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 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | intro e | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ Epi (Sigma... | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rw [epi_iff_surjective] at e | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Epi (Sig... | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | let i : ∐ X ≅ finiteCoproduct X :=
(colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _) | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | intro b | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | obtain ⟨t,rfl⟩ := e b | case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | let q := i.hom t | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | refine ⟨q.1,q.2,?_⟩ | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 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
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rw [this] | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | show _ = (i.inv ≫ Sigma.desc π) (i.hom t) | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | suffices i.inv ≫ Sigma.desc π = finiteCoproduct.desc X π by
rw [this] ; rfl | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rw [Iso.inv_comp_eq] | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | apply colimit.hom_ext | case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Fu... | case tfae_2_to_3.intro.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfa... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rintro ⟨a⟩ | case tfae_2_to_3.intro.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : ... | case tfae_2_to_3.intro.w.mk
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ t... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 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
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e... | case tfae_2_to_3.intro.w.mk
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | ext | case tfae_2_to_3.intro.w.mk
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e... | case tfae_2_to_3.intro.w.mk.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)... | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rfl | case tfae_2_to_3.intro.w.mk.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_2_to_3.intro.w.mk.w
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | simp only [i.hom_inv_id] | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rfl | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rw [this] | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | rfl | α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
e : Function.Surjective ↑(Sig... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Epi.lean | ExtrDisc.effectiveEpiFamily_tfae | [161, 1] | [195, 14] | apply effectiveEpiFamily_of_jointly_surjective | case tfae_3_to_1
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_2_to_3 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tfae_3_to_1
α✝ : Type
inst✝¹ : Fintype α✝
B✝ : ExtrDisc
X✝ : α✝ → ExtrDisc
π✝ : (a : α✝) → X✝ a ⟶ B✝
surj : ∀ (b : CoeSort.coe B✝), ∃ a x, ↑(π✝ a) x = b
α : Type
inst✝ : Fintype α
B : ExtrDisc
X : α → ExtrDisc
π : (a : α) → X a ⟶ B
tfae_1_to_2✝ tfae_1_to... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.toFiniteCoproductCompFromFiniteCoproduct | [27, 1] | [30, 48] | ext | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z) | case h.w
α : Type
inst✝ : Fintype α
Z : α → CompHaus
b✝ : α
x✝ : (forget CompHaus).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 → CompHaus
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.toFiniteCoproductCompFromFiniteCoproduct | [27, 1] | [30, 48] | simp [toFiniteCoproduct, fromFiniteCoproduct] | case h.w
α : Type
inst✝ : Fintype α
Z : α → CompHaus
b✝ : α
x✝ : (forget CompHaus).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 → CompHaus
b✝ : α
x✝ : (forget CompHaus).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 | CompHaus/ExplicitLimits.lean | CompHaus.FromFiniteCoproductComptToFiniteCoproduct | [33, 1] | [36, 48] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z) | α : Type
inst✝ : Fintype α
Z : α → CompHaus
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 → CompHaus
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.FromFiniteCoproductComptToFiniteCoproduct | [33, 1] | [36, 48] | simp [toFiniteCoproduct, fromFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → CompHaus
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 → CompHaus
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 | CompHaus/ExplicitLimits.lean | CompHaus.IsIsotoFiniteCoproduct | [54, 1] | [56, 44] | simp | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ 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 → CompHaus
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.IsIsotoFiniteCoproduct | [54, 1] | [56, 44] | simp | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ 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 → CompHaus
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.IsIsofromFiniteCoproduct | [58, 1] | [60, 42] | simp | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ 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 → CompHaus
⊢ fromFiniteCoproduct Z ≫ toFiniteCoproduct Z = 𝟙 (finiteCoproduct Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.IsIsofromFiniteCoproduct | [58, 1] | [60, 42] | simp | α : Type
inst✝ : Fintype α
Z : α → CompHaus
⊢ 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 → CompHaus
⊢ toFiniteCoproduct Z ≫ fromFiniteCoproduct Z = 𝟙 (∐ Z)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.Sigma.ιCompToFiniteCoproduct | [63, 1] | [65, 27] | simp [toFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → CompHaus
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 → CompHaus
a : α
⊢ Sigma.ι Z a ≫ toFiniteCoproduct Z = finiteCoproduct.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ιCompFromFiniteCoproduct | [68, 1] | [70, 29] | simp [fromFiniteCoproduct] | α : Type
inst✝ : Fintype α
Z : α → CompHaus
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 → CompHaus
a : α
⊢ ι Z a ≫ fromFiniteCoproduct Z = Sigma.ι Z a
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_injective | [90, 1] | [93, 43] | intro x y hxy | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a : α
⊢ Function.Injective ↑(ι Z a) | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a : α
x y : (forget CompHaus).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✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a : α
⊢ Function.Injective ↑(ι Z a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_injective | [90, 1] | [93, 43] | exact eq_of_heq (Sigma.ext_iff.mp hxy).2 | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a : α
x y : (forget CompHaus).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✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a : α
x y : (forget CompHaus).obj (Z a)
hxy : ↑(ι Z a) x = ↑(ι Z a) y
⊢ x = y
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_jointly_surjective | [95, 1] | [97, 28] | exact ⟨R.fst, R.snd, rfl⟩ | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
R : ↑(finiteCoproduct Z).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✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
R : ↑(finiteCoproduct Z).toTop
⊢ ∃ a r, R = ↑(ι Z a) r
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_desc_apply | [99, 1] | [104, 21] | intro x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
⊢ ∀ (x : (forget CompHaus).obj (Z a)), ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).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✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
⊢ ∀ (x : (forget CompHaus).obj (Z a)), ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_desc_apply | [99, 1] | [104, 21] | change (ι Z a ≫ desc Z π) _ = _ | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).obj (Z a)
⊢ ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).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✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).obj (Z a)
⊢ ↑(desc Z π) (↑(ι Z a) x) = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.ι_desc_apply | [99, 1] | [104, 21] | simp only [ι_desc] | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).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✝ → CompHaus
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
a : α
x : (forget CompHaus).obj (Z a)
⊢ ↑(ι Z a ≫ desc Z π) x = ↑(π a) x
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.finiteCoproduct.range_eq | [106, 1] | [108, 9] | rw [h] | α✝ : Type
inst✝¹ : Fintype α✝
Z✝ : α✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
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✝ → CompHaus
α : Type
inst✝ : Fintype α
Z : α → CompHaus
a b : α
h : a = b
⊢ Set.range ↑(ι Z a) = Set.range ↑(ι Z b)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.toExplicitCompFromExcplict | [127, 1] | [132, 37] | refine' Limits.pullback.hom_ext (k := (toExplicit f i ≫ fromExplicit f i)) _ _ | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ toExplicit f i ≫ fromExplicit f i = 𝟙 (Limits.pullback f i) | case refine'_1
α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ (toExplicit f ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ toExplicit f i ≫ fromExplicit f i = 𝟙 (Limits.pullback f i)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.toExplicitCompFromExcplict | [127, 1] | [132, 37] | simp [toExplicit, fromExplicit] | case refine'_1
α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | CompHaus/ExplicitLimits.lean | CompHaus.toExplicitCompFromExcplict | [127, 1] | [132, 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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | CompHaus/ExplicitLimits.lean | CompHaus.fromExcplictComptoExplicit | [135, 1] | [137, 101] | simp [toExplicit, fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | CompHaus/ExplicitLimits.lean | CompHaus.fromExcplictComptoExplicit | [135, 1] | [137, 101] | simp [toExplicit, fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | CompHaus/ExplicitLimits.lean | CompHaus.fst_comp_fromExplicit | [151, 1] | [154, 70] | dsimp [fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ pullback.fst f i = fromExplicit f i ≫ Limits.pullback.fst | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ pullback.fst f i = fromExplicit f i ≫ Limits.pullback.fst
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.fst_comp_fromExplicit | [151, 1] | [154, 70] | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | CompHaus/ExplicitLimits.lean | CompHaus.snd_comp_fromExplicit | [156, 1] | [159, 70] | dsimp [fromExplicit] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ pullback.snd f i = fromExplicit f i ≫ Limits.pullback.snd | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
i : Y ⟶ Z
⊢ pullback.snd f i = fromExplicit f i ≫ Limits.pullback.snd
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitLimits.lean | CompHaus.snd_comp_fromExplicit | [156, 1] | [159, 70] | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app] | α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
Z✝ : α → CompHaus
X Y Z : CompHaus
f : X ⟶ Z
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
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | let ζ := finiteCoproduct.desc _ (fun a => pullback.snd f (π a) ≫ finiteCoproduct.ι Z a ) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a)) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
⊢ Function.Injective ↑(finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.fst f (π a))
T... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | let σ := finiteCoproduct.desc _ ((fun a => pullback.fst f (π a))) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have comm : ζ ≫ β = σ ≫ f := by
refine' finiteCoproduct.hom_ext _ _ _ (fun a => _)
simp [← Category.assoc, finiteCoproduct.ι_desc, pullback.condition] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | intro R₁ R₂ hR | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have himage : (ζ ≫ β) R₁ = (ζ ≫ β) R₂ := by
rw [comm]; change f (σ R₁) = f (σ R₂); rw [hR] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | replace himage := congr_arg (inv β) himage | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | change ((ζ ≫ β ≫ inv β) R₁) = ((ζ ≫ β ≫ inv β) R₂) at himage | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [IsIso.hom_inv_id, Category.comp_id] at himage | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨a₁, r₁, h₁⟩ := finiteCoproduct.ι_jointly_surjective R₁ | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | case intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨a₂, r₂, h₂⟩ := finiteCoproduct.ι_jointly_surjective R₂ | case intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π... | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduc... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have ha₁ : a₁ = R₁.fst := (congrArg Sigma.fst h₁).symm | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
fi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have ha₂ : a₂ = R₂.fst := (congrArg Sigma.fst h₂).symm | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
fi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have ha : a₁ = a₂ := by rwa [ha₁, ha₂] | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
fi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have : R₁ ∈ Set.range (finiteCoproduct.ι _ a₂) | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
fi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | obtain ⟨xr', hr'⟩ := this | case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullb... | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
fi... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← hr', h₂] at hR | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have hf : ∀ (a : α), Function.Injective
((finiteCoproduct.ι _ a) ≫ (finiteCoproduct.desc _ ((fun a => pullback.fst f (π a))))) | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have := hf a₂ hR | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← hr', h₂, this] | case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a =>... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | simp [← Category.assoc, finiteCoproduct.ι_desc, pullback.condition] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [comm] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | change f (σ R₁) = f (σ R₂) | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [hR] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | constructor <;> rfl | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← this.1, ← this.2, himage] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rwa [ha₁, ha₂] | α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ finiteCopro... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.range_eq ha, h₁] | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | simp only [Set.mem_range, exists_apply_eq_apply] | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | intro a | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | simp only [finiteCoproduct.ι_desc] | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | intro x y h | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have h₁ := h | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | apply_fun f at h | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | change (pullback.fst f (π a) ≫ f) x = _ at h | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have h' := h.symm | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | change (pullback.fst f (π a) ≫ f) y = _ at h' | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [pullback.condition] at h' | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have : Function.Injective (π a) | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have h₂ := this h' | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | suffices : x.val = y.val | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | exact Prod.ext h₁ h₂.symm | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | intro r s hrs | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.ι_desc_apply] at hrs | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have hrs' := hrs.symm | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | rw [← finiteCoproduct.ι_desc_apply] at hrs' | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | have : Function.Injective (finiteCoproduct.desc (fun a ↦ Z a) π) | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | exact (finiteCoproduct.ι_injective a (this hrs')).symm | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | apply Function.Bijective.injective | case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ f... | case this.hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | exact ConcreteCategory.bijective_of_isIso _ | case this.hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.de... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | CompHaus/ExplicitSheaves.lean | CompHaus.extensivity_injective | [10, 1] | [65, 23] | exact Subtype.ext this | case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (fun a => pullback f (π a)) fun a => pullback.snd f (π a) ≫ fin... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
α : Type
inst✝ : Fintype α
X : CompHaus
Z : α → CompHaus
π : (a : α) → Z a ⟶ X
Y : CompHaus
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
ζ : (finiteCoproduct fun a => pullback f (π a)) ⟶ finiteCoproduct Z :=
finiteCoproduct.desc (f... |
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