-- Mathlib/Probability/Independence/Basic.lean
/-- The probability of an intersection of preimages conditioning on another intersection factors
into a product. -/
lemma cond_iInter [Finite ι] (hY : ∀ i, Measurable (Y i))
    (hindep : iIndepFun (fun _ ↦ mα.prod mβ) (fun i ω ↦ (X i ω, Y i ω)) μ)
    (hf : ∀ i ∈ s, MeasurableSet[mα.comap (X i)] (f i))
    (hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0) (ht : ∀ i, MeasurableSet (t i)) :
    μ[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, μ[f i | Y i in t i]

Proof state:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ (μ (⋂ i, Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) = (∏ i : ι, μ (Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i)

Tactic:
rw [hindep.meas_iInter]

Result:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)

Proof state:
case intro
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
⊢ μ[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, μ[f i | Y i ⁻¹' t i]

Tactic:
let g (i' : ι) := if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'

Result:
case intro
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ μ[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, μ[f i | Y i ⁻¹' t i]

Proof state:
case h
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
i : ι
hi : i ∈ s
A : Set α
hA : MeasurableSet A
hA' : X i ⁻¹' A = f i
x✝ : Ω
⊢ x✝ ∈ (fun ω => (X i ω, Y i ω)) ⁻¹' A ×ˢ univ ↔ x✝ ∈ f i

Tactic:
simp [← hA']

Result:
no goals

Proof state:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)

Tactic:
exact fun i ↦ ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp [eq_comm]⟩

Result:
no goals

Proof state:
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
i : ι
hi : i ∈ s
A : Set α
hA : MeasurableSet A
hA' : X i ⁻¹' A = f i
⊢ (fun ω => (X i ω, Y i ω)) ⁻¹' A ×ˢ univ = f i

Tactic:
ext

Result:
case h
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
i : ι
hi : i ∈ s
A : Set α
hA : MeasurableSet A
hA' : X i ⁻¹' A = f i
x✝ : Ω
⊢ x✝ ∈ (fun ω => (X i ω, Y i ω)) ⁻¹' A ×ˢ univ ↔ x✝ ∈ f i

Proof state:
case neg
ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun x => mα.prod mβ) (fun i ω => (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' => if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
i : ι
hi : i ∉ s
⊢ MeasurableSet (Y i ⁻¹' t i)

Tactic:
exact ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp [eq_comm]⟩

Result:
no goals