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
    

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
⊢ ∀ B ∈ C0, Set.Nonempty (AA ∩ B)

Next tactic:
rintro B ⟨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⟩
    

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
Q : Set X.A
hQ : Q ∈ F
⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q)

Next tactic:
have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)
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)
    

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
Q : Set X.A
hQ : Q ∈ F
this : Set.Nonempty (Q ∩ Compactum.cl A)
⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q)

Next tactic:
rcases this with ⟨q, hq1, P, hq2, hq3⟩
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⟩
    

Proof state:
case intro.intro.intro.intro.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
Q : Set X.A
hQ : Q ∈ F
q : X.A
hq1 : q ∈ Q
P : Ultrafilter X.A
hq2 : P ∈ Compactum.basic A
hq3 : str X P = q
⊢ Set.Nonempty (AA ∩ str X ⁻¹' Q)

Next tactic:
refine' ⟨P, hq2, _⟩
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, _⟩
    

Proof state:
case intro.intro.intro.intro.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
Q : Set X.A
hQ : Q ∈ F
q : X.A
hq1 : q ∈ Q
P : Ultrafilter X.A
hq2 : P ∈ Compactum.basic A
hq3 : str X P = q
⊢ P ∈ str X ⁻¹' Q

Next tactic:
rw [← hq3] at hq1
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
    

Proof state:
case intro.intro.intro.intro.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
Q : Set X.A
hQ : Q ∈ F
q : X.A
P : Ultrafilter X.A
hq1 : str X P ∈ Q
hq2 : P ∈ Compactum.basic A
hq3 : str X P = q
⊢ P ∈ str X ⁻¹' Q

Next tactic:
simpa
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.
  

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)
⊢ str X F ∈ Compactum.cl A

Next tactic:
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⟩
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
    

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)
this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)
⊢ str X F ∈ Compactum.cl A

Next tactic:
obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this
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
    

Proof state:
case 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)
this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ G.sets
⊢ str X F ∈ Compactum.cl A

Next tactic:
use X.join G
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
    

Proof state:
case h
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)
this : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ G.sets
⊢ join X G ∈ Compactum.basic A ∧ str X (join X G) = str X F

Next tactic:
have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, 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⟩)
    

Proof state:
case h
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)
this✝ : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ G.sets
this : Ultrafilter.map (str X) G = F
⊢ join X G ∈ Compactum.basic A ∧ str X (join X G) = str X F

Next tactic:
rw [join_distrib, this]
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]
    

Proof state:
case h
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)
this✝ : ∀ (T : fsu), ι T ⊆ C1 → Set.Nonempty (⋂₀ ι T)
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ G.sets
this : Ultrafilter.map (str X) G = F
⊢ join X G ∈ Compactum.basic A ∧ str X F = str X F

Next tactic:
exact ⟨h1 (Or.inl rfl), rfl⟩