mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.CWComplex.Classical.Subcomplex

Goal state before:
pp: case pos
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
n : ℕ
x✝ : 0 < n
j : cell C n
hj : j ∈ E.I n
⊢ Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)

pp: case neg
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
n : ℕ
x✝ : 0 < n
j : cell C n
hj : j ∉ E.I n
⊢ Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)

Tactic:
·

Goal state after:
case pos
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
n : ℕ
x✝ : 0 < n
j : cell C n
hj : j ∈ E.I n
⊢ Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)
Module: Mathlib.Topology.CWComplex.Classical.Subcomplex

Goal state before:
pp: case neg
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
n : ℕ
x✝ : 0 < n
j : cell C n
hj : j ∉ E.I n
⊢ Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)

Tactic:
·

Goal state after:
case neg
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : T2Space X
inst✝ : RelCWComplex C D
E : Subcomplex C
A : Set X
hA : A ⊆ ↑E
h : (∀ (n : ℕ) (j : ↑(E.I n)), IsClosed[t] (A ∩ ↑(map n ↑j) '' closedBall 0 1)) ∧ IsClosed[t] (A ∩ D)
n : ℕ
x✝ : 0 < n
j : cell C n
hj : j ∉ E.I n
⊢ Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)
Module: Mathlib.Topology.CWComplex.Classical.Finite

Goal state before:
pp: X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
⊢ ∀ᶠ (n : ℕ) in Filter.atTop, IsEmpty (cell C n)

Tactic:
simp only [Filter.eventually_atTop]

Goal state after:
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)
Module: Mathlib.Topology.CWComplex.Classical.Finite

Goal state before:
pp: case inl
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
h✝ : IsEmpty ((n : ℕ) × cell C n)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)

pp: case inr
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
h✝ : Nonempty ((n : ℕ) × cell C n)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)

Tactic:
·

Goal state after:
case inl
X : Type u_2
inst✝¹ : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
finite : _root_.Finite ((n : ℕ) × cell C n)
h✝ : IsEmpty ((n : ℕ) × cell C n)
⊢ ∃ a, ∀ (b : ℕ), a ≤ b → IsEmpty (cell C b)
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case mp
X : Compactum
S : Set X.A
⊢ IsOpen Sᶜ → ∀ (F : Ultrafilter X.A), S ∈ F → X.str F ∈ S

pp: case mpr
X : Compactum
S : Set X.A
⊢ (∀ (F : Ultrafilter X.A), S ∈ F → X.str F ∈ S) → IsOpen Sᶜ

Tactic:
·

Goal state after:
case mp
X : Compactum
S : Set X.A
⊢ IsOpen Sᶜ → ∀ (F : Ultrafilter X.A), S ∈ F → X.str F ∈ S
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case mpr
X : Compactum
S : Set X.A
⊢ (∀ (F : Ultrafilter X.A), S ∈ F → X.str F ∈ S) → IsOpen Sᶜ

Tactic:
·

Goal state after:
case mpr
X : Compactum
S : Set X.A
⊢ (∀ (F : Ultrafilter X.A), S ∈ F → X.str F ∈ S) → IsOpen Sᶜ
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: X : Compactum
A B : Set X.A
⊢ basic (A ∩ B) = basic A ∩ basic B

Tactic:
ext G

Goal state after:
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ basic (A ∩ B) ↔ G ∈ basic A ∩ basic B
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case mp
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ basic (A ∩ B) → G ∈ basic A ∩ basic B

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

Tactic:
·

Goal state after:
case mp
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ basic (A ∩ B) → G ∈ basic A ∩ basic B
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case mpr
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ basic A ∩ basic B → G ∈ basic (A ∩ B)

Tactic:
·

Goal state after:
case mpr
X : Compactum
A B : Set X.A
G : Ultrafilter X.A
⊢ G ∈ basic A ∩ basic B → G ∈ basic (A ∩ B)
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case inl
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 ↦ ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻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, 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 : P ∈ C0
⊢ P.Nonempty

pp: case inr
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 ↦ ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻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, 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

Tactic:
·

Goal state after:
case inl
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 ↦ ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻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, 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 : P ∈ C0
⊢ P.Nonempty
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case inr
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 ↦ ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻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, 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

Tactic:
·

Goal state after:
case inr
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 ↦ ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻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, 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
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: 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 = 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 ∈ 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 : S ∈ T0
⊢ S.Nonempty

pp: case inr
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 = 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 ∈ 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

Tactic:
·

Goal state after:
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 = 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 ∈ 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 : S ∈ T0
⊢ S.Nonempty
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case inr
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 = 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 ∈ 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

Tactic:
·

Goal state after:
case inr
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 = 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 ∈ 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
Module: Mathlib.Topology.Category.Profinite.Product

Goal state before:
pp: case refine_1
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a b : (fun X ↦ ↑X.toTop) (indexCone hC).pt
h :
  (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a =
    (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) b
⊢ a = b

pp: case refine_2
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
⊢ ∃ a_1, (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a_1 = a

Tactic:
·

Goal state after:
case refine_1
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a b : (fun X ↦ ↑X.toTop) (indexCone hC).pt
h :
  (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a =
    (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) b
⊢ a = b
Module: Mathlib.Topology.Category.Profinite.Product

Goal state before:
pp: case refine_2
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
⊢ ∃ a_1, (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a_1 = a

Tactic:
·

Goal state after:
case refine_2
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
⊢ ∃ a_1, (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a_1 = a
Module: Mathlib.Topology.Category.Profinite.Product

Goal state before:
pp: case refine_2
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
b : ↑C
hb : ∀ (J : Finset ι), (π_app C fun x ↦ x ∈ J) b = ↑a (op J)
⊢ ∃ a_1, (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a_1 = a

pp: ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
⊢ ∃ x, ∀ (J : Finset ι), (π_app C fun x ↦ x ∈ J) x = ↑a (op J)

Tactic:
·

Goal state after:
case refine_2
ι : Type u
X : ι → Type
inst✝² : (i : ι) → TopologicalSpace (X i)
C : Set ((i : ι) → X i)
J K : ι → Prop
inst✝¹ : ∀ (i : ι), T2Space (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
hC : IsCompact C
this : CompactSpace ↑C
a : (fun X ↦ ↑X.toTop) (limitCone (indexFunctor hC)).pt
b : ↑C
hb : ∀ (J : Finset ι), (π_app C fun x ↦ x ∈ J) b = ↑a (op J)
⊢ ∃ a_1, (ConcreteCategory.hom ((limitConeIsLimit (indexFunctor hC)).lift (indexCone hC))) a_1 = a
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case cond
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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

pp: case H
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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 ∈ finiteInterClosure (insert AA T0)

Tactic:
·

Goal state after:
case cond
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case cond.univ_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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

pp: case cond.inter_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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 : Set (Ultrafilter X.A)⦄, s ∈ T0 → ∀ ⦃t : Set (Ultrafilter X.A)⦄, t ∈ T0 → s ∩ t ∈ T0

Tactic:
·

Goal state after:
case cond.univ_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case 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 = 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 ∈ 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
⊢ x✝ ∈ Set.univ → x✝ ∈ basic Set.univ

Tactic:
·

Goal state after:
case 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 = 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 ∈ 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
⊢ x✝ ∈ Set.univ → x✝ ∈ basic Set.univ
Module: Mathlib.Topology.Category.Profinite.Nobeling.Span

Goal state before:
pp: case pos
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
f : ↑(π C fun x ↦ x ∈ s) → ↥s → Bool := fun x j ↦ ↑x ↑j
x : I → Bool
hx : x ∈ C
y : I → Bool
hy : y ∈ C
h : f ⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ = f ⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩
i : I
hi : i ∈ s
⊢ ↑⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ i = ↑⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩ i

pp: case neg
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
f : ↑(π C fun x ↦ x ∈ s) → ↥s → Bool := fun x j ↦ ↑x ↑j
x : I → Bool
hx : x ∈ C
y : I → Bool
hy : y ∈ C
h : f ⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ = f ⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩
i : I
hi : i ∉ s
⊢ ↑⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ i = ↑⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩ i

Tactic:
·

Goal state after:
case pos
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
f : ↑(π C fun x ↦ x ∈ s) → ↥s → Bool := fun x j ↦ ↑x ↑j
x : I → Bool
hx : x ∈ C
y : I → Bool
hy : y ∈ C
h : f ⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ = f ⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩
i : I
hi : i ∈ s
⊢ ↑⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ i = ↑⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩ i
Module: Mathlib.Topology.Category.Profinite.Nobeling.Span

Goal state before:
pp: case neg
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
f : ↑(π C fun x ↦ x ∈ s) → ↥s → Bool := fun x j ↦ ↑x ↑j
x : I → Bool
hx : x ∈ C
y : I → Bool
hy : y ∈ C
h : f ⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ = f ⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩
i : I
hi : i ∉ s
⊢ ↑⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ i = ↑⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩ i

Tactic:
·

Goal state after:
case neg
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
f : ↑(π C fun x ↦ x ∈ s) → ↥s → Bool := fun x j ↦ ↑x ↑j
x : I → Bool
hx : x ∈ C
y : I → Bool
hy : y ∈ C
h : f ⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ = f ⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩
i : I
hi : i ∉ s
⊢ ↑⟨Proj (fun x ↦ x ∈ s) x, ⋯⟩ i = ↑⟨Proj (fun x ↦ x ∈ s) y, ⋯⟩ i
Module: Mathlib.Topology.Category.Compactum

Goal state before:
pp: case cond.inter_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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 : Set (Ultrafilter X.A)⦄, s ∈ T0 → ∀ ⦃t : Set (Ultrafilter X.A)⦄, t ∈ T0 → s ∩ t ∈ T0

Tactic:
·

Goal state after:
case cond.inter_mem
X : Compactum
F : Ultrafilter X.A
x : X.A
fsu : Type u_1 := ⋯
ssu : Type u_1 := ⋯
ι : fsu → ssu := ⋯
T0 : ssu := ⋯
AA : Set (Ultrafilter X.A) := ⋯
T1 : ssu := ⋯
T2 : Set (Set (Ultrafilter X.A)) := ⋯
cond : ↑F ≤ 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 : Set (Ultrafilter X.A)⦄, s ∈ T0 → ∀ ⦃t : Set (Ultrafilter X.A)⦄, t ∈ T0 → s ∩ t ∈ T0
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
I J : (n : ℕ) → Set (cell C n)
hIJ : D ∪ ⋃ n, ⋃ j, openCell n ↑j = D ∪ ⋃ n, ⋃ j, openCell n ↑j
n : ℕ
i : cell C n
hi : i ∈ I n
hJ : i ∉ J n
h : openCell n i ⊆ D ∪ ⋃ n, ⋃ j, openCell n ↑j
m : ℕ
j : ↑(J m)
⊢ ⟨n, i⟩ ≠ ⟨m, ↑j⟩

Tactic:
by lia

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Span

Goal state before:
pp: case pos
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
x y : ↑(π C fun x ↦ x ∈ s)
h : y = x
b : I
left✝ : b ∈ s.sort fun x1 x2 ↦ x1 ≥ x2
hh : ↑x b = true
⊢ (e (π C fun x ↦ x ∈ s) b) x = 1

pp: case neg
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
x y : ↑(π C fun x ↦ x ∈ s)
h : y = x
b : I
left✝ : b ∈ s.sort fun x1 x2 ↦ x1 ≥ x2
hh : ¬↑x b = true
⊢ (1 - e (π C fun x ↦ x ∈ s) b) x = 1

Tactic:
·

Goal state after:
case pos
I : Type u
C : Set (I → Bool)
inst✝ : LinearOrder I
s : Finset I
x y : ↑(π C fun x ↦ x ∈ s)
h : y = x
b : I
left✝ : b ∈ s.sort fun x1 x2 ↦ x1 ≥ x2
hh : ↑x b = true
⊢ (e (π C fun x ↦ x ∈ s) b) x = 1