l3lab/ntp-mathlib huggingface dataset:
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    

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
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 Set.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

Next tactic:
suffices h_forall : ∀ n ∈ s, f n = ft1 n
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · 

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
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 Set.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
⊢ t1 = ⋂ n ∈ s, f n

Next tactic:
rw [ht1_eq]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      

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
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 Set.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 ∈ s, ft1 x = ⋂ n ∈ s, f n

Next tactic:
ext x
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      

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
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 Set.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

Next tactic:
simp_rw [Set.mem_iInter]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      

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
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 Set.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 ∈ if i = a then t2 else if i ∈ s then ft1 i else Set.univ

Next tactic:
conv => lhs; intro i hns; rw [← h_forall i hns]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => 

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
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 Set.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 ∈ if i = a then t2 else if i ∈ s then ft1 i else Set.univ

Next tactic:
lhs; intro i hns; rw [← h_forall i hns]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => 

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
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 Set.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 ∈ if i = a then t2 else if i ∈ s then ft1 i else Set.univ

Next tactic:
lhs
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; 

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
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 Set.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

Next tactic:
intro i hns
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; 

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
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 Set.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 : ι
hns : i ∈ s
| x ∈ ft1 i

Next tactic:
rw [← h_forall i hns]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    

Proof state:
case h_forall
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
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 Set.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

Next tactic:
intro n hnS
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    intro n hnS
    

Proof state:
case h_forall
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
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 Set.univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
n : ι
hnS : n ∈ s
⊢ f n = ft1 n

Next tactic:
have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    intro n hnS
    have hn_ne_a : n ≠ a := by 

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
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 Set.univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
n : ι
hnS : n ∈ s
⊢ n ≠ a

Next tactic:
rintro rfl
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; 

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)
S : Finset ι
hp_ind : iIndepSets π κ
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
n : ι
hnS : n ∈ s
haS : n ∉ S
ht2_mem_pia : t2 ∈ π n
f : ι → Set Ω := fun n_1 => if n_1 = n then t2 else if n_1 ∈ s then ft1 n_1 else Set.univ
h_f_mem : ∀ n_1 ∈ insert n s, f n_1 ∈ π n_1
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
⊢ False

Next tactic:
exact haS (hs_mem hnS)
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
    

Proof state:
case h_forall
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
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 Set.univ
h_f_mem : ∀ n ∈ insert a s, f n ∈ π n
h_f_mem_pi : ∀ n ∈ s, f n ∈ π n
n : ι
hnS : n ∈ s
hn_ne_a : n ≠ a
⊢ f n = ft1 n

Next tactic:
simp_rw [if_pos hnS, if_neg hn_ne_a]
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 := by
  rintro t1 t2 ⟨s, hs_mem, ft1, hft1_mem, ht1_eq⟩ ht2_mem_pia
  rw [Finset.coe_subset] at hs_mem
  classical
  let f := fun n => ite (n = a) t2 (ite (n ∈ s) (ft1 n) Set.univ)
  have h_f_mem : ∀ n ∈ insert a s, f n ∈ π n := by
    intro n hn_mem_insert
    dsimp only
    cases' Finset.mem_insert.mp hn_mem_insert with hn_mem hn_mem
    · simp [hn_mem, ht2_mem_pia]
    · have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hn_mem)
      simp [hn_ne_a, hn_mem, hft1_mem n hn_mem]
  have h_f_mem_pi : ∀ n ∈ s, f n ∈ π n := fun x hxS => h_f_mem x (by simp [hxS])
  have h_t1 : t1 = ⋂ n ∈ s, f n := by
    suffices h_forall : ∀ n ∈ s, f n = ft1 n
    · rw [ht1_eq]
      ext x
      simp_rw [Set.mem_iInter]
      conv => lhs; intro i hns; rw [← h_forall i hns]
    intro n hnS
    have hn_ne_a : n ≠ a := by rintro rfl; exact haS (hs_mem hnS)
    simp_rw [if_pos hnS, if_neg hn_ne_a]
  

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
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 Set.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
⊢ ∀ᵐ (a : α) ∂μ, ↑↑(κ a) (t1 ∩ t2) = ↑↑(κ a) t1 * ↑↑(κ a) t2

Next tactic:
have h_μ_t1 : ∀ᵐ a' ∂μ, κ a' t1 = ∏ n in s, κ a' (f n) := by
    filter_upwards [hp_ind s h_f_mem_pi] with a' ha'
    rw [h_t1, ← ha']