-- Mathlib/Probability/Kernel/Disintegration/Density.lean
lemma setIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
    (a : α) {s : Set β} (hs : MeasurableSet s) {A : Set γ} (hA : MeasurableSet A) :
    ∫ x in A, density κ ν a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal

Proof state:
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s

Tactic:
rw [prod_diff_prod, compl_eq_univ_diff]

Result:
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ (univ \ A) ×ˢ s = univ ×ˢ (s \ s) ∪ (univ \ A) ×ˢ s

Proof state:
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
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal

Tactic:
have : IsFiniteKernel κ := isFiniteKernel_of_isFiniteKernel_fst (h := isFiniteKernel_of_le hκν)

Result:
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
⊢ ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal

Proof state:
case iUnion.e_a.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) (f i ×ˢ s)).toReal
⊢ ∀ (i : ℕ), MeasurableSet (f i ×ˢ s)

Tactic:
exact fun i ↦ (hf i).prod hs

Result:
no goals

Proof state:
case iUnion.e_a.hn
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) (f i ×ˢ s)).toReal
⊢ Pairwise (Function.onFun Disjoint fun i => f i ×ˢ s)

Tactic:
exact hf_disj.mono fun _ _ h ↦ h.set_prod_left _ _

Result:
no goals

Proof state:
case compl
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

Tactic:
have : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s := by
  rw [prod_diff_prod, compl_eq_univ_diff]
  simp

Result:
case compl
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
this : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s
⊢ ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

Proof state:
case compl
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ∫ (x : γ) in A, κ.density ν a x s ∂ν a + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ∫ (x : γ), κ.density ν a x s ∂ν a
⊢ ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

Tactic:
rw [hA_eq, integral_density hκν a hs] at h

Result:
case compl
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}
A : Set γ
hA : MeasurableSet A
hA_eq : ∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal
h : ((κ a) (A ×ˢ s)).toReal + ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (univ ×ˢ s)).toReal
⊢ ∫ (x : γ) in Aᶜ, κ.density ν a x s ∂ν a = ((κ a) (Aᶜ ×ˢ s)).toReal

Proof state:
case empty
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}
⊢ ∫ (x : γ) in ∅, κ.density ν a x s ∂ν a = ((κ a) (∅ ×ˢ s)).toReal

Tactic:
simp

Result:
no goals