awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
u_ind :
  ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u = ⋃ a ∈ A, ⋂ t ∈ a, t
compl :
  ∀ (A : Set (Set (Set α))) (A_fin : A.Finite) (mem_A : ∀ a ∈ A, a.Finite) (hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜)
    (u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t),
    ∃ (A : Set (Set (Set α))),
      A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t
⊢ ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t

Tactic:
rcases u_ind with ⟨A, A_fin, mem_A, hA, u_eq⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
⊢ Finite.{u_1 + 1} (↑A : Type u_1)

Tactic:
have := finite_coe_iff.2 A_fin
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this : Finite.{u_1 + 1} (↑A : Type u_1)
⊢ ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)

Tactic:
have := fun a : A ↦ finite_coe_iff.2 <| mem_A a.1 a.2
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
⊢ ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t

Tactic:
⟨{{(f a).1ᶜ | a : A} | f : (Π a : A, ↑a)}, finite_coe_iff.1 inferInstance,
      fun a ⟨f, hf⟩ ↦ hf ▸ finite_coe_iff.1 inferInstance, fun a ha t ht ↦ ?_, ?_⟩

Premises used: Compl.compl, setOf, Set.Finite, Exists, Eq.rec, inferInstance, Set.finite_coe_iff, _private.0.MeasureTheory.mem_generateSetAlgebra_elim.match_1, And.intro, Exists.intro, Iff.mp, Subtype.val, Eq
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
compl :
  ∀ (a : Set (Set α))
    (ha :
      Membership.mem (γ := Set (Set (Set α)))
        {x : Set (Set α) |
          ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
            {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
        a)
    (t : Set α) (ht : t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
compl₁ :
  uᶜ =
    ⋃ (a : Set (Set α)),
      ⋃ (_ :
        Membership.mem (γ := Set (Set (Set α)))
          {x : Set (Set α) |
            ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
              {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
          a),
        ⋂ t ∈ a, t
⊢ ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t

Tactic:
refine ⟨{{(f a).1ᶜ | a : A} | f : (Π a : A, ↑a)}, finite_coe_iff.1 inferInstance,
      fun a ⟨f, hf⟩ ↦ hf ▸ finite_coe_iff.1 inferInstance, fun a ha t ht ↦ ?_, ?_⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
a : Set (Set α)
ha :
  Membership.mem (γ := Set (Set (Set α)))
    {x : Set (Set α) |
      ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
        {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
    a
t : Set α
ht : t ∈ a
compl :
  ∀ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1))
    (ht : Membership.mem (γ := Set (Set α)) {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} t),
    t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
⊢ t ∈ 𝒜 ∨ tᶜ ∈ 𝒜

Tactic:
rcases ha with ⟨f, rfl⟩

Premises used: rfl
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
t : Set α
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
ht : Membership.mem (γ := Set (Set α)) {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} t
compl : ∀ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ ∈ 𝒜 ∨ (↑(f a) : Set α)ᶜᶜ ∈ 𝒜
⊢ t ∈ 𝒜 ∨ tᶜ ∈ 𝒜

Tactic:
rcases ht with ⟨a, rfl⟩

Premises used: rfl
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
a : (↑A : Type u_1)
compl : (↑(f a) : Set α) ∈ 𝒜 ∨ (↑(f a) : Set α)ᶜ ∈ 𝒜
⊢ (↑(f a) : Set α)ᶜ ∈ 𝒜 ∨ (↑(f a) : Set α)ᶜᶜ ∈ 𝒜

Tactic:
rw [compl_compl, or_comm]

Premises used: compl_compl, or_comm
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
a : (↑A : Type u_1)
⊢ (↑(f a) : Set α) ∈ 𝒜 ∨ (↑(f a) : Set α)ᶜ ∈ 𝒜

Tactic:
hA a.1 a.2 (f a).1 (f a).2

Premises used: Subtype.val, Subtype.property
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
a : (↑A : Type u_1)
⊢ (↑(f a) : Set α) ∈ 𝒜 ∨ (↑(f a) : Set α)ᶜ ∈ 𝒜

Tactic:
exact hA a.1 a.2 (f a).1 (f a).2

Premises used: Subtype.val, Subtype.property
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
compl :
  ∀ (x : α),
    x ∈ uᶜ ↔
      Membership.mem (γ := Set α)
        (⋃ (a : Set (Set α)),
          ⋃ (_ :
            Membership.mem (γ := Set (Set (Set α)))
              {x : Set (Set α) |
                ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
                  {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
              a),
            ⋂ t ∈ a, t)
        x
⊢ uᶜ =
    ⋃ (a : Set (Set α)),
      ⋃ (_ :
        Membership.mem (γ := Set (Set (Set α)))
          {x : Set (Set α) |
            ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
              {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
          a),
        ⋂ t ∈ a, t

Tactic:
ext x
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
compl :
  (∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2) ↔
    ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
      ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
        (i :
          (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i_1),
        x ∈ i_1
⊢ x ∈ uᶜ ↔
    Membership.mem (γ := Set α)
      (⋃ (a : Set (Set α)),
        ⋃ (_ :
          Membership.mem (γ := Set (Set (Set α)))
            {x : Set (Set α) |
              ∃ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
                {x : Set α | ∃ (a : (↑A : Type u_1)), (↑(f a) : Set α)ᶜ = x} = x}
            a),
          ⋂ t ∈ a, t)
      x

Tactic:
simp only [u_eq, compl_iUnion, compl_iInter, mem_iInter, mem_iUnion, mem_compl_iff,
        exists_prop, Subtype.exists, mem_setOf_eq, iUnion_exists, iUnion_iUnion_eq',
        iInter_exists]

Premises used: Set.compl_iInter, Set.mem_iUnion, Subtype.exists, Set.mem_setOf_eq, exists_prop, Set.mem_iInter, Set.compl_iUnion, Set.iInter_exists, Set.iUnion_exists, Set.mem_compl_iff, Set.iUnion_iUnion_eq'
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
compl :
  (∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2) →
    ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
      ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
        (i :
          (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i_1),
        x ∈ i_1
compl₁ :
  (∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
      ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
        (i :
          (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i_1),
        x ∈ i_1) →
    ∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2
⊢ (∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2) ↔
    ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
      ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
        (i :
          (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i_1),
        x ∈ i_1

Tactic:
constructor
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
hx : ∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2
compl :
  ∀ (f : (i : Set (Set α)) → (i : i ∈ A) → Set α) (hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1),
    ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
      ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
        (i :
          (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i_1),
        x ∈ i_1
⊢ ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
    ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
      (i :
        (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                  (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
              Set α)ᶜ =
          i_1),
      x ∈ i_1

Tactic:
choose f hf using hx
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (i : Set (Set α)) → (i : i ∈ A) → Set α
hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1
x_1 : Set α
a : Set (Set α)
ha : a ∈ A
h :
  (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ =
    x_1
rw :
  Membership.mem (γ := Set α)
    (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ
    x
⊢ x ∈ x_1

Tactic:
rw [← h]
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (i : Set (Set α)) → (i : i ∈ A) → Set α
hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1
x_1 : Set α
a : Set (Set α)
ha : a ∈ A
h :
  (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ =
    x_1
⊢ Membership.mem (γ := Set α)
    (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ
    x

Tactic:
(hf a ha).2

Premises used: Set.Mem, setOf, And.right, Not
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (i : Set (Set α)) → (i : i ∈ A) → Set α
hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1
x_1 : Set α
a : Set (Set α)
ha : a ∈ A
h :
  (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ =
    x_1
⊢ Membership.mem (γ := Set α)
    (↑((fun (x_2 : (↑A : Type u_1)) ↦
              (match (motive := (x : (↑A : Type u_1)) → (↑(↑x : Set (Set α)) : Type u_1)) x_2 with
                | ⟨a, ha⟩ =>
                  Subtype.mk (p := fun (x : Set α) ↦
                    x ∈ (↑(Subtype.mk (p := fun (x : Set (Set α)) ↦ x ∈ A) a ha) : Set (Set α))) (f a ha) ⋯ :
                (↑(↑x_2 : Set (Set α)) : Type u_1)))
            (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) a ⋯)) :
        Set α)ᶜ
    x

Tactic:
exact (hf a ha).2

Premises used: And.right
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (i : Set (Set α)) → (i : i ∈ A) → Set α
hf : ∀ (i : Set (Set α)) (i_1 : i ∈ A), f i i_1 ∈ i ∧ x ∉ f i i_1
⊢ ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
    ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
      (i :
        (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                  (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
              Set α)ᶜ =
          i_1),
      x ∈ i_1

Tactic:
exact ⟨fun ⟨a, ha⟩ ↦ ⟨f a ha, (hf a ha).1⟩, fun _ a ha h ↦ by rw [← h]; exact (hf a ha).2⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
hx :
  ∃ (i : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)),
    ∀ (i_1 : Set α) (i_2 : Set (Set α)) (i_3 : i_2 ∈ A)
      (i :
        (↑(i (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) :
                  (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_2 ⋯) : Set (Set α)) : Type u_1)) :
              Set α)ᶜ =
          i_1),
      x ∈ i_1
compl :
  ∀ (f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1))
    (hf :
      ∀ (i : Set α) (i_1 : Set (Set α)) (i_2 : i_1 ∈ A)
        (i_3 :
          (↑(f (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) :
                    (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) : Set (Set α)) : Type u_1)) :
                Set α)ᶜ =
            i),
        x ∈ i)
    (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2
⊢ ∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2

Tactic:
rcases hx with ⟨f, hf⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
hf :
  ∀ (i : Set α) (i_1 : Set (Set α)) (i_2 : i_1 ∈ A)
    (i_3 :
      (↑(f (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) :
                (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) : Set (Set α)) : Type u_1)) :
            Set α)ᶜ =
        i),
    x ∈ i
⊢ ∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2

Tactic:
fun a ha ↦ ⟨f ⟨a, ha⟩, (f ⟨a, ha⟩).2, hf (f ⟨a, ha⟩)ᶜ a ha rfl⟩

Premises used: Compl.compl, Subtype.mk, And.intro, Exists.intro, Subtype.val, rfl, Subtype.property
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this_1 : Finite.{u_1 + 1} (↑A : Type u_1)
this : ∀ (a : (↑A : Type u_1)), Finite.{u_1 + 1} (↑(↑a : Set (Set α)) : Type u_1)
x : α
f : (a : (↑A : Type u_1)) → (↑(↑a : Set (Set α)) : Type u_1)
hf :
  ∀ (i : Set α) (i_1 : Set (Set α)) (i_2 : i_1 ∈ A)
    (i_3 :
      (↑(f (Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) :
                (↑(↑(Subtype.mk (p := fun (a : Set (Set α)) ↦ a ∈ A) i_1 ⋯) : Set (Set α)) : Type u_1)) :
            Set α)ᶜ =
        i),
    x ∈ i
⊢ ∀ (i : Set (Set α)) (i_1 : i ∈ A), ∃ i_2 ∈ i, x ∉ i_2

Tactic:
exact fun a ha ↦ ⟨f ⟨a, ha⟩, (f ⟨a, ha⟩).2, hf (f ⟨a, ha⟩)ᶜ a ha rfl⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u v : Set α
hs : generateSetAlgebra 𝒜 u
ht : generateSetAlgebra 𝒜 v
u_ind :
  ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u = ⋃ a ∈ A, ⋂ t ∈ a, t
v_ind :
  ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ v = ⋃ a ∈ A, ⋂ t ∈ a, t
union :
  ∀ (Au : Set (Set (Set α))) (Au_fin : Au.Finite) (mem_Au : ∀ a ∈ Au, a.Finite)
    (hAu : ∀ a ∈ Au, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) (u_eq : u = ⋃ a ∈ Au, ⋂ t ∈ a, t),
    ∃ (A : Set (Set (Set α))),
      A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u ∪ v = ⋃ a ∈ A, ⋂ t ∈ a, t
⊢ ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u ∪ v = ⋃ a ∈ A, ⋂ t ∈ a, t

Tactic:
rcases u_ind with ⟨Au, Au_fin, mem_Au, hAu, u_eq⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u v : Set α
hs : generateSetAlgebra 𝒜 u
ht : generateSetAlgebra 𝒜 v
v_ind :
  ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ v = ⋃ a ∈ A, ⋂ t ∈ a, t
Au : Set (Set (Set α))
Au_fin : Au.Finite
mem_Au : ∀ a ∈ Au, a.Finite
hAu : ∀ a ∈ Au, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ Au, ⋂ t ∈ a, t
union :
  ∀ (Av : Set (Set (Set α))) (Av_fin : Av.Finite) (mem_Av : ∀ a ∈ Av, a.Finite)
    (hAv : ∀ a ∈ Av, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) (v_eq : v = ⋃ a ∈ Av, ⋂ t ∈ a, t),
    ∃ (A : Set (Set (Set α))),
      A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u ∪ v = ⋃ a ∈ A, ⋂ t ∈ a, t
⊢ ∃ (A : Set (Set (Set α))),
    A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u ∪ v = ⋃ a ∈ A, ⋂ t ∈ a, t

Tactic:
rcases v_ind with ⟨Av, Av_fin, mem_Av, hAv, v_eq⟩
-- Module: Mathlib.MeasureTheory.SetAlgebra
Goal:
α : Type u_1
𝒜 : Set (Set α)
s u v : Set α
hs : generateSetAlgebra 𝒜 u
ht : generateSetAlgebra 𝒜 v
Au : Set (Set (Set α))
Au_fin : Au.Finite
mem_Au : ∀ a ∈ Au, a.Finite
hAu : ∀ a ∈ Au, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ Au, ⋂ t ∈ a, t
Av : Set (Set (Set α))
Av_fin : Av.Finite
mem_Av : ∀ a ∈ Av, a.Finite
hAv : ∀ a ∈ Av, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
v_eq : v = ⋃ a ∈ Av, ⋂ t ∈ a, t
rw : (⋃ a ∈ Au, ⋂ t ∈ a, t) ∪ v = ⋃ a ∈ Au ∪ Av, ⋂ t ∈ a, t
rw₁ : Eq.{u_1 + 1} (α := Set α) ((⋃ a ∈ Au, ⋂ t ∈ a, t) ∪ ⋃ a ∈ Av, ⋂ t ∈ a, t) (⋃ a ∈ Au ∪ Av, ⋂ t ∈ a, t)
rw₂ rw₃ : Eq.{u_1 + 1} (α := Set α) (⋃ x ∈ Au ∪ Av, ⋂ t ∈ x, t) (⋃ a ∈ Au ∪ Av, ⋂ t ∈ a, t)
⊢ u ∪ v = ⋃ a ∈ Au ∪ Av, ⋂ t ∈ a, t

Tactic:
rw [u_eq, v_eq, ← biUnion_union]

Premises used: Set.biUnion_union