l3lab/ntp-mathlib huggingface dataset:
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
⊢ ∀ S ∈ T0, Set.Nonempty (AA ∩ S)

Next tactic:
rintro S ⟨S, hS, rfl⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    

Proof state:
case intro.intro
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
S : Set X.A
hS : S ∈ F
⊢ Set.Nonempty (AA ∩ Compactum.basic S)

Next tactic:
rcases claim2 _ hS with ⟨G, hG, hG2⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    

Proof state:
case intro.intro.intro.intro
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
S : Set X.A
hS : S ∈ F
G : Ultrafilter X.A
hG : G ∈ Compactum.basic S
hG2 : str X G = x
⊢ Set.Nonempty (AA ∩ Compactum.basic S)

Next tactic:
exact ⟨G, hG2, hG⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
⊢ str X F = x

Next tactic:
have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
⊢ ∀ S ∈ T0, Set.Nonempty S

Next tactic:
rintro S ⟨S, hS, rfl⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    

Proof state:
case intro.intro
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
S : Set X.A
hS : S ∈ F
⊢ Set.Nonempty (Compactum.basic S)

Next tactic:
exact ⟨F, hS⟩
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
  -- Every element of T2 is nonempty.
  

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
claim5 : ∀ S ∈ T0, Set.Nonempty S
⊢ str X F = x

Next tactic:
have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      cases' this _ hS with h h
      · exact claim5 S h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim4 Q hQ
    intro S hS
    apply finiteInterClosure_insert
    · constructor
      · use Set.univ
        refine' ⟨Filter.univ_sets _, _⟩
        ext
        refine' ⟨_, by tauto⟩
        · intro
          apply Filter.univ_sets
      · exact claim3
    · exact hS
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
  -- Every element of T2 is nonempty.
  have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
claim5 : ∀ S ∈ T0, Set.Nonempty S
⊢ ∀ S ∈ T2, Set.Nonempty S

Next tactic:
suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      cases' this _ hS with h h
      · exact claim5 S h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim4 Q hQ
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
  -- Every element of T2 is nonempty.
  have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
claim5 : ∀ S ∈ T0, Set.Nonempty S
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
⊢ ∀ S ∈ T2, Set.Nonempty S

Next tactic:
intro S hS
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
  -- Every element of T2 is nonempty.
  have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      

Proof state:
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
claim5 : ∀ S ∈ T0, Set.Nonempty S
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ Set.Nonempty S

Next tactic:
cases' this _ hS with h h
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by
  -- 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 T0 : ssu := { S | ∃ A ∈ F, S = basic A }
  let AA := X.str ⁻¹' {x}
  let T1 := insert AA T0
  let T2 := finiteInterClosure T1
  intro cond
  -- If F contains a closed set A, then x is contained in A.
  have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
    intro A hA h
    by_contra H
    rw [le_nhds_iff] at cond
    specialize cond Aᶜ H hA.isOpen_compl
    rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_not_mem] at cond
    contradiction
  -- If A ∈ F, then x ∈ cl A.
  have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
    intro A hA
    exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
  -- T0 is closed under intersections.
  have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
    rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩
    exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩
  -- For every S ∈ T0, the intersection AA ∩ S is nonempty.
  have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
  -- Every element of T0 is nonempty.
  have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
  -- Every element of T2 is nonempty.
  have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      cases' this _ hS with h h
      · 

Proof state:
case inl
X : Compactum
F : Ultrafilter X.A
x : X.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
T0 : ssu := {S | ∃ A ∈ F, S = Compactum.basic A}
AA : Set (Ultrafilter X.A) := str X ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ Compactum.cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, Set.Nonempty (AA ∩ S)
claim5 : ∀ S ∈ T0, Set.Nonempty S
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
S : Set (Ultrafilter X.A)
hS : S ∈ T2
h : S ∈ T0
⊢ Set.Nonempty S

Next tactic:
exact claim5 S h