥(Iic c)) → X ↑i), f (updateFinset x₀ (Iic c) c_1) ∂(partialTraj κ a c) ((Iic a).restrict x₀) =
    ∫⁻ (z : (i : ↥(Iic c)) → X ↑i), f (updateFinset x₀ (Iic c) z) ∂(partialTraj κ a c) ((Iic a).restrict x₀)

pp: 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)

Tactic:
]

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: 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

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))

Tactic:
]

Goal state after:
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

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: 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
⊢ (Measure.pi fun i ↦ μ ↑i) s = ((fun I ↦ Measure.pi fun i ↦ μ ↑i) I) s

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)

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: 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
⊢ ∫⁻ (x_1 : (i : ↥(Ioc b c)) → X ↑i),
      f (Function.updateFinset x (Ioc b c) x_1) ∂((partialTraj κ b c).map (restrict₂ ⋯)) (frestrictLe b x) =
    f x

Tactic:
]

Goal state after:
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
⊢ ∫⁻ (x_1 : (i : ↥(Ioc b c)) → X ↑i),
      f (Function.updateFinset x (Ioc b c) x_1) ∂((partialTraj κ b c).map (restrict₂ ⋯)) (frestrictLe b x) =
    f x
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
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
h_ae : κ.rnDeriv η a =ᵐ[η a] ∂κ a/∂η a
⊢ κ a = κ a

Tactic:
]

Goal state after:
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)
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel (κ.withDensity f)
hf : Measurable (Function.uncurry f)
a : α
h_ae : (κ.withDensity f).rnDeriv κ a =ᵐ[κ a] ∂(κ.withDensity f) a/∂κ a
hf' : ∀ (a : α), Measurable (f a)
x : γ
hx1 : (κ.withDensity f).rnDeriv κ a x = (∂(κ.withDensity f) a/∂κ a) x
hx2 : (∂(κ a).withDensity (f a)/∂κ a) x = f a x
⊢ f a x = f a x

Tactic:
]

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

Goal state before:
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_ac : κ a ≪ η a
⊢ (∀ᵐ (b : γ) ∂η a, κ.rnDeriv η a b = 1) ↔ ∂κ a/∂η a =ᵐ[η a] 1

Tactic:
]

Goal state after:
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_ac : κ a ≪ η a
⊢ (∀ᵐ (b : γ) ∂η a, κ.rnDeriv η a b = 1) ↔ ∂κ a/∂η a =ᵐ[η a] 1
Module: Mathlib.Probability.Kernel.RadonNikodym

Goal state before:
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_ac : κ a ≪ η a
c : γ
hc : κ.rnDeriv η a c = (∂κ a/∂η a) c
⊢ (∂κ a/∂η a) c = 1 ↔ (∂κ a/∂η a) c = 1

Tactic:
]

Goal state after:
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 c f

Tactic:
]

Goal state after:
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
⊢ 0 ⟂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
⊢ 0 ⟂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
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
rw [compProd_of_not_sfinite _ _ hμ]

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
⊢ 0 ⟂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 0

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 0
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
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
s : Set (α × β) := κ.mutuallySingularSet η
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (η a) (Prod.mk a ⁻¹' s) ∂ν = 0 ∧ ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' sᶜ) ∂μ = 0

Tactic:
]

Goal state after:
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
s : Set (α × β) := κ.mutuallySingularSet η
hs : MeasurableSet s
⊢ ∫⁻ (a : α), (η a) (Prod.mk a ⁻¹' s) ∂ν = 0 ∧ ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' sᶜ) ∂μ = 0
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 c f

Tactic:
]

Goal state after:
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
⊢ f = f

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: 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
s : Set (α × β) := κ.mutuallySingularSet η
hs : MeasurableSet s
h_eq : ∀ (a : α), Prod.mk a ⁻¹' s = κ.mutuallySingularSetSlice η a
a : α
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Probability.ProductMeasure

Goal state before:
pp: case e_f
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)
ν : Measure ((i : ι) → X i)
hν : ∀ (s : Finset ι) (t : (i : ι) → Set (X i)), (∀ (i : ι), MeasurableSet (t i)) → ν ((↑s).pi t) = ∏ i ∈ s, (μ i) (t i)
s : Finset ι
t : (i : ↥s) → Set (X ↑i)
ht : ∀ (i : ↥s), MeasurableSet (t i)
i : ↥s
⊢ (μ ↑i) (t ⟨↑i, ⋯⟩) = (μ ↑i) (t i)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.AbsolutelyContinuous

Goal state before:
pp: 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
s : Set (α × β) := κ.mutuallySingularSet η
hs : MeasurableSet s
h_eq : ∀ (a : α), Prod.mk a ⁻¹' s = κ.mutuallySingularSetSlice η a
h1 : ∀ (a : α), (η a) (Prod.mk a ⁻¹' s) = 0
a : α
ha : κ a ⟂ₘ η a
⊢ κ a ⟂ₘ η a

Tactic:
]

Goal state after:
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
s : Set (α × β) := κ.mutuallySingularSet η
hs : MeasurableSet s
h_eq : ∀ (a : α), Prod.mk a ⁻¹' s = κ.mutuallySingularSetSlice η a
h1 : ∀ (a : α), (η a) (Prod.mk a ⁻¹' s) = 0
a : α
ha : κ a ⟂ₘ η a
⊢ κ a ⟂ₘ η a
Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Goal state before:
pp: case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
b : 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∞
hf : DependsOn f ↑(Iic b)
mf : Measurable f
x y : (i : ℕ) → X i
hxy : ∀ i ∈ ↑(Iic a), x i = y i
hba : b ≤ a
⊢ f x = f y

Tactic:
]

Goal state after:
case inl
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
b : 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∞
hf : DependsOn f ↑(Iic b)
mf : Measurable f
x y : (i : ℕ) → X i
hxy : ∀ i ∈ ↑(Iic a), x i = y i
hba : b ≤ a
⊢ f x = f y