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/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact ⟨x, mem_diff_of_mem xa xuv⟩ | case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsCl... | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact (c a).diff (uo.union vo) | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed S... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact ((c a).diff (uo.union vo)).isClosed | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed S... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases n with ⟨x, n⟩ | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v :... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [mem_iInter, mem_diff, forall_and, forall_const] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rw [← mem_iInter] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [suv n.1, not_true, imp_false] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact n.2 | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | generalize hs : (fun a ↦ closure (r '' Ici a)) = s | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have m : Antitone s := by
intro a b ab; rw [← hs]; exact closure_mono (monotone_image (Ici_subset_Ici.mpr ab)) | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have d : Directed Superset s := by
intro a b; exact ⟨a ⊔ b, m le_sup_left, m le_sup_right⟩ | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have p : ∀ a, IsPreconnected (s a) := by
intro a; rw [← hs]; exact ((p _).image _ rc.continuousOn).closure | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have c : ∀ a, IsCompact (s a) := by
intro a; rw [← hs]; exact isClosed_closure.isCompact | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have e : {x | MapClusterPt x atTop r} = ⋂ a, s a := by
ext x
simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm P, ← hs]
apply forall_congr'; intro t
simp only [@forall_comm P, mem_inter_iff, mem_image, mem_Ici, @and_comm (_ ∈ t),
exists_exists_and_eq_an... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [e] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact IsPreconnected.directed_iInter d p c | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a b ab | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact closure_mono (monotone_image (Ici_subset_Ici.mpr ab)) | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a b | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact ⟨a ⊔ b, m le_sup_left, m le_sup_right⟩ | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact ((p _).image _ rc.continuousOn).closure | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact isClosed_closure.isCompact | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | ext x | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm P, ← hs] | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | apply forall_congr' | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro t | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | simp only [@forall_comm P, mem_inter_iff, mem_image, mem_Ici, @and_comm (_ ∈ t),
exists_exists_and_eq_and, Filter.frequently_atTop, exists_prop] | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | set r' : Pᵒᵈ → X := r | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | have rc' : Continuous r' := rc | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | have p' : ∀ a : Pᵒᵈ, IsPreconnected (Ici a) := fun a ↦ p a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | exact IsPreconnected.limits_atTop p' rc' | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | by_cases ab : ¬a < b | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [not_not] at ab | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬¬a < b
⊢ IsPreconnected {x | MapClusterPt x (�... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | generalize hs : (fun t : Ioc a b ↦ closure (r '' Ioc a t)) = s | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have n : Nonempty (Ioc a b) := ⟨b, right_mem_Ioc.mpr ab⟩ | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have m : Monotone s := by
intro a b ab; rw [← hs]; refine closure_mono (monotone_image ?_)
exact Ioc_subset_Ioc (le_refl _) (Subtype.coe_le_coe.mpr ab) | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have d : Directed Superset s := fun a b ↦ ⟨min a b, m (min_le_left _ _), m (min_le_right _ _)⟩ | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have p : ∀ t, IsPreconnected (s t) := by
intro ⟨t, m⟩; rw [← hs]; refine (isPreconnected_Ioc.image _ (rc.mono ?_)).closure
simp only [mem_Ioc] at m
simp only [Subtype.coe_mk, Ioc_subset_Ioc_iff m.1, m.2, le_refl, true_and_iff] | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have c : ∀ t, IsCompact (s t) := by intro t; rw [← hs]; exact isClosed_closure.isCompact | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [e] | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact IsPreconnected.directed_iInter d p c | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Ioc_eq_empty ab, nhdsWithin_empty, MapClusterPt, Filter.map_bot, ClusterPt.bot,
setOf_false, isPreconnected_empty] | case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro a b ab | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | refine closure_mono (monotone_image ?_) | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact Ioc_subset_Ioc (le_refl _) (Subtype.coe_le_coe.mpr ab) | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro ⟨t, m⟩ | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | refine (isPreconnected_Ioc.image _ (rc.mono ?_)).closure | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] at m | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Subtype.coe_mk, Ioc_subset_Ioc_iff m.1, m.2, le_refl, true_and_iff] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro t | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact isClosed_closure.isCompact | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply Set.ext | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro x | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm _ (Set X), ← hs] | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply forall_congr' | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro u | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [@forall_comm _ (u ∈ 𝓝 x)] | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply forall_congr' | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro _ | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_inter_iff, Filter.frequently_iff, nhdsWithin_Ioc_eq_nhdsWithin_Ioi ab] | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | constructor | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro h ⟨t, m⟩ | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have tm : Ioc a t ∈ 𝓝[Ioi a] a := by
apply Ioc_mem_nhdsWithin_Ioi
simp only [mem_Ioc] at m; simp only [mem_Ico]; use le_refl _, m.1 | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases h tm with ⟨v, vm, vu⟩ | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact ⟨r v, vu, mem_image_of_mem _ vm⟩ | case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply Ioc_mem_nhdsWithin_Ioi | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] at m | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ico] | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use le_refl _, m.1 | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro h v vm | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : Continuous... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.mp vm with ⟨w, wa, wv⟩ | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : Continuous... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioi] at wa | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have m : min w b ∈ Ioc a b := by simp only [mem_Ioc]; use lt_min wa ab, min_le_right _ _ | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases h ⟨_, m⟩ with ⟨x, xu, rx⟩ | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Subtype.coe_mk, mem_image, mem_Ioc, le_min_iff] at rx | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases rx with ⟨c, ⟨ac, cw, _⟩, cx⟩ | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use c, wv (mem_Ioc.mpr ⟨ac, cw⟩) | case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b... | case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rwa [cx] | case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use lt_min wa ab, min_le_right _ _ | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | generalize hu : (fun x : s ↦ (x : X)) ⁻¹' t = u | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
⊢ s ⊆ interior t | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ s ⊆ int... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have uo : IsOpen u := by
rw [← subset_interior_iff_isOpen]; intro ⟨x, m⟩ h
simp only [mem_preimage, Subtype.coe_mk, ← hu] at h
have n := op ⟨m, h⟩
simp only [mem_interior_iff_mem_nhds, preimage_coe_mem_nhds_subtype, Subtype.coe_mk,
← hu] at n ⊢
exact nhdsWithin_le_nhds n | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ s ⊆ int... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have uc : IsClosed u := by
rw [← closure_eq_iff_isClosed]; refine subset_antisymm ?_ subset_closure
rw [← hu]
refine _root_.trans (continuous_subtype_val.closure_preimage_subset _) ?_
intro ⟨x, m⟩ h; exact cl ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have p : IsPreconnected (univ : Set s) := (Subtype.preconnectedSpace sp).isPreconnected_univ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | cases' disjoint_or_subset_of_isClopen p ⟨uc, uo⟩ with h h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← subset_interior_iff_isOpen] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ IsOpen ... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ int... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | intro ⟨x, m⟩ h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ int... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [mem_preimage, Subtype.coe_mk, ← hu] at h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have n := op ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [mem_interior_iff_mem_nhds, preimage_coe_mem_nhds_subtype, Subtype.coe_mk,
← hu] at n ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.