url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final | [354, 1] | [363, 16] | exact hF' E | case hFecs
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
B
⊢ EqualizerCondition (F ⋙ coyoneda.obj E.op) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hFecs
A : Type (u + 2)
inst✝ : Category A
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts F
hF' : ∀ (E : A), EqualizerCondition (F ⋙ coyoneda.obj E.op)
E : A
B : CompHaus
S : Presieve B
hS :
S ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
B
⊢ EqualizerCondition (F ⋙ coyoneda.obj E.op)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final' | [365, 1] | [373, 38] | rw [Presheaf.isSheaf_iff_isSheaf_forget (coherentTopology CompHaus) F G,
isSheaf_iff_isSheaf_of_type, ← extensiveRegular_generates_coherent, Presieve.isSheaf_coverage] | A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
⊢ Presheaf.IsSheaf (coherentTopology CompHaus) F | A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
X →
IsSheafFor (F ⋙ G) R | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
⊢ Presheaf.IsSheaf (coherentTopology CompHaus) F
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final' | [365, 1] | [373, 38] | intro B S' hS | A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
X →
IsSheafFor (F ⋙ G) R | A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
B : CompHaus
S' : Presieve B
hS :
S' ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
B
⊢ IsSheafFor (F ⋙ G) S' | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
⊢ ∀ {X : CompHaus} (R : Presieve X),
R ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
X →
IsSheafFor (F ⋙ G) R
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.final' | [365, 1] | [373, 38] | exact isSheafFor_of_Dagur hS hF hF' | A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
B : CompHaus
S' : Presieve B
hS :
S' ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
B
⊢ IsSheafFor (F ⋙ G) S' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type (u + 2)
inst✝³ : Category A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
hF : PreservesFiniteProducts (F ⋙ G)
hF' : EqualizerCondition (F ⋙ G)
B : CompHaus
S' : Presieve B
hS :
S' ∈
Coverage.covering
(ExtensiveRegularCoverage' CompHaus
(_ : ∀ {X Y Z : CompHaus} (f : Y ⟶ X) (π : Z ⟶ X) [inst : Epi π], Epi Limits.pullback.fst)
(_ :
∀ {α : Type} [inst : Fintype α] {X : CompHaus} {Z : α → CompHaus} (π : (a : α) → Z a ⟶ X) {Y : CompHaus}
(f : Y ⟶ X) (x : IsIso (Sigma.desc π)), IsIso (Sigma.desc fun x_1 => Limits.pullback.fst)))
B
⊢ IsSheafFor (F ⋙ G) S'
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TopCat.isOpen_iso | [16, 1] | [23, 13] | let ff := TopCat.homeoOfIso f | X Y : TopCat
U : Set ↑X
f : X ≅ Y
h : IsOpen U
⊢ IsOpen (↑f.hom '' U) | X Y : TopCat
U : Set ↑X
f : X ≅ Y
h : IsOpen U
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
⊢ IsOpen (↑f.hom '' U) | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
U : Set ↑X
f : X ≅ Y
h : IsOpen U
⊢ IsOpen (↑f.hom '' U)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TopCat.isOpen_iso | [16, 1] | [23, 13] | rw [← Homeomorph.isOpen_image ff] at h | X Y : TopCat
U : Set ↑X
f : X ≅ Y
h : IsOpen U
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
⊢ IsOpen (↑f.hom '' U) | X Y : TopCat
U : Set ↑X
f : X ≅ Y
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
h : IsOpen (↑ff '' U)
⊢ IsOpen (↑f.hom '' U) | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
U : Set ↑X
f : X ≅ Y
h : IsOpen U
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
⊢ IsOpen (↑f.hom '' U)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TopCat.isOpen_iso | [16, 1] | [23, 13] | assumption | X Y : TopCat
U : Set ↑X
f : X ≅ Y
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
h : IsOpen (↑ff '' U)
⊢ IsOpen (↑f.hom '' U) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
U : Set ↑X
f : X ≅ Y
ff : ↑X ≃ₜ ↑Y := homeoOfIso f
h : IsOpen (↑ff '' U)
⊢ IsOpen (↑f.hom '' U)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | have surj' : Function.Surjective f.hom | X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
⊢ TotallyDisconnectedSpace ↑Y | case surj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
⊢ Function.Surjective ↑f.hom
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
⊢ TotallyDisconnectedSpace ↑Y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | have inj' : Function.Injective f.hom | X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y | case inj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ Function.Injective ↑f.hom
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | constructor | X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ IsTotallyDisconnected Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ TotallyDisconnectedSpace ↑Y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | unfold _root_.IsTotallyDisconnected | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ IsTotallyDisconnected Set.univ | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ ∀ (t : Set ↑Y), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ IsTotallyDisconnected Set.univ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | intro t _ ht₂ | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ ∀ (t : Set ↑Y), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton t | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
⊢ ∀ (t : Set ↑Y), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply Set.subsingleton_of_preimage surj' | case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton t | case isTotallyDisconnected_univ.hs
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | replace k := k.isTotallyDisconnected_univ | case isTotallyDisconnected_univ.hs
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : IsTotallyDisconnected Set.univ
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | unfold _root_.IsTotallyDisconnected at k | case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : IsTotallyDisconnected Set.univ
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : IsTotallyDisconnected Set.univ
⊢ Set.Subsingleton (↑f.hom ⁻¹' t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply k | case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t) | case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ↑f.hom ⁻¹' t ⊆ Set.univ
case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Set.Subsingleton (↑f.hom ⁻¹' t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | tauto | case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ↑f.hom ⁻¹' t ⊆ Set.univ
case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t) | case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ↑f.hom ⁻¹' t ⊆ Set.univ
case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply IsPreconnected.preimage_of_open_map | case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t) | case isTotallyDisconnected_univ.hs.a.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected t
case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected (↑f.hom ⁻¹' t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | assumption | case isTotallyDisconnected_univ.hs.a.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected t
case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hs
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsPreconnected t
case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | assumption | case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hinj
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ Function.Injective ↑f.hom
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | tauto | case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hsf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ t ⊆ Set.range ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply (TopCat.homeoOfIso f).surjective | case surj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
⊢ Function.Surjective ↑f.hom | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case surj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
⊢ Function.Surjective ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply (TopCat.homeoOfIso f).injective | case inj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ Function.Injective ↑f.hom | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inj'
X Y : TopCat
k : TotallyDisconnectedSpace ↑X
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
⊢ Function.Injective ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | unfold IsOpenMap | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ∀ (U : Set ↑X), IsOpen U → IsOpen (↑f.hom '' U) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ IsOpenMap ↑f.hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | intro U hU | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ∀ (U : Set ↑X), IsOpen U → IsOpen (↑f.hom '' U) | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
U : Set ↑X
hU : IsOpen U
⊢ IsOpen (↑f.hom '' U) | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
⊢ ∀ (U : Set ↑X), IsOpen U → IsOpen (↑f.hom '' U)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | TotallyDisconnectedSpace.ofIso | [25, 1] | [57, 8] | apply TopCat.isOpen_iso _ hU | case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
U : Set ↑X
hU : IsOpen U
⊢ IsOpen (↑f.hom '' U) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case isTotallyDisconnected_univ.hs.a.hf
X Y : TopCat
f : X ≅ Y
surj' : Function.Surjective ↑f.hom
inj' : Function.Injective ↑f.hom
t : Set ↑Y
a✝ : t ⊆ Set.univ
ht₂ : IsPreconnected t
k : ∀ (t : Set ↑X), t ⊆ Set.univ → IsPreconnected t → Set.Subsingleton t
U : Set ↑X
hU : IsOpen U
⊢ IsOpen (↑f.hom '' U)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.finiteCoproduct.hom_ext | [103, 1] | [108, 10] | ext ⟨a,x⟩ | α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
⊢ f = g | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : ↑(X a).toCompHaus.toTop
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
⊢ f = g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.finiteCoproduct.hom_ext | [103, 1] | [108, 10] | specialize h a | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : ↑(X a).toCompHaus.toTop
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ι X a ≫ f = ι X a ≫ g
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | Please generate a tactic in lean4 to solve the state.
STATE:
case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : ↑(X a).toCompHaus.toTop
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.finiteCoproduct.hom_ext | [103, 1] | [108, 10] | apply_fun (fun q => q x) at h | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ι X a ≫ f = ι X a ≫ g
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(ι X a ≫ f) x = ↑(ι X a ≫ g) x
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | Please generate a tactic in lean4 to solve the state.
STATE:
case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ι X a ≫ f = ι X a ≫ g
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.finiteCoproduct.hom_ext | [103, 1] | [108, 10] | exact h | case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(ι X a ≫ f) x = ↑(ι X a ≫ g) x
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.mk
α : Type
inst✝ : Fintype α
X : α → Profinite
B : Profinite
f g : finiteCoproduct X ⟶ B
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(ι X a ≫ f) x = ↑(ι X a ≫ g) x
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_continuous | [213, 1] | [217, 25] | apply Continuous.quotient_lift | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Continuous (ιFun π) | case h
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Continuous fun x =>
match x with
| { fst := a, snd := x } => ↑(π a) x | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Continuous (ιFun π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_continuous | [213, 1] | [217, 25] | apply continuous_sigma | case h
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Continuous fun x =>
match x with
| { fst := a, snd := x } => ↑(π a) x | case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ ∀ (i : α),
Continuous fun a =>
match { fst := i, snd := a } with
| { fst := a, snd := x } => ↑(π a) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Continuous fun x =>
match x with
| { fst := a, snd := x } => ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_continuous | [213, 1] | [217, 25] | intro a | case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ ∀ (i : α),
Continuous fun a =>
match { fst := i, snd := a } with
| { fst := a, snd := x } => ↑(π a) x | case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ Continuous fun a_1 =>
match { fst := a, snd := a_1 } with
| { fst := a, snd := x } => ↑(π a) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ ∀ (i : α),
Continuous fun a =>
match { fst := i, snd := a } with
| { fst := a, snd := x } => ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_continuous | [213, 1] | [217, 25] | exact (π a).continuous | case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ Continuous fun a_1 =>
match { fst := a, snd := a_1 } with
| { fst := a, snd := x } => ↑(π a) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ Continuous fun a_1 =>
match { fst := a, snd := a_1 } with
| { fst := a, snd := x } => ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_injective | [219, 1] | [223, 23] | rintro ⟨⟨a,x⟩⟩ ⟨⟨b,y⟩⟩ (h : π _ _ = π _ _) | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Function.Injective (ιFun π) | case mk.mk.mk.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Quot.mk Setoid.r { fst := a, snd := x } = Quot.mk Setoid.r { fst := b, snd := y } | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Function.Injective (ιFun π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_injective | [219, 1] | [223, 23] | apply Quotient.sound' | case mk.mk.mk.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Quot.mk Setoid.r { fst := a, snd := x } = Quot.mk Setoid.r { fst := b, snd := y } | case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Setoid.r { fst := a, snd := x } { fst := b, snd := y } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Quot.mk Setoid.r { fst := a, snd := x } = Quot.mk Setoid.r { fst := b, snd := y }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_injective | [219, 1] | [223, 23] | refine ⟨pullback (π a) (π b), ⟨⟨x,y⟩,h⟩, pullback.fst _ _, pullback.snd _ _, ?_, rfl, rfl⟩ | case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Setoid.r { fst := a, snd := x } { fst := b, snd := y } | case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst = pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ Setoid.r { fst := a, snd := x } { fst := b, snd := y }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_injective | [219, 1] | [223, 23] | ext ⟨_, h⟩ | case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst = pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst | case mk.mk.mk.mk.a.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h✝ : ↑(π a) x = ↑(π b) y
val✝ : ↑(X a).toCompHaus.toTop × ↑(X b).toCompHaus.toTop
h : val✝ ∈ {xy | ↑(π a) xy.fst = ↑(π b) xy.snd}
⊢ ↑(pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst) { val := val✝, property := h } =
↑(pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst) { val := val✝, property := h } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.a
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h : ↑(π a) x = ↑(π b) y
⊢ pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst = pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_injective | [219, 1] | [223, 23] | exact h | case mk.mk.mk.mk.a.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h✝ : ↑(π a) x = ↑(π b) y
val✝ : ↑(X a).toCompHaus.toTop × ↑(X b).toCompHaus.toTop
h : val✝ ∈ {xy | ↑(π a) xy.fst = ↑(π b) xy.snd}
⊢ ↑(pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst) { val := val✝, property := h } =
↑(pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst) { val := val✝, property := h } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.mk.mk.mk.a.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a₁✝ : Quotient (relation π)
a : α
x : ↑((fun a => X a) a).toCompHaus.toTop
a₂✝ : Quotient (relation π)
b : α
y : ↑((fun a => X a) b).toCompHaus.toTop
h✝ : ↑(π a) x = ↑(π b) y
val✝ : ↑(X a).toCompHaus.toTop × ↑(X b).toCompHaus.toTop
h : val✝ ∈ {xy | ↑(π a) xy.fst = ↑(π b) xy.snd}
⊢ ↑(pullback.fst (π a) (π b) ≫ π { fst := a, snd := x }.fst) { val := val✝, property := h } =
↑(pullback.snd (π a) (π b) ≫ π { fst := b, snd := y }.fst) { val := val✝, property := h }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_surjective | [225, 1] | [228, 34] | intro b | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Function.Surjective (ιFun π) | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
⊢ ∃ a, ιFun π a = b | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ Function.Surjective (ιFun π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_surjective | [225, 1] | [228, 34] | obtain ⟨a,x,h⟩ := surj b | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
⊢ ∃ a, ιFun π a = b | case intro.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(π a) x = b
⊢ ∃ a, ιFun π a = b | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
⊢ ∃ a, ιFun π a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.ιFun_surjective | [225, 1] | [228, 34] | refine ⟨Quotient.mk _ ⟨a,x⟩, h⟩ | case intro.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(π a) x = b
⊢ ∃ a, ιFun π a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
b : ↑B.toCompHaus.toTop
a : α
x : ↑(X a).toCompHaus.toTop
h : ↑(π a) x = b
⊢ ∃ a, ιFun π a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.CompHaus.TotallyDisconnectedSpace_ofIsIso | [252, 1] | [257, 41] | have f : X.toTop ≅ Y.toTop | X✝¹ Y✝ B✝ : Profinite
f✝ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f : X ≅ Y
⊢ TotallyDisconnectedSpace ↑Y.toTop | case f
X✝¹ Y✝ B✝ : Profinite
f✝ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f : X ≅ Y
⊢ X.toTop ≅ Y.toTop
X✝¹ Y✝ B✝ : Profinite
f✝¹ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f✝ : X ≅ Y
f : X.toTop ≅ Y.toTop
⊢ TotallyDisconnectedSpace ↑Y.toTop | Please generate a tactic in lean4 to solve the state.
STATE:
X✝¹ Y✝ B✝ : Profinite
f✝ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f : X ≅ Y
⊢ TotallyDisconnectedSpace ↑Y.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.CompHaus.TotallyDisconnectedSpace_ofIsIso | [252, 1] | [257, 41] | apply TotallyDisconnectedSpace.ofIso f | X✝¹ Y✝ B✝ : Profinite
f✝¹ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f✝ : X ≅ Y
f : X.toTop ≅ Y.toTop
⊢ TotallyDisconnectedSpace ↑Y.toTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝¹ Y✝ B✝ : Profinite
f✝¹ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f✝ : X ≅ Y
f : X.toTop ≅ Y.toTop
⊢ TotallyDisconnectedSpace ↑Y.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.CompHaus.TotallyDisconnectedSpace_ofIsIso | [252, 1] | [257, 41] | exact compHausToTop.mapIso f | case f
X✝¹ Y✝ B✝ : Profinite
f✝ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f : X ≅ Y
⊢ X.toTop ≅ Y.toTop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case f
X✝¹ Y✝ B✝ : Profinite
f✝ : X✝¹ ⟶ B✝
g : Y✝ ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X✝ : α → Profinite
π : (a : α) → X✝ a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X Y : CompHaus
k : TotallyDisconnectedSpace ↑X.toTop
f : X ≅ Y
⊢ X.toTop ≅ Y.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.π'_comp_ι_hom | [313, 1] | [316, 6] | ext | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π' π surj a ≫ (ι₂Iso π surj).hom = π a | case w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π' π surj a ≫ (ι₂Iso π surj).hom) x✝ = ↑(π a) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π' π surj a ≫ (ι₂Iso π surj).hom = π a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.π'_comp_ι_hom | [313, 1] | [316, 6] | rfl | case w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π' π surj a ≫ (ι₂Iso π surj).hom) x✝ = ↑(π a) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π' π surj a ≫ (ι₂Iso π surj).hom) x✝ = ↑(π a) x✝
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.π_comp_ι_inv | [318, 1] | [321, 31] | rw [Iso.comp_inv_eq] | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a ≫ (ι₂Iso π surj).inv = π' π surj a | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a = π' π surj a ≫ (ι₂Iso π surj).hom | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a ≫ (ι₂Iso π surj).inv = π' π surj a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.EffectiveEpiFamily.π_comp_ι_inv | [318, 1] | [321, 31] | exact π'_comp_ι_hom _ surj _ | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a = π' π surj a ≫ (ι₂Iso π surj).hom | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
surj : ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
a : α
⊢ π a = π' π surj a ≫ (ι₂Iso π surj).hom
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | tfae_have 1 → 2 | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | tfae_have 2 → 3 | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | tfae_have 3 → 1 | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | case tfae_3_to_1
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | tfae_finish | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
tfae_3_to_1 : (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
⊢ TFAE [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b]
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | intro | case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π) | case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
⊢ EffectiveEpiFamily X π → Epi (Sigma.desc π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | infer_instance | case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_1_to_2
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
✝ : EffectiveEpiFamily X π
⊢ Epi (Sigma.desc π)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | intro e | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rw [epi_iff_surjective] at e | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | let i : ∐ X ≅ finiteCoproduct X :=
(colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _) | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | intro b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
b : ↑B.toCompHaus.toTop
⊢ ∃ a x, ↑(π a) x = b | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
⊢ ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | obtain ⟨t,rfl⟩ := e b | case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
b : ↑B.toCompHaus.toTop
⊢ ∃ a x, ↑(π a) x = b | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
b : ↑B.toCompHaus.toTop
⊢ ∃ a x, ↑(π a) x = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | let q := i.hom t | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | refine ⟨q.1,q.2,?_⟩ | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ∃ a x, ↑(π a) x = ↑(Sigma.desc π) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | have : t = i.inv (i.hom t) := show t = (i.hom ≫ i.inv) t by simp only [i.hom_inv_id] ; rfl | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rw [this] | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | show _ = (i.inv ≫ Sigma.desc π) (i.hom t) | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t)) | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(Sigma.desc π) (↑i.inv (↑i.hom t))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
rw [this] ; rfl | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rw [Iso.inv_comp_eq] | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | apply colimit.hom_ext | case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π | case tfae_2_to_3.intro.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ∀ (j : Discrete α),
colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ Sigma.desc π = i.hom ≫ finiteCoproduct.desc X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rintro ⟨a⟩ | case tfae_2_to_3.intro.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ∀ (j : Discrete α),
colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π | case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
⊢ ∀ (j : Discrete α),
colimit.ι (Discrete.functor X) j ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) j ≫ i.hom ≫ finiteCoproduct.desc X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | simp only [Discrete.functor_obj, colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app,
colimit.comp_coconePointUniqueUpToIso_hom_assoc] | case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π | case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ colimit.ι (Discrete.functor X) { as := a } ≫ Sigma.desc π =
colimit.ι (Discrete.functor X) { as := a } ≫ i.hom ≫ finiteCoproduct.desc X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | ext | case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π | case tfae_2_to_3.intro.w.mk.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
⊢ π a = (finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rfl | case tfae_2_to_3.intro.w.mk.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk.w
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this : t = ↑i.inv (↑i.hom t)
a : α
x✝ : (forget Profinite).obj (X a)
⊢ ↑(π a) x✝ = ↑((finiteCoproduct.cocone X).ι.app { as := a } ≫ finiteCoproduct.desc X π) x✝
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | simp only [i.hom_inv_id] | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ t = ↑(i.hom ≫ i.inv) t | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ t = ↑(𝟙 (∐ X)) t | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ t = ↑(i.hom ≫ i.inv) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rfl | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ t = ↑(𝟙 (∐ X)) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
⊢ t = ↑(𝟙 (∐ X)) t
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rw [this] | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this✝ : t = ↑i.inv (↑i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t) | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this✝ : t = ↑i.inv (↑i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t) | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this✝ : t = ↑i.inv (↑i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ ↑(π q.fst) q.snd = ↑(i.inv ≫ Sigma.desc π) (↑i.hom t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | rfl | X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this✝ : t = ↑i.inv (↑i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sigma.desc π)
i : ∐ X ≅ finiteCoproduct X :=
IsColimit.coconePointUniqueUpToIso (colimit.isColimit (Discrete.functor X)) (finiteCoproduct.isColimit X)
t : (forget Profinite).obj (∐ fun b => X b)
q : (fun x => (forget Profinite).obj (finiteCoproduct X)) t := ↑i.hom t
this✝ : t = ↑i.inv (↑i.hom t)
this : i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π
⊢ ↑(π q.fst) q.snd = ↑(finiteCoproduct.desc X π) (↑i.hom t)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/Epi.lean | Profinite.effectiveEpiFamily_tfae | [361, 1] | [394, 14] | apply effectiveEpiFamily_of_jointly_surjective | case tfae_3_to_1
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_3_to_1
X✝ Y B✝ : Profinite
f : X✝ ⟶ B✝
g : Y ⟶ B✝
α : Type
inst✝ : Fintype α
B : Profinite
X : α → Profinite
π : (a : α) → X a ⟶ B
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 : Epi (Sigma.desc π) → ∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b
⊢ (∀ (b : ↑B.toCompHaus.toTop), ∃ a x, ↑(π a) x = b) → EffectiveEpiFamily X π
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_reflects_mono | [11, 1] | [17, 49] | constructor | C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
⊢ Mono f | case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
⊢ ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
⊢ Mono f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_reflects_mono | [11, 1] | [17, 49] | intros W φ ψ h | case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
⊢ ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h | case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ φ = ψ | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
⊢ ∀ {Z : C} (g h : Z ⟶ X), g ≫ f = h ≫ f → g = h
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_reflects_mono | [11, 1] | [17, 49] | apply e.functor.map_injective | case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ φ = ψ | case right_cancellation.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ = e.functor.map ψ | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ φ = ψ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_reflects_mono | [11, 1] | [17, 49] | apply hef.1 | case right_cancellation.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ = e.functor.map ψ | case right_cancellation.a.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ ≫ e.functor.map f = e.functor.map ψ ≫ e.functor.map f | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ = e.functor.map ψ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_reflects_mono | [11, 1] | [17, 49] | rw [← Functor.map_comp, ← Functor.map_comp, h] | case right_cancellation.a.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ ≫ e.functor.map f = e.functor.map ψ ≫ e.functor.map f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a.a
C : Type u
inst✝¹ : Category C
D : Type v
inst✝ : Category D
X Y : C
f : X ⟶ Y
e : C ≌ D
hef : Mono (e.functor.map f)
W : C
φ ψ : W ⟶ X
h : φ ≫ f = ψ ≫ f
⊢ e.functor.map φ ≫ e.functor.map f = e.functor.map ψ ≫ e.functor.map f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | constructor | C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
⊢ Mono (e.functor.map f) | case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
⊢ ∀ {Z : D} (g h : Z ⟶ e.functor.obj X), g ≫ e.functor.map f = h ≫ e.functor.map f → g = h | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
⊢ Mono (e.functor.map f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | intros W φ ψ h | case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
⊢ ∀ {Z : D} (g h : Z ⟶ e.functor.obj X), g ≫ e.functor.map f = h ≫ e.functor.map f → g = h | case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ φ = ψ | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
⊢ ∀ {Z : D} (g h : Z ⟶ e.functor.obj X), g ≫ e.functor.map f = h ≫ e.functor.map f → g = h
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | apply Functor.map_injective e.inverse | case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ φ = ψ | case right_cancellation.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ = e.inverse.map ψ | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ φ = ψ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | apply Mono.right_cancellation (f := (Equivalence.unitInv e).app X) | case right_cancellation.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ = e.inverse.map ψ | case right_cancellation.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ ≫ (Equivalence.unitInv e).app X = e.inverse.map ψ ≫ (Equivalence.unitInv e).app X | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ = e.inverse.map ψ
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | apply Mono.right_cancellation (f := f) | case right_cancellation.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ ≫ (Equivalence.unitInv e).app X = e.inverse.map ψ ≫ (Equivalence.unitInv e).app X | case right_cancellation.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f = (e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.inverse.map φ ≫ (Equivalence.unitInv e).app X = e.inverse.map ψ ≫ (Equivalence.unitInv e).app X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | apply e.functor.map_injective | case right_cancellation.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f = (e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f | case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.functor.map ((e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f) =
e.functor.map ((e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f) | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f = (e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | simp only [Functor.map_comp] | case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.functor.map ((e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f) =
e.functor.map ((e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f) | case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.functor.map (e.inverse.map φ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f =
(e.functor.map (e.inverse.map ψ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ e.functor.map ((e.inverse.map φ ≫ (Equivalence.unitInv e).app X) ≫ f) =
e.functor.map ((e.inverse.map ψ ≫ (Equivalence.unitInv e).app X) ≫ f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CategoryTheoryExercises/Solutions/Solution3.lean | equiv_preserves_mono | [19, 1] | [30, 8] | simpa | case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.functor.map (e.inverse.map φ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f =
(e.functor.map (e.inverse.map ψ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right_cancellation.a.a.a.a
C : Type u
inst✝² : Category C
D : Type v
inst✝¹ : Category D
X Y : C
f : X ⟶ Y
inst✝ : Mono f
e : C ≌ D
W : D
φ ψ : W ⟶ e.functor.obj X
h : φ ≫ e.functor.map f = ψ ≫ e.functor.map f
⊢ (e.functor.map (e.inverse.map φ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f =
(e.functor.map (e.inverse.map ψ) ≫ e.functor.map ((Equivalence.unitInv e).app X)) ≫ e.functor.map f
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | Multiset.inf_eq_top | [13, 1] | [15, 7] | rw [le_antisymm_iff] | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf = ⊤ ↔ ∀ a ∈ m, a = ⊤ | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf = ⊤ ↔ ∀ a ∈ m, a = ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | Multiset.inf_eq_top | [13, 1] | [15, 7] | simp | α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : SemilatticeInf α
inst✝ : OrderTop α
m : Multiset α
⊢ m.inf ≤ ⊤ ∧ ⊤ ≤ m.inf ↔ ∀ a ∈ m, a = ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | intro hc | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head? c ≠ ⊤ | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head? c ≠ ⊤
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | have := hc ▸ (head?_mem_toMultiset_map c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head?_ne_top | [52, 1] | [56, 15] | simp at this | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
hc : head? c = ⊤
this : ⊤ ∈ Multiset.map WithTop.some (toMultiset c)
⊢ False
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_def | [58, 1] | [61, 71] | simpa using MinHeap.coe_head_eq_head? c h | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ ↑(head c h) = ↑((head? c).untop ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ ↑(head c h) = ↑((head? c).untop ⋯)
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem_toMultiset | [63, 1] | [66, 20] | obtain ⟨x, hx₁, hx₂⟩ := Multiset.mem_map.mp (head?_mem_toMultiset_map c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ toMultiset c | case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ toMultiset c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem_toMultiset | [63, 1] | [66, 20] | simp [hx₁, ← hx₂] | case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
x : α
hx₁ : x ∈ toMultiset c
hx₂ : ↑x = head? c
⊢ head c h ∈ toMultiset c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_mem | [68, 1] | [70, 52] | exact mem_toMultiset.mp (head_mem_toMultiset c h) | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
h : ¬isEmpty c = true
⊢ head c h ∈ c
TACTIC:
|
https://github.com/negiizhao/Algorithm.git | ae2bb2d101ce546ffe90434d718f919ef346c2ea | Algorithm/Data/Classes/MinHeap.lean | MinHeap.head_le | [72, 1] | [74, 53] | simpa [WithTop.untop_le_iff] using head?_le c x hx | C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
x : α
hx : x ∈ c
⊢ head c ⋯ ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
C : Type u_1
α : Type u_2
inst✝² : Preorder α
inst✝¹ : IsTotalPreorder α fun x x_1 => x ≤ x_1
inst✝ : MinHeap C α
c : C
x : α
hx : x ∈ c
⊢ head c ⋯ ≤ x
TACTIC:
|
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