l3lab/ntp-mathlib huggingface dataset:
theorem associated_of_factorization_eq (a b : α) (ha : a ≠ 0) (hb : b ≠ 0)
    (h : factorization a = factorization b) : Associated a b := by
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : factorization a = factorization b
⊢ a ~ᵤ b

Next tactic:
simp_rw [factorization, AddEquiv.apply_eq_iff_eq] at h
theorem associated_of_factorization_eq (a b : α) (ha : a ≠ 0) (hb : b ≠ 0)
    (h : factorization a = factorization b) : Associated a b := by
  simp_rw [factorization, AddEquiv.apply_eq_iff_eq] at h
  

Proof state:
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : NormalizationMonoid α
inst✝ : DecidableEq α
a b : α
ha : a ≠ 0
hb : b ≠ 0
h : normalizedFactors a = normalizedFactors b
⊢ a ~ᵤ b

Next tactic:
rwa [associated_iff_normalizedFactors_eq_normalizedFactors ha hb]
theorem squareCylinders_eq_iUnion_image (C : ∀ i, Set (Set (α i))) :
    squareCylinders C = ⋃ s : Finset ι, (fun t ↦ (s : Set ι).pi t) '' univ.pi C := by
  

Proof state:
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
⊢ squareCylinders C = ⋃ s, (fun t => pi (↑s) t) '' pi univ C

Next tactic:
ext1 f
theorem squareCylinders_eq_iUnion_image (C : ∀ i, Set (Set (α i))) :
    squareCylinders C = ⋃ s : Finset ι, (fun t ↦ (s : Set ι).pi t) '' univ.pi C := by
  ext1 f
  

Proof state:
case h
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
f : Set ((i : ι) → α i)
⊢ f ∈ squareCylinders C ↔ f ∈ ⋃ s, (fun t => pi (↑s) t) '' pi univ C

Next tactic:
simp only [squareCylinders, mem_iUnion, mem_image, mem_univ_pi, exists_prop, mem_setOf_eq,
    eq_comm (a := f)]
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  

Proof state:
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
⊢ IsPiSystem (squareCylinders C)

Next tactic:
rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    intro i
    have : (t₁' i ∩ t₂' i).Nonempty := by
      obtain ⟨f, hf⟩ := hst_nonempty
      rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, mem_inter_iff, mem_pi, mem_pi] at hf
      refine ⟨f i, ⟨?_, ?_⟩⟩
      · by_cases hi₁ : i ∈ s₁
        · exact hf.1 i hi₁
        · rw [h1' i hi₁]
          exact mem_univ _
      · by_cases hi₂ : i ∈ s₂
        · exact hf.2 i hi₂
        · rw [h2' i hi₂]
          exact mem_univ _
    refine hC i _ ?_ _ ?_ this
    · by_cases hi₁ : i ∈ s₁
      · rw [← h1 i hi₁]
        exact h₁ i (mem_univ _)
      · rw [h1' i hi₁]
        exact hC_univ i
    · by_cases hi₂ : i ∈ s₂
      · rw [← h2 i hi₂]
        exact h₂ i (mem_univ _)
      · rw [h2' i hi₂]
        exact hC_univ i
  · rw [Finset.coe_union]
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
⊢ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂ ∈ squareCylinders C

Next tactic:
rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  

Proof state:
case intro.intro.intro.intro.intro.intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
⊢ (pi (↑s₁ ∪ ↑s₂) fun i => t₁' i ∩ t₂' i) ∈ squareCylinders C

Next tactic:
refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
⊢ (fun i => t₁' i ∩ t₂' i) ∈ pi univ C

Next tactic:
rw [mem_univ_pi]
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    

Proof state:
case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
⊢ ∀ (i : ι), t₁' i ∩ t₂' i ∈ C i

Next tactic:
intro i
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    intro i
    

Proof state:
case intro.intro.intro.intro.intro.intro.refine_1
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
⊢ t₁' i ∩ t₂' i ∈ C i

Next tactic:
have : (t₁' i ∩ t₂' i).Nonempty := by
      obtain ⟨f, hf⟩ := hst_nonempty
      rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, mem_inter_iff, mem_pi, mem_pi] at hf
      refine ⟨f i, ⟨?_, ?_⟩⟩
      · by_cases hi₁ : i ∈ s₁
        · exact hf.1 i hi₁
        · rw [h1' i hi₁]
          exact mem_univ _
      · by_cases hi₂ : i ∈ s₂
        · exact hf.2 i hi₂
        · rw [h2' i hi₂]
          exact mem_univ _
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    intro i
    have : (t₁' i ∩ t₂' i).Nonempty := by
      

Proof state:
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
hst_nonempty : Set.Nonempty (pi (↑s₁) t₁ ∩ pi (↑s₂) t₂)
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
⊢ Set.Nonempty (t₁' i ∩ t₂' i)

Next tactic:
obtain ⟨f, hf⟩ := hst_nonempty
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    intro i
    have : (t₁' i ∩ t₂' i).Nonempty := by
      obtain ⟨f, hf⟩ := hst_nonempty
      

Proof state:
case intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
f : (i : ι) → α i
hf : f ∈ pi (↑s₁) t₁ ∩ pi (↑s₂) t₂
⊢ Set.Nonempty (t₁' i ∩ t₂' i)

Next tactic:
rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, mem_inter_iff, mem_pi, mem_pi] at hf
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) := by
  rintro S₁ ⟨s₁, t₁, h₁, rfl⟩ S₂ ⟨s₂, t₂, h₂, rfl⟩ hst_nonempty
  classical
  let t₁' := s₁.piecewise t₁ (fun i ↦ univ)
  let t₂' := s₂.piecewise t₂ (fun i ↦ univ)
  have h1 : ∀ i ∈ (s₁ : Set ι), t₁ i = t₁' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h1' : ∀ i ∉ (s₁ : Set ι), t₁' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  have h2 : ∀ i ∈ (s₂ : Set ι), t₂ i = t₂' i :=
    fun i hi ↦ (Finset.piecewise_eq_of_mem _ _ _ hi).symm
  have h2' : ∀ i ∉ (s₂ : Set ι), t₂' i = univ :=
    fun i hi ↦ Finset.piecewise_eq_of_not_mem _ _ _ hi
  rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, ← union_pi_inter h1' h2']
  refine ⟨s₁ ∪ s₂, fun i ↦ t₁' i ∩ t₂' i, ?_, ?_⟩
  · rw [mem_univ_pi]
    intro i
    have : (t₁' i ∩ t₂' i).Nonempty := by
      obtain ⟨f, hf⟩ := hst_nonempty
      rw [Set.pi_congr rfl h1, Set.pi_congr rfl h2, mem_inter_iff, mem_pi, mem_pi] at hf
      

Proof state:
case intro
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set ((fun i => α i) i)
h₁ : t₁ ∈ pi univ C
s₂ : Finset ι
t₂ : (i : ι) → Set ((fun i => α i) i)
h₂ : t₂ ∈ pi univ C
t₁' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₁ t₁ fun i => univ
t₂' : (i : ι) → Set ((fun i => α i) i) := Finset.piecewise s₂ t₂ fun i => univ
h1 : ∀ i ∈ ↑s₁, t₁ i = t₁' i
h1' : ∀ i ∉ ↑s₁, t₁' i = univ
h2 : ∀ i ∈ ↑s₂, t₂ i = t₂' i
h2' : ∀ i ∉ ↑s₂, t₂' i = univ
i : ι
f : (i : ι) → α i
hf : (∀ i ∈ ↑s₁, f i ∈ t₁' i) ∧ ∀ i ∈ ↑s₂, f i ∈ t₂' i
⊢ Set.Nonempty (t₁' i ∩ t₂' i)

Next tactic:
refine ⟨f i, ⟨?_, ?_⟩⟩