l3lab/ntp-mathlib huggingface dataset:
theorem isClosed_iff {X : Compactum} (S : Set X) :
    IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
  rw [← isOpen_compl_iff]
  constructor
  · intro cond F h
    by_contra c
    specialize cond F c
    rw [compl_mem_iff_not_mem] at cond
    contradiction
  · intro h1 F h2
    specialize h1 F
    cases' F.mem_or_compl_mem S with h h
    

Proof state:
case mpr.inl
X : Compactum
S : Set X.A
F : Ultrafilter X.A
h2 : str X F ∈ Sᶜ
h1 : S ∈ F → str X F ∈ S
h : S ∈ F
⊢ Sᶜ ∈ F
case mpr.inr
X : Compactum
S : Set X.A
F : Ultrafilter X.A
h2 : str X F ∈ Sᶜ
h1 : S ∈ F → str X F ∈ S
h : Sᶜ ∈ F
⊢ Sᶜ ∈ F

Next tactic:
exacts [absurd (h1 h) h2, h]
instance {X : Compactum} : CompactSpace X := by
  

Proof state:
X : Compactum
⊢ CompactSpace X.A

Next tactic:
constructor
instance {X : Compactum} : CompactSpace X := by
  constructor
  

Proof state:
case isCompact_univ
X : Compactum
⊢ IsCompact Set.univ

Next tactic:
rw [isCompact_iff_ultrafilter_le_nhds]
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  

Proof state:
case isCompact_univ
X : Compactum
⊢ ∀ (f : Ultrafilter X.A), ↑f ≤ 𝓟 Set.univ → ∃ x ∈ Set.univ, ↑f ≤ 𝓝 x

Next tactic:
intro F _
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  

Proof state:
case isCompact_univ
X : Compactum
F : Ultrafilter X.A
a✝ : ↑F ≤ 𝓟 Set.univ
⊢ ∃ x ∈ Set.univ, ↑F ≤ 𝓝 x

Next tactic:
refine' ⟨X.str F, by tauto, _⟩
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine' ⟨X.str F, by 

Proof state:
X : Compactum
F : Ultrafilter X.A
a✝ : ↑F ≤ 𝓟 Set.univ
⊢ str X F ∈ Set.univ

Next tactic:
tauto
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine' ⟨X.str F, by tauto, _⟩
  

Proof state:
case isCompact_univ
X : Compactum
F : Ultrafilter X.A
a✝ : ↑F ≤ 𝓟 Set.univ
⊢ ↑F ≤ 𝓝 (str X F)

Next tactic:
rw [le_nhds_iff]
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine' ⟨X.str F, by tauto, _⟩
  rw [le_nhds_iff]
  

Proof state:
case isCompact_univ
X : Compactum
F : Ultrafilter X.A
a✝ : ↑F ≤ 𝓟 Set.univ
⊢ ∀ (s : Set X.A), str X F ∈ s → IsOpen s → s ∈ ↑F

Next tactic:
intro S h1 h2
instance {X : Compactum} : CompactSpace X := by
  constructor
  rw [isCompact_iff_ultrafilter_le_nhds]
  intro F _
  refine' ⟨X.str F, by tauto, _⟩
  rw [le_nhds_iff]
  intro S h1 h2
  

Proof state:
case isCompact_univ
X : Compactum
F : Ultrafilter X.A
a✝ : ↑F ≤ 𝓟 Set.univ
S : Set X.A
h1 : str X F ∈ S
h2 : IsOpen S
⊢ S ∈ ↑F

Next tactic:
exact h2 F h1
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  

Proof state:
X : Compactum
A B : Set X.A
⊢ Compactum.basic (A ∩ B) = Compactum.basic A ∩ Compactum.basic B

Next tactic:
ext G
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  

Proof state:
case h
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ Compactum.basic (A ∩ B) ↔ G ∈ Compactum.basic A ∩ Compactum.basic B

Next tactic:
constructor
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · 

Proof state:
case h.mp
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ Compactum.basic (A ∩ B) → G ∈ Compactum.basic A ∩ Compactum.basic B

Next tactic:
intro hG
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    

Proof state:
case h.mp
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
hG : G ∈ Compactum.basic (A ∩ B)
⊢ G ∈ Compactum.basic A ∩ Compactum.basic B

Next tactic:
constructor
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> 

Proof state:
case h.mp.left
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
hG : G ∈ Compactum.basic (A ∩ B)
⊢ G ∈ Compactum.basic A

Next tactic:
filter_upwards [hG] with _
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> 

Proof state:
case h.mp.right
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
hG : G ∈ Compactum.basic (A ∩ B)
⊢ G ∈ Compactum.basic B

Next tactic:
filter_upwards [hG] with _
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    

Proof state:
case h
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
hG : G ∈ Compactum.basic (A ∩ B)
a✝ : X.A
⊢ a✝ ∈ A ∩ B → a✝ ∈ A
case h X : Compactum A B : Set X.A G : Ultrafilter X.A hG : G ∈ Compactum.basic (A ∩ B) a✝ : X.A ⊢ a✝ ∈ A ∩ B → a✝ ∈ B

Next tactic:
exacts [And.left, And.right]
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · 

Proof state:
case h.mpr
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ Compactum.basic A ∩ Compactum.basic B → G ∈ Compactum.basic (A ∩ B)

Next tactic:
rintro ⟨h1, h2⟩
private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by
  ext G
  constructor
  · intro hG
    constructor <;> filter_upwards [hG] with _
    exacts [And.left, And.right]
  · rintro ⟨h1, h2⟩
    

Proof state:
case h.mpr.intro
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
h1 : G ∈ Compactum.basic A
h2 : G ∈ Compactum.basic B
⊢ G ∈ Compactum.basic (A ∩ B)

Next tactic:
exact inter_mem h1 h2
private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha =>
  ⟨X.incl a, ha, by 

Proof state:
X : Compactum
A : Set X.A
a : X.A
ha : a ∈ A
⊢ str X (incl X a) = a

Next tactic:
simp
private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
  

Proof state:
X : Compactum
A : Set X.A
⊢ Compactum.cl (Compactum.cl A) ⊆ Compactum.cl A

Next tactic:
rintro _ ⟨F, hF, 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.
  

Proof state:
case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
⊢ str X F ∈ Compactum.cl A

Next tactic:
let fsu := Finset (Set (Ultrafilter X))
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))
  

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))
⊢ str X F ∈ Compactum.cl A

Next tactic:
let ssu := Set (Set (Ultrafilter X))
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))
  

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))
⊢ str X F ∈ Compactum.cl A

Next tactic:
let ι : fsu → ssu := fun x ↦ ↑x
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
  

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
⊢ str X F ∈ Compactum.cl A

Next tactic:
let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
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 }
  

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}
⊢ str X F ∈ Compactum.cl A

Next tactic:
let AA := { G : Ultrafilter X | A ∈ 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 }
  

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}
⊢ str X F ∈ Compactum.cl A

Next tactic:
let C1 := insert AA C0
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
  

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
⊢ str X F ∈ Compactum.cl A

Next tactic:
let C2 := finiteInterClosure 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.
  

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
⊢ str X F ∈ Compactum.cl A

Next tactic:
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
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
    

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
⊢ ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0

Next tactic:
rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, 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⟩
    

Proof state:
case 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
Q : Set X.A
hQ : Q ∈ F
R : Set X.A
hR : R ∈ F
⊢ str X ⁻¹' Q ∩ str X ⁻¹' R ∈ C0

Next tactic:
use Q ∩ R
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
    

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
Q : Set X.A
hQ : Q ∈ F
R : Set X.A
hR : R ∈ F
⊢ Q ∩ R ∈ F ∧ str X ⁻¹' (Q ∩ R) = str X ⁻¹' Q ∩ str X ⁻¹' R

Next tactic:
simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
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]
    

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
Q : Set X.A
hQ : Q ∈ F
R : Set X.A
hR : R ∈ F
⊢ Q ∩ R ∈ F

Next tactic:
exact inter_sets _ hQ hR
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.
  

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
⊢ str X F ∈ Compactum.cl A

Next tactic:
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
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
    

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
⊢ ∀ B ∈ C0, Set.Nonempty 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⟩
    

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

Next tactic:
obtain ⟨q⟩ := Filter.nonempty_of_mem 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
    

Proof state:
case 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
Q : Set X.A
hQ : Q ∈ F
q : X.A
h✝ : q ∈ Q
⊢ Set.Nonempty (str X ⁻¹' Q)

Next tactic:
use X.incl q
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
    

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
Q : Set X.A
hQ : Q ∈ F
q : X.A
h✝ : q ∈ Q
⊢ incl X q ∈ 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.
  

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
⊢ str X F ∈ Compactum.cl A

Next tactic:
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