awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ basic (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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
C0 : ssu := {Z : Set (Ultrafilter X.A) | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G : Ultrafilter X.A | 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, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
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
⊢ P.Nonempty
have inr : ∀ (Q : Set (Ultrafilter X.A)) (hQ : Q ∈ C0) (hP : AA ∩ Q ∈ C2), (AA ∩ Q).Nonempty
Tactic: rcases h with ⟨Q, hQ, rfl⟩
Cleared hypotheses: P, hP, h
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ basic (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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
C0 : ssu := {Z : Set (Ultrafilter X.A) | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G : Ultrafilter X.A | 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, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
⊢ ∀ P ∈ C2, P.Nonempty
have suffices : ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q
Tactic: suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by
      intro P hP
      rcases this P hP with h | h
      · exact claim2 _ h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim3 _ hQ
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ basic (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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
C0 : ssu := {Z : Set (Ultrafilter X.A) | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G : Ultrafilter X.A | 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, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
⊢ ∀ (T : fsu), ι T ⊆ C1 → (⋂₀ ι T).Nonempty
have claim6 : ∀ P ∈ C2, P.Nonempty
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
      rcases 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
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ basic (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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
C0 : ssu := {Z : Set (Ultrafilter X.A) | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G : Ultrafilter X.A | 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, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
claim6 : ∀ P ∈ C2, P.Nonempty
T : fsu
hT : ι T ⊆ C1
⊢ (⋂₀ ι T).Nonempty
have suffices : ⋂₀ ι T ∈ C2
Tactic: suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ basic (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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
C0 : ssu := {Z : Set (Ultrafilter X.A) | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G : Ultrafilter X.A | 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, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
claim4 : FiniteInter (finiteInterClosure C1)
claim5 : FiniteInter C0
claim6 : ∀ P ∈ C2, P.Nonempty
T : fsu
hT : ι T ⊆ C1
⊢ ⋂₀ ι T ∈ C2
have apply : (↑T : Set (Set (Ultrafilter X.A))) ⊆ finiteInterClosure C1
Tactic: apply claim4.finiteInter_mem T
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
X : Compactum
A : Set X.A
⊢ IsClosed (cl A)
have rw : ∀ (F : Ultrafilter X.A), cl A ∈ F → X.str F ∈ cl A
Tactic: rw [isClosed_iff]
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
A : Set X.A
hA : IsClosed A
h : A ∈ F
⊢ x ∈ A
have by_contra : ∀ (H : x ∉ A), False
Tactic: by_contra H
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
A : Set X.A
hA : IsClosed A
h : A ∈ F
H : x ∉ A
⊢ False
have rw : ∀ (cond : ∀ (s : Set X.A), x ∈ s → IsOpen s → s ∈ (↑F : Filter X.A)) (A : Set X.A) (hA : IsClosed A) (h : A ∈ F)
  (H : x ∉ A), False
Tactic: rw [le_nhds_iff] at cond
Cleared hypotheses: cond, A, hA, h, H
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ∀ (s : Set X.A), x ∈ s → IsOpen s → s ∈ (↑F : Filter X.A)
A : Set X.A
hA : IsClosed A
h : A ∈ F
H : x ∉ A
⊢ False
have cond₁ : Aᶜ ∈ (↑F : Filter X.A)
Tactic: specialize cond Aᶜ H hA.isOpen_compl
Cleared hypotheses: cond
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
A : Set X.A
hA : IsClosed A
h : A ∈ F
H : x ∉ A
cond : Aᶜ ∈ (↑F : Filter X.A)
⊢ False
have cond₁ : A ∉ F
Tactic: rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_notMem] at cond
Cleared hypotheses: cond
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
⊢ X.str F = x
have claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
Tactic: 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_notMem] at cond
    contradiction
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
⊢ X.str F = x
have claim2 : ∀ A ∈ F, x ∈ cl A
Tactic: 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))
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
⊢ X.str F = x
have claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
Tactic: 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]⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
S : Set X.A
hS : S ∈ F
⊢ (AA ∩ basic S).Nonempty
have rcases : ∀ (G : Ultrafilter X.A) (hG : G ∈ basic S) (hG2 : X.str G = x), (AA ∩ basic S).Nonempty
Tactic: rcases claim2 _ hS with ⟨G, hG, hG2⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
⊢ X.str F = x
have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
Tactic: have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by
    rintro S ⟨S, hS, rfl⟩
    rcases claim2 _ hS with ⟨G, hG, hG2⟩
    exact ⟨G, hG2, hG⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
⊢ X.str F = x
have claim5 : ∀ S ∈ T0, S.Nonempty
Tactic: have claim5 : ∀ S ∈ T0, Set.Nonempty S := by
    rintro S ⟨S, hS, rfl⟩
    exact ⟨F, hS⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ S.Nonempty
have inl : ∀ (h : S ∈ T0), S.Nonempty
Tactic: rcases this _ hS with h | h
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
S : Set (Ultrafilter X.A)
hS : S ∈ T2
inl : ∀ (h : S ∈ T0), S.Nonempty
⊢ S.Nonempty
have inr : ∀ (h : ∃ Q ∈ T0, S = AA ∩ Q), S.Nonempty
Tactic: rcases this _ hS with h | h
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
this : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
S : Set (Ultrafilter X.A)
hS : S ∈ T2
h : ∃ Q ∈ T0, S = AA ∩ Q
⊢ S.Nonempty
have inr : ∀ (Q : Set (Ultrafilter X.A)) (hQ : Q ∈ T0) (hS : AA ∩ Q ∈ T2), (AA ∩ Q).Nonempty
Tactic: rcases h with ⟨Q, hQ, rfl⟩
Cleared hypotheses: S, hS, h
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
⊢ ∀ S ∈ T2, S.Nonempty
have suffices : ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
Tactic: suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      rcases this _ hS with h | h
      · exact claim5 S h
      · rcases h with ⟨Q, hQ, rfl⟩
        exact claim4 Q hQ
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q
have cond₁ : FiniteInter T0
Tactic: apply finiteInterClosure_insert
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ FiniteInter T0
have cond₁ : Set.univ ∈ T0
Tactic: constructor
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ Set.univ ∈ T0
have h : Set.univ ∈ F ∧ Eq.{u_1 + 1} (α := Set (Ultrafilter X.A)) Set.univ (basic Set.univ)
Tactic: use Set.univ
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ Set.univ ∈ F ∧ Eq.{u_1 + 1} (α := Set (Ultrafilter X.A)) Set.univ (basic Set.univ)
have h : Eq.{u_1 + 1} (α := Set (Ultrafilter X.A)) Set.univ (basic Set.univ)
Tactic: refine ⟨Filter.univ_sets _, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
⊢ Eq.{u_1 + 1} (α := Set (Ultrafilter X.A)) Set.univ (basic Set.univ)
have h : ∀ (x : Ultrafilter X.A),
  Membership.mem (γ := Set (Ultrafilter X.A)) Set.univ x ↔
    Membership.mem (γ := Set (Ultrafilter X.A)) (basic Set.univ) x
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
x_1 : Ultrafilter X.A
⊢ Membership.mem (γ := Set (Ultrafilter X.A)) Set.univ x_1 ↔
    Membership.mem (γ := Set (Ultrafilter X.A)) (basic Set.univ) x_1
have h : Membership.mem (γ := Set (Ultrafilter X.A)) Set.univ x_1 →
  Membership.mem (γ := Set (Ultrafilter X.A)) (basic Set.univ) x_1
Tactic: refine ⟨?_, by tauto⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (hypothesis)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
⊢ X.str F = x
have claim6 : ∀ S ∈ T2, S.Nonempty
Tactic: have claim6 : ∀ S ∈ T2, Set.Nonempty S := by
    suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by
      intro S hS
      rcases 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
-- Lean 4 proof step from Mathlib.Topology.Category.Compactum (goal)
Goal:
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 : fsu) ↦ (↑x : Set (Set (Ultrafilter X.A)))
T0 : ssu := {S : Set (Ultrafilter X.A) | ∃ A ∈ F, S = basic A}
AA : Set (Ultrafilter X.A) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : (↑F : Filter X.A) ≤ 𝓝 x
claim1 : ∀ (A : Set X.A), IsClosed A → A ∈ F → x ∈ A
claim2 : ∀ A ∈ F, x ∈ cl A
claim3 : ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0
claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
claim6 : ∀ S ∈ T2, S.Nonempty
this : ∀ (F : fsu), (↑F : Set (Set (Ultrafilter X.A))) ⊆ T1 → (⋂₀ ι F).Nonempty
⊢ X.str F = x
have obtain : ∀ (G : Ultrafilter (Ultrafilter X.A)) (h1 : T1 ⊆ (↑G : Filter (Ultrafilter X.A)).sets), X.str F = x
Tactic: obtain ⟨G, h1⟩ := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this