-- Mathlib/Probability/Independence/Kernel.lean
theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
    (hp_ind : iIndepSets π κ μ) (haS : a ∉ S) :
    IndepSets (piiUnionInter π S) (π a) κ μ

Proof state:
case inl
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
n : ι
hn_mem_insert : n ∈ insert a s
hn_mem : n = a
⊢ (if n = a then t2 else if n ∈ s then ft1 n else univ) ∈ π n

Tactic:
simp [hn_mem, ht2_mem_pia]

Result:
no goals

Proof state:
case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_forall : ∀ n ∈ s, f n = ft1 n
x : Ω
⊢ (∀ i ∈ s, x ∈ ft1 i) ↔ ∀ i ∈ s, x ∈ f i

Tactic:
conv => lhs; intro i hns; rw [← h_forall i hns]

Result:
no goals

Proof state:
case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])

Result:
case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Proof state:
case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_t1 : t1 = ⋂ n ∈ s, f n
h_μ_t1 : ∀ᵐ (a' : α) ∂μ, (κ a') t1 = ∏ n ∈ s, (κ a') (f n)
h_t2 : t2 = f a
h_μ_inter : ∀ᵐ (a' : α) ∂μ, (κ a') (t1 ∩ t2) = ∏ n ∈ insert a s, (κ a') (f n)
has : a ∉ s
a' : α
ha1 : (κ a') t1 = ∏ n ∈ s, (κ a') (f n)
ha2 : (κ a') (t1 ∩ t2) = ∏ n ∈ insert a s, (κ a') (f n)
⊢ (κ a') (t1 ∩ t2) = (κ a') t1 * (κ a') t2

Tactic:
rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1]

Result:
no goals

Proof state:
case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_forall : ∀ n ∈ s, f n = ft1 n
x : Ω
⊢ x ∈ ⋂ x ∈ s, ft1 x ↔ x ∈ ⋂ n ∈ s, f n

Tactic:
simp_rw [Set.mem_iInter]

Result:
case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
h_forall : ∀ n ∈ s, f n = ft1 n
x : Ω
⊢ (∀ i ∈ s, x ∈ ft1 i) ↔ ∀ i ∈ s, x ∈ f i

Proof state:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ t1 = ⋂ n ∈ s, f n

Tactic:
suffices h_forall : ∀ n ∈ s, f n = ft1 n by
  rw [ht1_eq]
  ext x
  simp_rw [Set.mem_iInter]
  conv => lhs; intro i hns; rw [← h_forall i hns]

Result:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ ∀ n ∈ s, f n = ft1 n

Proof state:
case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2

Tactic:
let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)

Result:
case intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
π : ι → Set (Set Ω)
a : ι
S : Finset ι
hp_ind : iIndepSets π κ μ
haS : a ∉ S
t1 t2 : Set Ω
s : Finset ι
hs_mem : s ⊆ S
ft1 : ι → Set Ω
hft1_mem : ∀ x ∈ s, ft1 x ∈ π x
ht1_eq : t1 = ⋂ x ∈ s, ft1 x
ht2_mem_pia : t2 ∈ π a
f : ι → Set Ω := fun n => if n = a then t2 else if n ∈ s then ft1 n else univ
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2