l3lab/ntp-mathlib huggingface dataset:
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  

Proof state:
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)

Next tactic:
have claim4 := finiteInterClosure_finiteInter C1
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  

Proof state:
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)

Next tactic:
have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  

Proof state:
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
⊢ ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)

Next tactic:
have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      cases' this P hP with h h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
    intro P hP
    exact claim5.finiteInterClosure_insert _ hP
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    

Proof state:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
⊢ ∀ P ∈ C2, Set.Nonempty P

Next tactic:
suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      cases' this P hP with h h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      

Proof state:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
this : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
⊢ ∀ P ∈ C2, Set.Nonempty P

Next tactic:
intro P hP
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      

Proof state:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
this : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
P : Set (Ultrafilter X.A)
hP : P ∈ C2
⊢ Set.Nonempty P

Next tactic:
cases' this P hP with h h
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      cases' this P hP with h h
      · 

Proof state:
case inl
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
this : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
P : Set (Ultrafilter X.A)
hP : P ∈ C2
h : P ∈ C0
⊢ Set.Nonempty P

Next tactic:
exact claim2 _ h
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      cases' this P hP with h h
      · exact claim2 _ h
      · 

Proof state:
case inr
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
this : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
P : Set (Ultrafilter X.A)
hP : P ∈ C2
h : ∃ Q ∈ C0, P = AA ∩ Q
⊢ Set.Nonempty P

Next tactic:
rcases h with ⟨Q, hQ, rfl⟩
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  rintro _ ⟨F, hF, rfl⟩
  -- Notation to be used in this proof.
  let fsu := Finset (Set (Ultrafilter X))
  let ssu := Set (Set (Ultrafilter X))
  let ι : fsu → ssu := fun x ↦ ↑x
  let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
  let AA := { G : Ultrafilter X | A ∈ G }
  let C1 := insert AA C0
  let C2 := finiteInterClosure C1
  -- C0 is closed under intersections.
  have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
    rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
    use Q ∩ R
    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
    exact inter_sets _ hQ hR
  -- All sets in C0 are nonempty.
  have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
    rintro B ⟨Q, hQ, rfl⟩
    obtain ⟨q⟩ := Filter.nonempty_of_mem hQ
    use X.incl q
    simpa
  -- The intersection of AA with every set in C0 is nonempty.
  have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by
    rintro B ⟨Q, hQ, rfl⟩
    have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
    rcases this with ⟨q, hq1, P, hq2, hq3⟩
    refine' ⟨P, hq2, _⟩
    rw [← hq3] at hq1
    simpa
  -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty.
  suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by
    obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
    use X.join G
    have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩)
    rw [join_distrib, this]
    exact ⟨h1 (Or.inl rfl), rfl⟩
  -- C2 is closed under finite intersections (by construction!).
  have claim4 := finiteInterClosure_finiteInter C1
  -- C0 is closed under finite intersections by claim1.
  have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩
  -- Every element of C2 is nonempty.
  have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by
    suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      cases' this P hP with h h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        

Proof state:
case inr.intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, str X ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, Set.Nonempty B
claim3 : ∀ B ∈ C0, Set.Nonempty (AA ∩ B)
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
this : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
Q : Set (Ultrafilter X.A)
hQ : Q ∈ C0
hP : AA ∩ Q ∈ C2
⊢ Set.Nonempty (AA ∩ Q)

Next tactic:
exact claim3 _ hQ