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/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_lt_0 | [25, 1] | [29, 27] | have h₁ := h.prop.left | h : fformat
⊢ 0 < (↑h).r | h : fformat
h₁ : 1 < (↑h).r
⊢ 0 < (↑h).r | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
⊢ 0 < (↑h).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_lt_0 | [25, 1] | [29, 27] | have h₂ : 0 < (1 : Int) := by trivial | h : fformat
h₁ : 1 < (↑h).r
⊢ 0 < (↑h).r | h : fformat
h₁ : 1 < (↑h).r
h₂ : 0 < 1
⊢ 0 < (↑h).r | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
h₁ : 1 < (↑h).r
⊢ 0 < (↑h).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_lt_0 | [25, 1] | [29, 27] | apply Int.lt_trans h₂ h₁ | h : fformat
h₁ : 1 < (↑h).r
h₂ : 0 < 1
⊢ 0 < (↑h).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
h₁ : 1 < (↑h).r
h₂ : 0 < 1
⊢ 0 < (↑h).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_lt_0 | [25, 1] | [29, 27] | trivial | h : fformat
h₁ : 1 < (↑h).r
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
h₁ : 1 < (↑h).r
⊢ 0 < 1
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | gt_zero_ne_zero | [32, 1] | [33, 31] | intros h h₁ | x : ℤ
⊢ 1 < x → x ≠ 0 | x : ℤ
h : 1 < x
h₁ : x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
⊢ 1 < x → x ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | gt_zero_ne_zero | [32, 1] | [33, 31] | simp [h₁] at h | x : ℤ
h : 1 < x
h₁ : x = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℤ
h : 1 < x
h₁ : x = 0
⊢ False
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ne_0 | [36, 1] | [39, 33] | intro h | ⊢ ∀ (fmt : fformat), (↑fmt).r ≠ 0 | h : fformat
⊢ (↑h).r ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ (fmt : fformat), (↑fmt).r ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ne_0 | [36, 1] | [39, 33] | have h₁ := h.prop.left | h : fformat
⊢ (↑h).r ≠ 0 | h : fformat
h₁ : 1 < (↑h).r
⊢ (↑h).r ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
⊢ (↑h).r ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ne_0 | [36, 1] | [39, 33] | apply gt_zero_ne_zero | h : fformat
h₁ : 1 < (↑h).r
⊢ (↑h).r ≠ 0 | case a
h : fformat
h₁ : 1 < (↑h).r
⊢ 1 < (↑h).r | Please generate a tactic in lean4 to solve the state.
STATE:
h : fformat
h₁ : 1 < (↑h).r
⊢ (↑h).r ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ne_0 | [36, 1] | [39, 33] | trivial | case a
h : fformat
h₁ : 1 < (↑h).r
⊢ 1 < (↑h).r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
h : fformat
h₁ : 1 < (↑h).r
⊢ 1 < (↑h).r
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_even | [42, 1] | [43, 29] | simp [fmt.prop.right.left] | fmt : fformat
⊢ (↑fmt).r % 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : fformat
⊢ (↑fmt).r % 2 = 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_prec_lt_0 | [46, 1] | [48, 10] | have h := fmt.prop.right.right | fmt : fformat
⊢ 0 < (↑fmt).p | fmt : fformat
h : 0 < (↑fmt).p
⊢ 0 < (↑fmt).p | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : fformat
⊢ 0 < (↑fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_prec_lt_0 | [46, 1] | [48, 10] | exact h | fmt : fformat
h : 0 < (↑fmt).p
⊢ 0 < (↑fmt).p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : fformat
h : 0 < (↑fmt).p
⊢ 0 < (↑fmt).p
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ipow_le_0 | [61, 1] | [69, 36] | simp [HPow.hPow, Pow.pow, pow_int] | fmt : fformat
e : ℤ
⊢ (↑fmt).r ^ e ≠ 0 | fmt : fformat
e : ℤ
⊢ ¬↑(↑fmt).r ^ e = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : fformat
e : ℤ
⊢ (↑fmt).r ^ e ≠ 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ipow_le_0 | [61, 1] | [69, 36] | cases e with
| ofNat n =>
simp [pow_ne_zero]
| negSucc n =>
simp_all only [zpow_negSucc, inv_eq_zero, add_pos_iff, or_true, pow_eq_zero_iff, Int.cast_eq_zero, fformat_radix_lt_1,
gt_zero_ne_zero, not_false_iff] | fmt : fformat
e : ℤ
⊢ ¬↑(↑fmt).r ^ e = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
fmt : fformat
e : ℤ
⊢ ¬↑(↑fmt).r ^ e = 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ipow_le_0 | [61, 1] | [69, 36] | simp [pow_ne_zero] | case ofNat
fmt : fformat
n : ℕ
⊢ ¬↑(↑fmt).r ^ Int.ofNat n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofNat
fmt : fformat
n : ℕ
⊢ ¬↑(↑fmt).r ^ Int.ofNat n = 0
TACTIC:
|
https://github.com/opencompl/HOLFloat-Lean.git | 6207518be26dcfc9980a63727bd1440cdbc6bb7a | HOLFloat/Fixed_theroem.lean | fformat_radix_ipow_le_0 | [61, 1] | [69, 36] | simp_all only [zpow_negSucc, inv_eq_zero, add_pos_iff, or_true, pow_eq_zero_iff, Int.cast_eq_zero, fformat_radix_lt_1,
gt_zero_ne_zero, not_false_iff] | case negSucc
fmt : fformat
n : ℕ
⊢ ¬↑(↑fmt).r ^ Int.negSucc n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case negSucc
fmt : fformat
n : ℕ
⊢ ¬↑(↑fmt).r ^ Int.negSucc n = 0
TACTIC:
|
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | by_cases G_empty : G = ∅ | A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 pos
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | simpa only [G_empty, image_empty] using empty_subset _ | case pos
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | intro a ha | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | rw [mem_closure_iff] | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | intro N hN ha' | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | rcases (G.mem_image ρ a).mp ha with ⟨e, he, rfl⟩ | case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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... | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : 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 | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | have non_empty : (G ∩ ρ⁻¹' N).Nonempty := ⟨e, mem_inter he <| mem_preimage.mpr ha'⟩ | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | have is_open : IsOpen <| G ∩ ρ⁻¹' N := hG.inter <| hN.preimage ρ_cont | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | have ne_top : ρ '' (G ∩ ρ⁻¹' N)ᶜ ≠ ⊤ := image_ne_top _ (compl_ne_univ.mpr non_empty)
is_open.isClosed_compl | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | rcases nonempty_compl.mpr ne_top with ⟨x, hx⟩ | case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjec... | case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjecti... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | have hx' : x ∈ (ρ '' Gᶜ)ᶜ := (compl_subset_compl.mpr <| image_subset ρ <|
compl_subset_compl.mpr <| G.inter_subset_left _) hx | case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.... | case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Su... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | rcases ρ_surj x with ⟨y, rfl⟩ | case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.... | case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Su... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | have hy : y ∈ G ∩ ρ⁻¹' N := not_mem_compl_iff.mp <| (not_imp_not.mpr <| mem_image_of_mem ρ) <|
(mem_compl_iff _ _).mp hx | case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Fun... | case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Fun... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Funct... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | image_subset_closure_compl_image_compl | [12, 1] | [31, 78] | exact ⟨ρ y, mem_inter (mem_preimage.mp <| mem_of_mem_inter_right hy) hx'⟩ | case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Fun... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.intro.intro
A : Type u_2
B : Type ?u.523
C : Type ?u.526
D : Type u_1
E : Set D
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace C
inst✝ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Funct... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | intro x y h_eq_im | A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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_to... | A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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_to... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_c... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | by_contra h | A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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_to... | A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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_to... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_c... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | rcases t2_separation h with ⟨G₁, G₂, hG₁, hG₂, hxG₁, hyG₂, hG⟩ | A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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_to... | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | Please generate a tactic in lean4 to solve the state.
STATE:
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_c... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have open1 : IsOpen ((ρ '' (G₁ᶜ))ᶜ) | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have open2 : IsOpen ((ρ '' (G₂ᶜ))ᶜ) | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have disj : Disjoint ((ρ '' (G₁ᶜ))ᶜ) ((ρ '' (G₂ᶜ))ᶜ) | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | replace disj := disjoint_of_extremally_disconnected disj open1 open2 | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have oups₁ := image_subset_closure_compl_image_compl ρ_cont ρ_surj image_ne_top hG₁ | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have oups₂ := image_subset_closure_compl_image_compl ρ_cont ρ_surj image_ne_top hG₂ | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have mem₁ : ρ x ∈ ρ '' G₁ | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case mem₁
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have mem₂ : ρ x ∈ ρ '' G₂ | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | have mem : ρ x ∈ (closure ((ρ '' G₁ᶜ)ᶜ)) ∩ (closure ((ρ '' G₂ᶜ)ᶜ)) := ⟨oups₁ mem₁, oups₂ mem₂⟩ | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | exact (disj.ne_of_mem mem.1 mem.2) rfl | case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_s... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | simp only [isOpen_compl_iff] | case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | apply IsCompact.isClosed | case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case open1
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | refine' IsCompact.image (IsClosed.isCompact _) ρ_cont | case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | simpa only [isClosed_compl_iff] | case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case open1.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | simp only [isOpen_compl_iff] | case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | Please generate a tactic in lean4 to solve the state.
STATE:
case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | apply IsCompact.isClosed | case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
... | case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case open2
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | refine' IsCompact.image (IsClosed.isCompact _) ρ_cont | case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | simpa only [isClosed_compl_iff] | case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case open2.hs
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | compact_to_T2_injective | [38, 1] | [69, 41] | rw [disjoint_compl_left_iff_subset] | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | refine' subset_trans (subset_image_compl ρ_surj) _ | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | simp only [compl_compl, image_subset_iff] | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | refine' subset_trans _ (subset_preimage_image _ _) | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | rw [← disjoint_compl_left_iff_subset] | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | simpa only [compl_compl] | case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case disj
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | exact mem_image_of_mem _ hxG₁ | case mem₁
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mem₁
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | rw [h_eq_im] | case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | Please generate a tactic in lean4 to solve the state.
STATE:
case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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 | compact_to_T2_injective | [38, 1] | [69, 41] | exact mem_image_of_mem _ hyG₂ | case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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
i... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mem₂
A : Type u_1
B : Type ?u.3855
C : Type ?u.3858
D : Type u_2
E : Set D
inst✝⁷ : TopologicalSpace A
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace C
inst✝⁴ : 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] | haveI : CompactSpace <| D' φ f := isCompact_iff_compactSpace.mp (IsClosed.isCompact
(isClosed_eq (Continuous.comp hφ continuous_fst) (Continuous.comp hf continuous_snd))) | 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] | let S := { E : Set (D' φ f) | CompactSpace E ∧ (π₁ φ f) '' E = univ} | 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 := zorn_superset S chain_cond | 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] | rcases this with ⟨E, ⟨⟨hE₁,hE₂⟩, hE₃⟩⟩ | 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 intro.intro.intro
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.Su... | 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] | use E | case intro.intro.intro
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.Su... | case intro.intro.intro
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.Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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.Surj... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | constructor | case intro.intro.intro
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.Su... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
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.Surj... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | constructor | case intro.intro.intro.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 : Funct... | 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 : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.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 : Functio... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro Ch h hCh | 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] | let M := sInter Ch | 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] | use M | 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] | constructor | 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 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 : 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.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] | constructor | case 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 : Function.Surjective π
imag... | case left.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 : Function.Surjective π... | 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.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] | rw [←isCompact_iff_compactSpace] | case left.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 : Function.Surjective π... | case left.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 : Function.Surjective π... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | apply IsClosed.isCompact | case left.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 : Function.Surjective π... | case left.left.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... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | apply isClosed_sInter | case left.left.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... | case left.left.h.a
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.Surjecti... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | intro N hN | case left.left.h.a
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.Surjecti... | case left.left.h.a
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.Surjecti... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left.h.a
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have N_comp := (h hN).1 | case left.left.h.a
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.Surjecti... | case left.left.h.a
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.Surjecti... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left.h.a
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [←isCompact_iff_compactSpace] at N_comp | case left.left.h.a
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.Surjecti... | case left.left.h.a
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.Surjecti... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left.h.a
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | exact IsCompact.isClosed N_comp | case left.left.h.a
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.Surjecti... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.left.h.a
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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | by_cases h₂ : Nonempty Ch | case left.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 ... | case pos
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case left.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
... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rotate_left 1 | case pos
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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] | simp at h₂ | case neg
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | rw [←eq_empty_iff_forall_not_mem] at h₂ | case neg
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | revert M | case neg
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | rw [h₂, sInter_empty] | case neg
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | simp | case neg
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... | case neg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 range_iff_surjective.mpr <| fun a => ⟨⟨⟨_, _⟩, (hf' <| φ a).choose_spec.symm⟩, rfl⟩ | case neg
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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] | ext x | case pos
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... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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] | refine' ⟨ fun _ => trivial , fun _ => _ ⟩ | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | rw [mem_image] | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | change Set.Nonempty {x_1 : D' φ f | x_1 ∈ M ∧ π₁ φ f x_1 = x} | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | let Z : Ch → Set (D' φ f) := fun X => X ∩ (π₁ _ _)⁻¹' {x} | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | suffices Set.Nonempty <| ⋂ (X : Ch), (X: Set (D' φ f)) ∩ (π₁ _ _)⁻¹' {x} by
convert this
rw [← iInter_inter, ← sInter_eq_iInter]
rw [←setOf_inter_eq_sep, inter_comm]
congr | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | have assumps : ∀ (i : ↑Ch), Set.Nonempty (Z i) ∧ IsCompact (Z i) ∧ IsClosed (Z i) := by
intro X
simp
have h₃ := (h (Subtype.mem X)).2
rw [←image_inter_nonempty_iff, h₃]
simp
have := (h (Subtype.mem X)).1
rw [←isCompact_iff_compactSpace] at this
have h_cl := IsClosed.inter (IsCompact.isClosed this)
... | case pos.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
ima... | case pos.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
ima... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed Z _
(fun X => (assumps X).1) (fun X => (assumps X).2.1) (fun X => (assumps X).2.2) | case pos.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
ima... | 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:
case pos.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... |
https://github.com/adamtopaz/CopenhagenMasterclass2023.git | a293ca1554f7e80d891fd4d86fb092c54d8a0a01 | ExtrDisc/Gleason.lean | exists_image_ne_top | [87, 1] | [172, 33] | change Directed _ ((fun X => X ∩ _) ∘ (fun X => X: ↑Ch → Set (D' φ f))) | 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] | refine' Directed.mono_comp _ _ _ | 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 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... | 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] | refine Iff.mp directedOn_iff_directed ?_ | case refine'_3
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... | case refine'_3
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_3
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] | refine IsChain.directedOn ?H | case refine'_3
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... | 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 refine'_3
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... |
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