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/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | dsimp [IsChain, Pairwise] at hCh ⊢ | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_n... | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_n... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun ⦃x⦄ a ⦃y⦄ a_1 a_2 => Or.comm.mp (hCh a a_1 a_2) | case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | convert this | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [← iInter_inter, ← sInter_eq_iInter] | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←setOf_inter_eq_sep, inter_comm] | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | congr | case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro X | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | simp | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have h₃ := (h (Subtype.mem X)).2 | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←image_inter_nonempty_iff, h₃] | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | simp | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have := (h (Subtype.mem X)).1 | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←isCompact_iff_compactSpace] at this | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have h_cl := IsClosed.inter (IsCompact.isClosed this)
(IsClosed.preimage (Continuous.comp continuous_fst continuous_subtype_val) <| T1Space.t1 x) | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact And.intro (IsClosed.isCompact h_cl) h_cl | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun x y => x ⊇ y | case refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact fun ⦃x y⦄ a => inter_subset_inter_left _ a | case refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro X hX | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact sInter_subset_of_mem hX | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact hE₁ | case intro.intro.intro.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Functi... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact hE₂ | case intro.intro.intro.right.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Fu... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro E₀ h₁ h₂ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : F... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | replace hE₃ := hE₃ E₀ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : F... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | by_contra h₃ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : F... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | replace hE₃ := hE₃ (And.intro h₂ h₃) (subset_of_ssubset h₁) | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : F... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact (ne_of_ssubset h₁) hE₃ | case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.right.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E✝ : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E✝ → A
ρ_cont : Continuous ρ
ρ_surj : F... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | constructor | A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top :... | case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
imag... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | refine' Continuous.comp _ _ | case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
imag... | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surject... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | refine' ContinuousOn.restrict _ | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surject... | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surject... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjectiv... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | apply Continuous.continuousOn | case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surject... | case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surje... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjectiv... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact Continuous.comp continuous_snd continuous_subtype_val | case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surje... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_1.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surject... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact (Homeomorph.continuous (Homeomorph.symm (ρ' hφ hf hf'))) | case left.refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surject... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.refine'_2
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjectiv... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | suffices : f ∘ (E' hφ hf hf').restrict (π₂ φ f) = φ ∘ (ρ' hφ hf hf').toFun | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | ext x | case this
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
imag... | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case this
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [Function.comp_apply, restrict_apply, Equiv.toFun_as_coe, Homeomorph.coe_toEquiv] | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | dsimp [π₂, ρ', compact_to_T2_homeomorph, Continuous.homeoOfEquivCompactToT2_toEquiv,
Continuous.homeoOfEquivCompactToT2, π₁] | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | have := x.val.prop | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | dsimp [D'] at this | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | exact this.symm | case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
im... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [← Function.comp.assoc] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | rw [this] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | gleason_diagram_commutes | [236, 1] | [257, 20] | simp only [Function.comp.assoc, Equiv.toFun_as_coe, Homeomorph.coe_toEquiv,
Equiv.invFun_as_coe, Homeomorph.coe_symm_toEquiv,
Homeomorph.self_comp_symm, Function.comp.right_id] | case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
ima... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.44712
E : Set D
inst✝⁹ : TopologicalSpace A
inst✝⁸ : TopologicalSpace B
inst✝⁷ : TopologicalSpace C
inst✝⁶ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | intro a | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
⊢ ∀ (a : α), OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
⊢ ∀ (a : α), OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have h₁ : OpenEmbedding (Sigma.desc i) :=
(ExtrDisc.homeoOfIso (asIso (Sigma.desc i))).openEmbedding | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have h₂ : OpenEmbedding (Sigma.ι Z a) := openEmbedding_ι _ _ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | have := OpenEmbedding.comp h₁ h₂ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | erw [← CategoryTheory.coe_comp (Sigma.ι Z a) (Sigma.desc i)] at this | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding (↑(Sigma.desc i) ∘ ↑(Sigma.ι Z a))
⊢ OpenEmbedding ↑(i a... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app] at this | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(Sigma.ι Z a ≫ Sigma.desc i)
⊢ OpenEmbedding ↑(i a)
TAC... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.openEmbedding_of_sigma_desc_iso | [15, 1] | [25, 13] | assumption | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
HIso : IsIso (Sigma.desc i)
a : α
h₁ : OpenEmbedding ↑(Sigma.desc i)
h₂ : OpenEmbedding ↑(Sigma.ι Z a)
this : OpenEmbedding ↑(i a)
⊢ OpenEmbedding ↑(i a)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isIso_of_bijective | [33, 1] | [36, 41] | exact CompHaus.isIso_of_bijective _ hf | X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
⊢ IsIso (toCompHaus.map f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
⊢ IsIso (toCompHaus.map f)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.isIso_of_bijective | [33, 1] | [36, 41] | apply isIso_of_fully_faithful toCompHaus | X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
this : IsIso (toCompHaus.map f)
⊢ IsIso f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : ExtrDisc
f : X ⟶ Y
hf : Function.Bijective ↑f
this : IsIso (toCompHaus.map f)
⊢ IsIso f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let ζ := finiteCoproduct.desc _ (fun a => pullback.snd f (hOpen a) ≫ finiteCoproduct.ι Z a ) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.d... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ Function.Injective
↑(finiteCoproduct.desc (fun a => (OpenE... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let α := finiteCoproduct.desc _ ((fun a => pullback.fst f (hOpen a))) | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoproduct.d... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbe... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | let β := finiteCoproduct.desc _ π | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | have comm : ζ ≫ β = α ≫ f := by
refine' finiteCoproduct.hom_ext _ _ _ (fun a => _)
simp [← Category.assoc, finiteCoproduct.ι_desc, ExtrDisc.pullback.condition] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | intro R₁ R₂ hR | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | have himage : (ζ ≫ β) R₁ = (ζ ≫ β) R₂ := by
rw [comm]; change f (α R₁) = f (α R₂); rw [hR] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | replace himage := congr_arg (inv β) himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | change ((ζ ≫ β ≫ inv β) R₁) = ((ζ ≫ β ≫ inv β) R₂) at himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [IsIso.hom_inv_id, Category.comp_id] at himage | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | exact Sigma.subtype_ext Hfst hR | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | simp [← Category.assoc, finiteCoproduct.ι_desc, ExtrDisc.pullback.condition] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [comm] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | change f (α R₁) = f (α R₂) | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [hR] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | constructor <;> rfl | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_injective | [38, 1] | [58, 34] | rw [← this.1, ← this.2, himage] | α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) ⟶ finiteCoproduct Z :=
finiteCoprod... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type
inst✝ : Fintype α✝
X : ExtrDisc
Z : α✝ → ExtrDisc
π : (a : α✝) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α✝), OpenEmbedding ↑(π a)
ζ : (finiteCoproduct fun a => (OpenEmbeddingCone f (_ : Ope... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | let β := finiteCoproduct.desc _ π | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(π a))).pt) fun a =>
ExtrDisc.pull... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso
(finiteCoproduct.desc (fun a => (... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
⊢ IsIso
(finiteCoproduct.desc (fun a => (OpenEmbeddingCone f... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | refine' isIso_of_bijective ⟨extensivity_injective f HIso hOpen, fun y => _⟩ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
⊢ IsIso
(finiteCoproduct.desc (fun a => (... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ∃ a,
↑(finiteCoproduc... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | refine' ⟨⟨(inv β (f y)).1, ⟨y, (inv β (f y)).2, _⟩⟩, rfl⟩ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ∃ a,
↑(finiteCoproduc... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ↑(π (↑(CategoryTheory.inv... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | have inj : Function.Injective (inv β) := by intros r s hrs
convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ ↑(π (↑(CategoryTheory.inv... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | apply inj | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injec... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | intro a | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injec... | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injec... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | simp only [IsIso.comp_inv_eq, finiteCoproduct.ι_desc] | case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | intros r s hrs | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
⊢ Function.Injective ↑(Cate... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
r s : CoeSort.coe X
hrs : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | convert congr_arg β hrs <;> change _ = (inv β ≫ β) _<;> simp only [IsIso.inv_hom_id]<;> rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
r s : CoeSort.coe X
hrs : ↑... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | change (_ ≫ inv β) _ = _ | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | rw [this] | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity_explicit | [60, 1] | [76, 56] | rfl | α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (fun a => Z a) π
y : CoeSort.coe Y
inj : Function.Injective ↑(... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
π : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
HIso : IsIso (finiteCoproduct.desc (fun a => Z a) π)
hOpen : ∀ (a : α), OpenEmbedding ↑(π a)
β : (finiteCoproduct fun a => Z a) ⟶ X := finiteCoproduct.desc (... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | have hOpen := openEmbedding_of_sigma_desc_iso H | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let θ := Sigma.mapIso (fun a => fromExplicitIso f (hOpen a)) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
⊢ IsIso (Sigma.desc fun x => Limits.pullback.fst)
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let δ := FromFiniteCoproductIso (fun a => (OpenEmbeddingCone f (hOpen a)).pt) | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | convert extensivity_explicit f HIso hOpen | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | refine' finiteCoproduct.hom_ext _ _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => p... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | simp [← Category.assoc, finiteCoproduct.ι_desc, fromExplicit] | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => p... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left θ.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left δ.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | let ε := ToFiniteCoproductIso Z | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | convert H | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | refine' Sigma.hom_ext _ _ (fun a => _) | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => p... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | simp [← Category.assoc] | case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => p... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.extensivity | [78, 1] | [95, 64] | apply IsIso.of_isIso_comp_left ε.hom | α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i a) :=
Sigma.mapIso fun a => fromExplicitIso f (_ : OpenEmb... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
x✝ : Fintype α
X : ExtrDisc
Z : α → ExtrDisc
i : (a : α) → Z a ⟶ X
Y : ExtrDisc
f : Y ⟶ X
H : IsIso (Sigma.desc i)
hOpen : ∀ (a : α), OpenEmbedding ↑(i a)
θ : (∐ fun a => (OpenEmbeddingCone f (_ : OpenEmbedding ↑(i a))).pt) ≅ ∐ fun a => pullback f (i... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | refine' fun P => ⟨(@fun X Y f e he => _)⟩ | ⊢ EverythingIsProjective ExtrDisc | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ EverythingIsProjective ExtrDisc
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | have proj : Projective (toCompHaus.obj P) := inferInstanceAs (Projective P.compHaus) | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | have : Epi (toCompHaus.map e) := by rw [CompHaus.epi_iff_surjective]
change Function.Surjective e
rwa [← ExtrDisc.epi_iff_surjective] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | set g := toCompHaus.preimage <| Projective.factorThru (toCompHaus.map f) (toCompHaus.map e) with hg | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ ∃ f', f' ≫ e = f | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
⊢ ∃ f', f' ≫ e = f
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | refine' ⟨g, toCompHaus.map_injective _⟩ | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ ∃ f', f' ≫ e = f | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ toCompHaus.map (g... | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorTh... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | rw [map_comp, hg, image_preimage, Projective.factorThru_comp] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
⊢ toCompHaus.map (g... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
this : Epi (toCompHaus.map e)
g : P ⟶ X := toCompHaus.preimage (Projective.factorThru (toCompHaus.map f) (toCompHaus.map e))
hg : g = toCompHaus.preimage (Projective.factorTh... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/ExplicitSheaves.lean | ExtrDisc.everything_proj | [97, 1] | [106, 64] | rw [CompHaus.epi_iff_surjective] | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Epi (toCompHaus.map e) | P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Function.Surjective ↑(toCompHaus.map e) | Please generate a tactic in lean4 to solve the state.
STATE:
P X Y : ExtrDisc
f : P ⟶ Y
e : X ⟶ Y
he : Epi e
proj : Projective (toCompHaus.obj P)
⊢ Epi (toCompHaus.map e)
TACTIC:
|
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