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 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | intro i | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∀ (a : α), S.arrows (π₀ a) | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.map (π₀ x)
i : α
⊢ S.arrows (π₀ i) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.map (π₀ x)
⊢ ∀ (a : α), S.arrows (π₀ a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily | [94, 1] | [141, 21] | exact (H₂ i).1 | case mpr.intro.intro.intro.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.map (π₀ x)
i : α
⊢ S.arrows (π₀ i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.intro.intro.right
X : ExtrDisc
S : Sieve X
α : Type
w✝ : Fintype α
Y : α → CompHaus
π : (a : α) → Y a ⟶ ExtrDisc.toCompHaus.obj X
H₁ : EffectiveEpiFamily Y π
Y₀ : α → ExtrDisc
π₀ : (x : α) → Y₀ x ⟶ X
f₀ : (x : α) → Y x ⟶ (Y₀ x).compHaus
H₂ : ∀ (x : α), S.arrows (π₀ x) ∧ π x = f₀ x ≫ ExtrDisc.toCompHaus.map (π₀ x)
i : α
⊢ S.arrows (π₀ i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coherentTopology_is_induced | [143, 1] | [147, 60] | ext X S | ⊢ coherentTopology ExtrDisc = CoverDense.inducedTopology coverDense | case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ coherentTopology ExtrDisc = CoverDense.inducedTopology coverDense
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coherentTopology_is_induced | [143, 1] | [147, 60] | rw [← coverDense.inducedTopology_Sieve_iff_EffectiveEpiFamily X] | case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X | case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
S ∈ GrothendieckTopology.sieves (CoverDense.inducedTopology coverDense) X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coherentTopology_is_induced | [143, 1] | [147, 60] | rw [← coherentTopology.Sieve_iff_hasEffectiveEpiFamily S] | case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : ExtrDisc
S : Sieve X
⊢ S ∈ GrothendieckTopology.sieves (coherentTopology ExtrDisc) X ↔
∃ α x Y π, EffectiveEpiFamily Y π ∧ ∀ (a : α), S.arrows (π a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverPreserving | [150, 1] | [156, 100] | rw [coherentTopology_is_induced] | ⊢ CoverPreserving (coherentTopology ExtrDisc) (coherentTopology CompHaus) ExtrDisc.toCompHaus | ⊢ CoverPreserving (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ CoverPreserving (coherentTopology ExtrDisc) (coherentTopology CompHaus) ExtrDisc.toCompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverPreserving | [150, 1] | [156, 100] | exact LocallyCoverDense.inducedTopology_coverPreserving (CoverDense.locallyCoverDense coverDense) | ⊢ CoverPreserving (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ CoverPreserving (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverLifting | [158, 1] | [164, 97] | rw [coherentTopology_is_induced] | ⊢ CoverLifting (coherentTopology ExtrDisc) (coherentTopology CompHaus) ExtrDisc.toCompHaus | ⊢ CoverLifting (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ CoverLifting (coherentTopology ExtrDisc) (coherentTopology CompHaus) ExtrDisc.toCompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Condensed/Equivalence.lean | Condensed.ExtrDiscCompHaus.coverLifting | [158, 1] | [164, 97] | exact LocallyCoverDense.inducedTopology_coverLifting (CoverDense.locallyCoverDense coverDense) | ⊢ CoverLifting (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ CoverLifting (CoverDense.inducedTopology coverDense) (coherentTopology CompHaus) ExtrDisc.toCompHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.lift_lifts | [242, 1] | [244, 39] | simp [lift] | X Y : CompHaus
Z : ExtrDisc
e : Z.compHaus ⟶ Y
f : X ⟶ Y
inst✝ : Epi f
⊢ lift e f ≫ f = e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : CompHaus
Z : ExtrDisc
e : Z.compHaus ⟶ Y
f : X ⟶ Y
inst✝ : Epi f
⊢ lift e f ≫ f = e
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | sigma_mk_preimage_image' | [260, 1] | [263, 11] | change Sigma.mk j ⁻¹' {⟨i, u⟩ | u ∈ U} = ∅ | α✝ : Type u_1
i j : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' (Sigma.mk i '' U) = ∅ | α✝ : Type u_1
i j : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
i j : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' (Sigma.mk i '' U) = ∅
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | sigma_mk_preimage_image' | [260, 1] | [263, 11] | simp [h] | α✝ : Type u_1
i j : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
i j : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
h : i ≠ j
⊢ Sigma.mk j ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = ∅
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | sigma_mk_preimage_image_eq_self | [265, 1] | [267, 7] | change Sigma.mk i ⁻¹' {⟨i, u⟩ | u ∈ U} = U | α✝ : Type u_1
i : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
⊢ Sigma.mk i ⁻¹' (Sigma.mk i '' U) = U | α✝ : Type u_1
i : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
⊢ Sigma.mk i ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = U | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
i : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
⊢ Sigma.mk i ⁻¹' (Sigma.mk i '' U) = U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | sigma_mk_preimage_image_eq_self | [265, 1] | [267, 7] | simp | α✝ : Type u_1
i : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
⊢ Sigma.mk i ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
i : α✝
β✝ : α✝ → Type u_2
U : Set (β✝ i)
⊢ Sigma.mk i ⁻¹' {x | ∃ u, u ∈ U ∧ { fst := i, snd := u } = x} = U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | ExtrDisc.finiteCoproduct.hom_ext | [344, 1] | [349, 10] | ext ⟨a,x⟩ | α : Type
inst✝ : Fintype α
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : CoeSort.coe (X a)
⊢ ↑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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
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 | ExtrDisc/Basic.lean | ExtrDisc.finiteCoproduct.hom_ext | [344, 1] | [349, 10] | specialize h a | case w.mk
α : Type
inst✝ : Fintype α
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : CoeSort.coe (X a)
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x } | case w.mk
α : Type
inst✝ : Fintype α
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
h : ∀ (a : α), ι X a ≫ f = ι X a ≫ g
a : α
x : CoeSort.coe (X a)
⊢ ↑f { fst := a, snd := x } = ↑g { fst := a, snd := x }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | ExtrDisc.finiteCoproduct.hom_ext | [344, 1] | [349, 10] | apply_fun (fun q => q x) at h | case w.mk
α : Type
inst✝ : Fintype α
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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 | ExtrDisc/Basic.lean | ExtrDisc.finiteCoproduct.hom_ext | [344, 1] | [349, 10] | exact h | case w.mk
α : Type
inst✝ : Fintype α
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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
B✝ : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π a) x = b
B : ExtrDisc
f g : finiteCoproduct X ⟶ B
a : α
x : CoeSort.coe (X a)
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 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | constructor | X : CompHaus
⊢ Projective X ↔ ExtremallyDisconnected ↑X.toTop | case mp
X : CompHaus
⊢ Projective X → ExtremallyDisconnected ↑X.toTop
case mpr
X : CompHaus
⊢ ExtremallyDisconnected ↑X.toTop → Projective X | Please generate a tactic in lean4 to solve the state.
STATE:
X : CompHaus
⊢ Projective X ↔ ExtremallyDisconnected ↑X.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | intro h | case mp
X : CompHaus
⊢ Projective X → ExtremallyDisconnected ↑X.toTop | case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected ↑X.toTop | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : CompHaus
⊢ Projective X → ExtremallyDisconnected ↑X.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | show ExtremallyDisconnected X.toExtrDisc | case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected ↑X.toTop | case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected (CoeSort.coe (toExtrDisc X)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected ↑X.toTop
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | infer_instance | case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected (CoeSort.coe (toExtrDisc X)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : CompHaus
h : Projective X
⊢ ExtremallyDisconnected (CoeSort.coe (toExtrDisc X))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | intro h | case mpr
X : CompHaus
⊢ ExtremallyDisconnected ↑X.toTop → Projective X | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
⊢ Projective X | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : CompHaus
⊢ ExtremallyDisconnected ↑X.toTop → Projective X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | let X' : ExtrDisc := ⟨X⟩ | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
⊢ Projective X | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
⊢ Projective X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | show Projective X'.compHaus | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X'.compHaus | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Basic.lean | CompHaus.Gleason | [402, 1] | [411, 58] | apply ExtrDisc.instProjectiveCompHausCategoryCompHaus | case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X'.compHaus | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : CompHaus
h : ExtremallyDisconnected ↑X.toTop
X' : ExtrDisc := ExtrDisc.mk X
⊢ Projective X'.compHaus
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | constructor | X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
⊢ ExtremallyDisconnected ↑U | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
⊢ ∀ (U_1 : Set ↑U), IsOpen U_1 → IsOpen (closure U_1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
⊢ ExtremallyDisconnected ↑U
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | intro V hV | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
⊢ ∀ (U_1 : Set ↑U), IsOpen U_1 → IsOpen (closure U_1) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
⊢ IsOpen (closure V) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
⊢ ∀ (U_1 : Set ↑U), IsOpen U_1 → IsOpen (closure U_1)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | have hV' : IsOpen (Subtype.val '' V) := hU.1.openEmbedding_subtype_val.isOpenMap V hV | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
⊢ IsOpen (closure V) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
⊢ IsOpen (closure V) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
⊢ IsOpen (closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | have := ExtremallyDisconnected.open_closure _ hV' | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
⊢ IsOpen (closure V) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (closure (Subtype.val '' V))
⊢ IsOpen (closure V) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
⊢ IsOpen (closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | rw [hU.2.closedEmbedding_subtype_val.closure_image_eq V] at this | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (closure (Subtype.val '' V))
⊢ IsOpen (closure V) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ IsOpen (closure V) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (closure (Subtype.val '' V))
⊢ IsOpen (closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | suffices hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' (closure V)) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ IsOpen (closure V) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (closure V)
case hhU
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ closure V = Subtype.val ⁻¹' (Subtype.val '' closure V) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ IsOpen (closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | exact ((closure V).preimage_image_eq Subtype.coe_injective).symm | case hhU
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ closure V = Subtype.val ⁻¹' (Subtype.val '' closure V) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hhU
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
⊢ closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | rw [hhU] | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (closure V) | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (Subtype.val ⁻¹' (Subtype.val '' closure V)) | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (closure V)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.clopen_extremallyDisconnected | [40, 1] | [50, 67] | exact isOpen_induced this | case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (Subtype.val ⁻¹' (Subtype.val '' closure V)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case open_closure
X : ExtrDisc
U : Set (CoeSort.coe X)
hU : IsClopen U
V : Set ↑U
hV : IsOpen V
hV' : IsOpen (Subtype.val '' V)
this : IsOpen (Subtype.val '' closure V)
hhU : closure V = Subtype.val ⁻¹' (Subtype.val '' closure V)
⊢ IsOpen (Subtype.val ⁻¹' (Subtype.val '' closure V))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.aux_forall_mem | [99, 1] | [102, 32] | rintro ⟨x, hx⟩ | X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ ∀ (x : ↑(↑f ⁻¹' Set.range ↑i)), ↑f ↑x ∈ Set.range ↑i | case mk
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
x : CoeSort.coe X
hx : x ∈ ↑f ⁻¹' Set.range ↑i
⊢ ↑f ↑{ val := x, property := hx } ∈ Set.range ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ ∀ (x : ↑(↑f ⁻¹' Set.range ↑i)), ↑f ↑x ∈ Set.range ↑i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.aux_forall_mem | [99, 1] | [102, 32] | simpa only [Set.mem_preimage] | case mk
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
x : CoeSort.coe X
hx : x ∈ ↑f ⁻¹' Set.range ↑i
⊢ ↑f ↑{ val := x, property := hx } ∈ Set.range ↑i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
x : CoeSort.coe X
hx : x ∈ ↑f ⁻¹' Set.range ↑i
⊢ ↑f ↑{ val := x, property := hx } ∈ Set.range ↑i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.aux_continuous_restrict | [104, 1] | [108, 21] | apply ContinuousOn.restrict | X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ Continuous (Set.restrict (↑f ⁻¹' Set.range ↑i) ↑f) | case a
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ ContinuousOn (↑f) (↑f ⁻¹' Set.range ↑i) | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ Continuous (Set.restrict (↑f ⁻¹' Set.range ↑i) ↑f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.aux_continuous_restrict | [104, 1] | [108, 21] | apply Continuous.continuousOn | case a
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ ContinuousOn (↑f) (↑f ⁻¹' Set.range ↑i) | case a.h
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ Continuous ↑f | Please generate a tactic in lean4 to solve the state.
STATE:
case a
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ ContinuousOn (↑f) (↑f ⁻¹' Set.range ↑i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.aux_continuous_restrict | [104, 1] | [108, 21] | exact f.continuous | case a.h
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ Continuous ↑f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
x✝ : OpenEmbedding ↑i
⊢ Continuous ↑f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | dsimp [lift, snd, OpenEmbeddingCone, OpenEmbeddingConeRightMap, ContinuousMap.comp, Set.restrict,
Set.codRestrict, OpenEmbedding.InvRange] | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
⊢ lift f hi a b w ≫ snd f hi = b | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
⊢ (ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }) =
b | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
⊢ lift f hi a b w ≫ snd f hi = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | ext z | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
⊢ (ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }) =
b | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
⊢ (ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }) =
b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | have := congr_fun (FunLike.ext'_iff.mp w.symm) z | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | have h : i (b z) = f (a z) := this | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | suffices : b z = (Homeomorph.ofEmbedding i hi.toEmbedding).symm
(⟨f (a z), by rw [← h] ; simp⟩) | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this✝ : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
this :
↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z
case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | apply_fun (Homeomorph.ofEmbedding i hi.toEmbedding) | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }) | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | simp only [Homeomorph.apply_symm_apply] | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }) | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) })
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | dsimp [Homeomorph.ofEmbedding] | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ { val := ↑i (↑b z), property := (_ : ∃ y, ↑i y = ↑i (↑b z)) } =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑(Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)) (↑b z) =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | simp_rw [h] | case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ { val := ↑i (↑b z), property := (_ : ∃ y, ↑i y = ↑i (↑b z)) } =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ { val := ↑i (↑b z), property := (_ : ∃ y, ↑i y = ↑i (↑b z)) } =
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | rw [← h] | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑f (↑a z) ∈ Set.range ↑i | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑i (↑b z) ∈ Set.range ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑f (↑a z) ∈ Set.range ↑i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | simp | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑i (↑b z) ∈ Set.range ↑i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
⊢ ↑i (↑b z) ∈ Set.range ↑i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.lift_snd | [199, 1] | [215, 14] | exact this.symm | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this✝ : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
this :
↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a : W ⟶ X
b : W ⟶ Y
w : a ≫ f = b ≫ i
z : (forget ExtrDisc).obj W
this✝ : ↑(b ≫ i) z = ↑(a ≫ f) z
h : ↑i (↑b z) = ↑f (↑a z)
this :
↑b z =
↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i)))
{ val := ↑f (↑a z), property := (_ : ↑f (↑a z) ∈ Set.range ↑i) }
⊢ ↑((ContinuousMap.mk fun z => { val := ↑a z, property := (_ : ↑a z ∈ ↑f ⁻¹' Set.range ↑i) }) ≫
ContinuousMap.mk
(↑(Homeomorph.symm (Homeomorph.ofEmbedding ↑i (_ : Embedding ↑i))) ∘ fun x =>
{ val := ↑f ↑x, property := (_ : ↑f ↑x ∈ Set.range ↑i) }))
z =
↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.hom_ext | [217, 1] | [223, 13] | ext z | X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
hfst : a ≫ fst f hi = b ≫ fst f hi
⊢ a = b | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
hfst : a ≫ fst f hi = b ≫ fst f hi
z : (forget ExtrDisc).obj W
⊢ ↑a z = ↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
hfst : a ≫ fst f hi = b ≫ fst f hi
⊢ a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.hom_ext | [217, 1] | [223, 13] | apply_fun (fun q => q z) at hfst | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
hfst : a ≫ fst f hi = b ≫ fst f hi
z : (forget ExtrDisc).obj W
⊢ ↑a z = ↑b z | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑a z = ↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
hfst : a ≫ fst f hi = b ≫ fst f hi
z : (forget ExtrDisc).obj W
⊢ ↑a z = ↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.hom_ext | [217, 1] | [223, 13] | apply Subtype.ext | case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑a z = ↑b z | case w.a
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑(↑a z) = ↑(↑b z) | Please generate a tactic in lean4 to solve the state.
STATE:
case w
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑a z = ↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.pullback.hom_ext | [217, 1] | [223, 13] | exact hfst | case w.a
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑(↑a z) = ↑(↑b z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.a
X Y Z W : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
a b : W ⟶ (OpenEmbeddingCone f hi).pt
z : (forget ExtrDisc).obj W
hfst : ↑(a ≫ fst f hi) z = ↑(b ≫ fst f hi) z
⊢ ↑(↑a z) = ↑(↑b z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.HasPullbackOpenEmbedding | [233, 1] | [237, 45] | constructor | X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ HasPullback f i | case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ Nonempty (LimitCone (cospan f i)) | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ HasPullback f i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.HasPullbackOpenEmbedding | [233, 1] | [237, 45] | use OpenEmbeddingCone f hi | case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ Nonempty (LimitCone (cospan f i)) | case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ IsLimit (OpenEmbeddingCone f hi) | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ Nonempty (LimitCone (cospan f i))
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.HasPullbackOpenEmbedding | [233, 1] | [237, 45] | exact ExtrDisc.OpenEmbeddingLimitCone f hi | case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ IsLimit (OpenEmbeddingCone f hi) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case exists_limit
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
⊢ IsLimit (OpenEmbeddingCone f hi)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.toExplicitCompFromExcplict | [264, 1] | [269, 37] | refine' Limits.pullback.hom_ext (k := (toExplicit f hi ≫ fromExplicit f hi)) _ _ | X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ toExplicit f hi ≫ fromExplicit f hi = 𝟙 (pullback f i) | case refine'_1
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.fst = 𝟙 (pullback f i) ≫ Limits.pullback.fst
case refine'_2
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.snd = 𝟙 (pullback f i) ≫ Limits.pullback.snd | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ toExplicit f hi ≫ fromExplicit f hi = 𝟙 (pullback f i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.toExplicitCompFromExcplict | [264, 1] | [269, 37] | simp [toExplicit, fromExplicit] | case refine'_1
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.fst = 𝟙 (pullback f i) ≫ Limits.pullback.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.fst = 𝟙 (pullback f i) ≫ Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.toExplicitCompFromExcplict | [264, 1] | [269, 37] | rw [Category.id_comp, Category.assoc, fromExplicit, Limits.pullback.lift_snd,
toExplicit, pullback.lift_snd] | case refine'_2
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.snd = 𝟙 (pullback f i) ≫ Limits.pullback.snd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (toExplicit f hi ≫ fromExplicit f hi) ≫ Limits.pullback.snd = 𝟙 (pullback f i) ≫ Limits.pullback.snd
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.fromExcplictComptoExplicit | [272, 1] | [274, 65] | simp [toExplicit, fromExplicit] | X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (fromExplicit f hi ≫ toExplicit f hi) ≫ pullback.fst f hi = 𝟙 (OpenEmbeddingCone f hi).pt ≫ pullback.fst f hi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y Z : ExtrDisc
f : X ⟶ Z
i : Y ⟶ Z
hi : OpenEmbedding ↑i
inst✝ : HasPullback f i
⊢ (fromExplicit f hi ≫ toExplicit f hi) ≫ pullback.fst f hi = 𝟙 (OpenEmbeddingCone f hi).pt ≫ pullback.fst f hi
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.compatibility | [342, 1] | [346, 34] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
⊢ δ hOpen f ≫ θ hOpen f ≫ η hOpen f = ζ hOpen f ≫ ε | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | Please generate a tactic in lean4 to solve the state.
STATE:
Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
⊢ δ hOpen f ≫ θ hOpen f ≫ η hOpen f = ζ hOpen f ≫ ε
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.compatibility | [342, 1] | [346, 34] | have := HasPullbackOpenEmbedding f (hOpen a) | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | Please generate a tactic in lean4 to solve the state.
STATE:
Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.compatibility | [342, 1] | [346, 34] | rw [← Category.assoc, δ, fromFiniteCoproduct, finiteCoproduct.ι_desc] | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ Sigma.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | Please generate a tactic in lean4 to solve the state.
STATE:
Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫
δ hOpen f ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Pullback.lean | CategoryTheory.ExtrDisc.compatibility | [342, 1] | [346, 34] | simp [ε, ζ, θ, η, fromExplicit] | Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ Sigma.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Y : ExtrDisc
α : Type
inst✝ : Fintype α
Z : α → ExtrDisc
X : ExtrDisc
i : (a : α) → Z a ⟶ X
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
f : Y ⟶ X
a : α
this : HasPullback f (i a)
⊢ Sigma.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ θ hOpen f ≫ η hOpen f =
finiteCoproduct.ι (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) a ≫ ζ hOpen f ≫ ε
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.toFiniteCoproductCompFromFiniteCoproduct | [24, 1] | [27, 48] | ext | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z) | case h.w
α : Type
inst✝ : Fintype α
Z : α → Profinite
b✝ : α
x✝ : (forget Profinite).obj (Z b✝)
⊢ ↑(Sigma.ι Z b✝ ≫ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z) x✝ = ↑(Sigma.ι Z b✝ ≫ 𝟙 (∐ Z)) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.toFiniteCoproductCompFromFiniteCoproduct | [24, 1] | [27, 48] | simp [toFiniteCoproduct, fromFiniteCoproduct] | case h.w
α : Type
inst✝ : Fintype α
Z : α → Profinite
b✝ : α
x✝ : (forget Profinite).obj (Z b✝)
⊢ ↑(Sigma.ι Z b✝ ≫ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z) x✝ = ↑(Sigma.ι Z b✝ ≫ 𝟙 (∐ Z)) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.w
α : Type
inst✝ : Fintype α
Z : α → Profinite
b✝ : α
x✝ : (forget Profinite).obj (Z b✝)
⊢ ↑(Sigma.ι Z b✝ ≫ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z) x✝ = ↑(Sigma.ι Z b✝ ≫ 𝟙 (∐ Z)) x✝
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.FromFiniteCoproductComptToFiniteCoproduct | [30, 1] | [33, 48] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z) | α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ finiteCoproduct.ι Z a ≫ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a ≫ 𝟙 (finiteCoproduct Z) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.FromFiniteCoproductComptToFiniteCoproduct | [30, 1] | [33, 48] | simp [toFiniteCoproduct, fromFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ finiteCoproduct.ι Z a ≫ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a ≫ 𝟙 (finiteCoproduct Z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ finiteCoproduct.ι Z a ≫ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a ≫ 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.IsIsotoFiniteCoproduct | [51, 1] | [53, 44] | simp | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.IsIsotoFiniteCoproduct | [51, 1] | [53, 44] | simp | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.IsIsofromFiniteCoproduct | [55, 1] | [57, 42] | simp | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.IsIsofromFiniteCoproduct | [55, 1] | [57, 42] | simp | α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.Sigma.ιCompToFiniteCoproduct | [60, 1] | [62, 27] | simp [toFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ Sigma.ι Z a ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ Sigma.ι Z a ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ιCompFromFiniteCoproduct | [65, 1] | [67, 29] | simp [fromFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ ι Z a ≫ fromFiniteCoproduct Z = Sigma.ι Z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ ι Z a ≫ fromFiniteCoproduct Z = Sigma.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_injective | [87, 1] | [90, 43] | intro x y hxy | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ Function.Injective ↑(ι Z a) | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
x y : (forget Profinite).obj (Z a)
hxy : ↑(ι Z a) x = ↑(ι Z a) y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
⊢ Function.Injective ↑(ι Z a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_injective | [87, 1] | [90, 43] | exact eq_of_heq (Sigma.ext_iff.mp hxy).2 | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
x y : (forget Profinite).obj (Z a)
hxy : ↑(ι Z a) x = ↑(ι Z a) y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a : α
x y : (forget Profinite).obj (Z a)
hxy : ↑(ι Z a) x = ↑(ι Z a) y
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_jointly_surjective | [92, 1] | [94, 28] | exact ⟨R.fst, R.snd, rfl⟩ | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
R : ↑(finiteCoproduct Z).toCompHaus.toTop
⊢ ∃ a r, R = ↑(ι Z a) r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
R : ↑(finiteCoproduct Z).toCompHaus.toTop
⊢ ∃ a r, R = ↑(ι Z a) r
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_desc_apply | [96, 1] | [101, 21] | intro x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
⊢ ∀ (x : (forget Profinite).obj (Z a)), ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
⊢ ∀ (x : (forget Profinite).obj (Z a)), ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_desc_apply | [96, 1] | [101, 21] | change (ι Z a ≫ desc Z π) _ = _ | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(ι Z a ≫ desc Z π) x = ↑(π a) x | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.ι_desc_apply | [96, 1] | [101, 21] | simp only [ι_desc] | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(ι Z a ≫ desc Z π) x = ↑(π a) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
X : Profinite
Z : α → Profinite
π : (a : α) → Z a ⟶ X
a : α
x : (forget Profinite).obj (Z a)
⊢ ↑(ι Z a ≫ desc Z π) x = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.finiteCoproduct.range_eq | [103, 1] | [105, 9] | rw [h] | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a b : α
h : a = b
⊢ Set.range ↑(ι Z a) = Set.range ↑(ι Z b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → Profinite
α : Type
inst✝ : Fintype α
Z : α → Profinite
a b : α
h : a = b
⊢ Set.range ↑(ι Z a) = Set.range ↑(ι Z b)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.condition | [113, 1] | [115, 22] | ext ⟨_,h⟩ | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ fst f g ≫ f = snd f g ≫ g | case w.mk
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
val✝ : ↑X.toCompHaus.toTop × ↑Y.toCompHaus.toTop
h : val✝ ∈ {xy | ↑f xy.fst = ↑g xy.snd}
⊢ ↑(fst f g ≫ f) { val := val✝, property := h } = ↑(snd f g ≫ g) { val := val✝, property := h } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ fst f g ≫ f = snd f g ≫ g
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.condition | [113, 1] | [115, 22] | exact h | case w.mk
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
val✝ : ↑X.toCompHaus.toTop × ↑Y.toCompHaus.toTop
h : val✝ ∈ {xy | ↑f xy.fst = ↑g xy.snd}
⊢ ↑(fst f g ≫ f) { val := val✝, property := h } = ↑(snd f g ≫ g) { val := val✝, property := h } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.mk
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
val✝ : ↑X.toCompHaus.toTop × ↑Y.toCompHaus.toTop
h : val✝ ∈ {xy | ↑f xy.fst = ↑g xy.snd}
⊢ ↑(fst f g ≫ f) { val := val✝, property := h } = ↑(snd f g ≫ g) { val := val✝, property := h }
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | ext z | α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
hfst : a ≫ fst f g = b ≫ fst f g
hsnd : a ≫ snd f g = b ≫ snd f g
⊢ a = b | case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
hfst : a ≫ fst f g = b ≫ fst f g
hsnd : a ≫ snd f g = b ≫ snd f g
z : (forget Profinite).obj Z
⊢ ↑a z = ↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
hfst : a ≫ fst f g = b ≫ fst f g
hsnd : a ≫ snd f g = b ≫ snd f g
⊢ a = b
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | apply_fun (fun q => q z) at hfst hsnd | case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
hfst : a ≫ fst f g = b ≫ fst f g
hsnd : a ≫ snd f g = b ≫ snd f g
z : (forget Profinite).obj Z
⊢ ↑a z = ↑b z | case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑a z = ↑b z | Please generate a tactic in lean4 to solve the state.
STATE:
case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
hfst : a ≫ fst f g = b ≫ fst f g
hsnd : a ≫ snd f g = b ≫ snd f g
z : (forget Profinite).obj Z
⊢ ↑a z = ↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | apply Subtype.ext | case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑a z = ↑b z | case w.a
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑(↑a z) = ↑(↑b z) | Please generate a tactic in lean4 to solve the state.
STATE:
case w
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑a z = ↑b z
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | apply Prod.ext | case w.a
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑(↑a z) = ↑(↑b z) | case w.a.h₁
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).fst = (↑(↑b z)).fst
case w.a.h₂
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).snd = (↑(↑b z)).snd | Please generate a tactic in lean4 to solve the state.
STATE:
case w.a
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ ↑(↑a z) = ↑(↑b z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | exact hfst | case w.a.h₁
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).fst = (↑(↑b z)).fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.a.h₁
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).fst = (↑(↑b z)).fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.pullback.hom_ext | [138, 1] | [146, 15] | exact hsnd | case w.a.h₂
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).snd = (↑(↑b z)).snd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.a.h₂
α : Type
inst✝ : Fintype α
Z✝¹ : α → Profinite
X Y Z✝ : Profinite
f : X ⟶ Z✝
g i : Y ⟶ Z✝
Z : Profinite
a b : Z ⟶ pullback f g
z : (forget Profinite).obj Z
hfst : ↑(a ≫ fst f g) z = ↑(b ≫ fst f g) z
hsnd : ↑(a ≫ snd f g) z = ↑(b ≫ snd f g) z
⊢ (↑(↑a z)).snd = (↑(↑b z)).snd
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.toExplicitCompFromExcplict | [177, 1] | [182, 37] | refine' Limits.pullback.hom_ext (k := (toExplicit f i ≫ fromExplicit f i)) _ _ | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ toExplicit f i ≫ fromExplicit f i = 𝟙 (Limits.pullback f i) | case refine'_1
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.fst = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.fst
case refine'_2
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.snd = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.snd | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ toExplicit f i ≫ fromExplicit f i = 𝟙 (Limits.pullback f i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.toExplicitCompFromExcplict | [177, 1] | [182, 37] | simp [toExplicit, fromExplicit] | case refine'_1
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.fst = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.fst = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.toExplicitCompFromExcplict | [177, 1] | [182, 37] | rw [Category.id_comp, Category.assoc, fromExplicit, Limits.pullback.lift_snd,
toExplicit, pullback.lift_snd] | case refine'_2
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.snd = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.snd | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (toExplicit f i ≫ fromExplicit f i) ≫ Limits.pullback.snd = 𝟙 (Limits.pullback f i) ≫ Limits.pullback.snd
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.fromExcplictComptoExplicit | [185, 1] | [187, 101] | simp [toExplicit, fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (fromExplicit f i ≫ toExplicit f i) ≫ pullback.fst f i = 𝟙 (pullback f i) ≫ pullback.fst f i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (fromExplicit f i ≫ toExplicit f i) ≫ pullback.fst f i = 𝟙 (pullback f i) ≫ pullback.fst f i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.fromExcplictComptoExplicit | [185, 1] | [187, 101] | simp [toExplicit, fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (fromExplicit f i ≫ toExplicit f i) ≫ pullback.snd f i = 𝟙 (pullback f i) ≫ pullback.snd f i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ (fromExplicit f i ≫ toExplicit f i) ≫ pullback.snd f i = 𝟙 (pullback f i) ≫ pullback.snd f i
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.fst_comp_fromExplicit | [201, 1] | [204, 70] | dsimp [fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.fst f i = fromExplicit f i ≫ Limits.pullback.fst | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.fst f i =
Limits.pullback.lift (pullback.fst f i) (pullback.snd f i) (_ : pullback.fst f i ≫ f = pullback.snd f i ≫ i) ≫
Limits.pullback.fst | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.fst f i = fromExplicit f i ≫ Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.fst_comp_fromExplicit | [201, 1] | [204, 70] | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app] | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.fst f i =
Limits.pullback.lift (pullback.fst f i) (pullback.snd f i) (_ : pullback.fst f i ≫ f = pullback.snd f i ≫ i) ≫
Limits.pullback.fst | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.fst f i =
Limits.pullback.lift (pullback.fst f i) (pullback.snd f i) (_ : pullback.fst f i ≫ f = pullback.snd f i ≫ i) ≫
Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | Profinite/ExplicitLimits.lean | Profinite.snd_comp_fromExplicit | [206, 1] | [209, 70] | dsimp [fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.snd f i = fromExplicit f i ≫ Limits.pullback.snd | α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.snd f i =
Limits.pullback.lift (pullback.fst f i) (pullback.snd f i) (_ : pullback.fst f i ≫ f = pullback.snd f i ≫ i) ≫
Limits.pullback.snd | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → Profinite
X Y Z : Profinite
f : X ⟶ Z
g i : Y ⟶ Z
⊢ pullback.snd f i = fromExplicit f i ≫ Limits.pullback.snd
TACTIC:
|
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