-- Mathlib/Topology/Category/Compactum.lean
theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x

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))
⊢ ↑F ≤ 𝓝 x → X.str F = x

Tactic:
let ι : fsu → ssu := fun x ↦ ↑x

Result:
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
⊢ ↑F ≤ 𝓝 x → X.str F = x

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
⊢ ↑F ≤ 𝓝 x → X.str F = x

Tactic:
let T0 : ssu := { S | ∃ A ∈ F, S = basic A }

Result:
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}
⊢ ↑F ≤ 𝓝 x → X.str F = x

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) := X.str ⁻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
⊢ X.str F = x

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))

Result:
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) := X.str ⁻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
⊢ X.str F = x

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) := X.str ⁻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
S1 : Set X.A
hS1 : S1 ∈ F
S2 : Set X.A
hS2 : S2 ∈ F
⊢ Compactum.basic S1 ∩ Compactum.basic S2 = Compactum.basic (S1 ∩ S2)

Tactic:
simp [basic_inter]

Result:
no goals

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) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
⊢ ↑F ≤ 𝓝 x → X.str F = x

Tactic:
let T2 := finiteInterClosure T1

Result:
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) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
⊢ ↑F ≤ 𝓝 x → X.str F = x

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) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
A : Set X.A
hA : IsClosed A
h : A ∈ F
⊢ x ∈ A

Tactic:
by_contra H

Result:
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) := X.str ⁻¹' {x}
T1 : ssu := insert AA T0
T2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure T1
cond : ↑F ≤ 𝓝 x
A : Set X.A
hA : IsClosed A
h : A ∈ F
H : x ∉ A
⊢ False

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) := X.str ⁻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, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
claim6 : ∀ S ∈ T2, S.Nonempty
T : fsu
hT : ↑T ⊆ T1
t : Set (Ultrafilter X.A)
ht : t ∈ ↑T
⊢ t ∈ finiteInterClosure T1

Tactic:
exact finiteInterClosure.basic (@hT t ht)

Result:
no goals

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) := X.str ⁻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
S1 : Set X.A
hS1 : S1 ∈ F
S2 : Set X.A
hS2 : S2 ∈ F
⊢ Compactum.basic S1 ∩ Compactum.basic S2 ∈ T0

Tactic:
exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩

Result:
no goals

Proof state:
case h.h
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) := X.str ⁻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, (AA ∩ S).Nonempty
claim5 : ∀ S ∈ T0, S.Nonempty
S : Set (Ultrafilter X.A)
hS : S ∈ T2
x✝ : Ultrafilter X.A
a✝ : x✝ ∈ Set.univ
⊢ x✝ ∈ Compactum.basic Set.univ

Tactic:
apply Filter.univ_sets

Result:
no goals

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) := X.str ⁻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
⊢ ∀ S1 ∈ T0, ∀ S2 ∈ T0, S1 ∩ S2 ∈ T0

Tactic:
rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩

Result:
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) := X.str ⁻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
S1 : Set X.A
hS1 : S1 ∈ F
S2 : Set X.A
hS2 : S2 ∈ F
⊢ Compactum.basic S1 ∩ Compactum.basic S2 ∈ T0