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stringlengths 6
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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 : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
x : X
xa : x ∈ s a
xuv : x ∉ u ∪ v
⊢ (s a \ (u ∪ v)).Nonempty
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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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 : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
x : X
xa : x ∈ s a
xuv : x ∉ u ∪ v
⊢ (s a \ (u ∪ v)).Nonempty
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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsCompact (s a \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
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 a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsCompact (s a \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsCompact (s a \ (u ∪ v))
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsClosed (s a \ (u ∪ v)) | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsClosed (s a \ (u ∪ v)) | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ IsClosed (s a \ (u ∪ v))
TACTIC:
|
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 : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
n : (⋂ a, s a \ (u ∪ v)).Nonempty
⊢ False | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ a, s a \ (u ∪ v)
⊢ False | 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
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
n : (⋂ a, s a \ (u ∪ v)).Nonempty
⊢ False
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ a, s a \ (u ∪ v)
⊢ False | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : (∀ (x_1 : I), x ∈ s x_1) ∧ x ∉ u ∪ v
⊢ False | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ a, s a \ (u ∪ v)
⊢ False
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : (∀ (x_1 : I), x ∈ s x_1) ∧ x ∉ u ∪ v
⊢ False | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ x ∉ u ∪ v
⊢ False | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : (∀ (x_1 : I), x ∈ s x_1) ∧ x ∉ u ∪ v
⊢ False
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ x ∉ u ∪ v
⊢ False | 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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ False
⊢ False | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ x ∉ u ∪ v
⊢ False
TACTIC:
|
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, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ False
⊢ False | 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 Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
x : X
n : x ∈ ⋂ i, s i ∧ False
⊢ False
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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_and, Filter.frequently_atTop, exists_prop] | 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected {x | MapClusterPt x atTop r} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected {x | MapClusterPt x atTop 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 : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected (⋂ a, s a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected {x | MapClusterPt x atTop r}
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected (⋂ a, s a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
e : {x | MapClusterPt x atTop r} = ⋂ a, s a
⊢ IsPreconnected (⋂ a, s a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
⊢ Antitone 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ s b ≤ s a | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
⊢ Antitone s
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ s b ≤ s 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ (fun a => closure (r '' Ici a)) b ≤ (fun a => closure (r '' Ici a)) a | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ s b ≤ s a
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ (fun a => closure (r '' Ici a)) b ≤ (fun a => closure (r '' Ici a)) a | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
a b : P
ab : a ≤ b
⊢ (fun a => closure (r '' Ici a)) b ≤ (fun a => closure (r '' Ici a)) a
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
⊢ Directed Superset 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
a b : P
⊢ ∃ z, s a ⊇ s z ∧ s b ⊇ s z | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
⊢ Directed Superset s
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
a b : P
⊢ ∃ z, s a ⊇ s z ∧ s b ⊇ s z | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
a b : P
⊢ ∃ z, s a ⊇ s z ∧ s b ⊇ s z
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
⊢ ∀ (a : P), IsPreconnected (s 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected (s a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
⊢ ∀ (a : P), IsPreconnected (s a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected (s 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected ((fun a => closure (r '' Ici a)) a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected (s a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected ((fun a => closure (r '' Ici a)) a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
a : P
⊢ IsPreconnected ((fun a => closure (r '' Ici a)) a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
⊢ ∀ (a : P), IsCompact (s 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact (s a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
⊢ ∀ (a : P), IsCompact (s a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact (s 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact ((fun a => closure (r '' Ici a)) a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact (s a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact ((fun a => closure (r '' Ici a)) a) | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
a : P
⊢ IsCompact ((fun a => closure (r '' Ici a)) a)
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
⊢ {x | MapClusterPt x atTop r} = ⋂ a, s a | 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ x ∈ {x | MapClusterPt x atTop r} ↔ x ∈ ⋂ a, s a | 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 : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
⊢ {x | MapClusterPt x atTop r} = ⋂ a, s a
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ x ∈ {x | MapClusterPt x atTop r} ↔ x ∈ ⋂ a, s a | 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : P) in atTop, r a ∈ s) ↔ ∀ b ∈ 𝓝 x, ∀ (a : P), ∃ x, x ∈ b ∩ r '' Ici a | 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✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ x ∈ {x | MapClusterPt x atTop r} ↔ x ∈ ⋂ a, s a
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : P) in atTop, r a ∈ s) ↔ ∀ b ∈ 𝓝 x, ∀ (a : P), ∃ x, x ∈ b ∩ r '' Ici a | 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ ∀ (a : Set X), (a ∈ 𝓝 x → ∃ᶠ (a_2 : P) in atTop, r a_2 ∈ a) ↔ a ∈ 𝓝 x → ∀ (a_1 : P), ∃ x, x ∈ a ∩ r '' Ici a_1 | 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✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : P) in atTop, r a ∈ s) ↔ ∀ b ∈ 𝓝 x, ∀ (a : P), ∃ x, x ∈ b ∩ r '' Ici a
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ ∀ (a : Set X), (a ∈ 𝓝 x → ∃ᶠ (a_2 : P) in atTop, r a_2 ∈ a) ↔ a ∈ 𝓝 x → ∀ (a_1 : P), ∃ x, x ∈ a ∩ r '' Ici a_1 | 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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
t : Set X
⊢ (t ∈ 𝓝 x → ∃ᶠ (a : P) in atTop, r a ∈ t) ↔ t ∈ 𝓝 x → ∀ (a : P), ∃ x, x ∈ t ∩ r '' Ici a | 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
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
⊢ ∀ (a : Set X), (a ∈ 𝓝 x → ∃ᶠ (a_2 : P) in atTop, r a_2 ∈ a) ↔ a ∈ 𝓝 x → ∀ (a_1 : P), ∃ x, x ∈ a ∩ r '' Ici a_1
TACTIC:
|
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), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
t : Set X
⊢ (t ∈ 𝓝 x → ∃ᶠ (a : P) in atTop, r a ∈ t) ↔ t ∈ 𝓝 x → ∀ (a : P), ∃ x, x ∈ t ∩ r '' Ici a | 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
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconnected (Ici a)
r : P → X
rc : Continuous r
s : P → Set X
hs : (fun a => closure (r '' Ici a)) = s
m : Antitone s
d : Directed Superset s
p : ∀ (a : P), IsPreconnected (s a)
c : ∀ (a : P), IsCompact (s a)
x : X
t : Set X
⊢ (t ∈ 𝓝 x → ∃ᶠ (a : P) in atTop, r a ∈ t) ↔ t ∈ 𝓝 x → ∀ (a : P), ∃ x, x ∈ t ∩ r '' Ici a
TACTIC:
|
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), IsPreconnected (Iic a)
r : P → X
rc : Continuous r
⊢ IsPreconnected {x | MapClusterPt x atBot 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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot r'} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Iic a)
r : P → X
rc : Continuous r
⊢ IsPreconnected {x | MapClusterPt x atBot r}
TACTIC:
|
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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot 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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot r'} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot r'}
TACTIC:
|
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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot 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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
p' : ∀ (a : Pᵒᵈ), IsPreconnected (Ici a)
⊢ IsPreconnected {x | MapClusterPt x atBot r'} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
⊢ IsPreconnected {x | MapClusterPt x atBot r'}
TACTIC:
|
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), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
p' : ∀ (a : Pᵒᵈ), IsPreconnected (Ici a)
⊢ IsPreconnected {x | MapClusterPt x atBot r'} | 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 : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnected (Iic a)
r : P → X
r' : Pᵒᵈ → X := r
rc : Continuous r'
rc' : Continuous r'
p' : ∀ (a : Pᵒᵈ), IsPreconnected (Ici a)
⊢ IsPreconnected {x | MapClusterPt x atBot r'}
TACTIC:
|
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[Ioc a b] a) r}
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[Ioc a b] a) 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)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 (𝓝[Ioc a b] a) r} | 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[Ioc a b] a) r} | 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 (Ioc a b)
ab : ¬¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
e : {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
e : {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
⊢ IsPreconnected (⋂ t, s t) | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
e : {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
e : {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
⊢ IsPreconnected (⋂ t, s t) | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
e : {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
⊢ IsPreconnected (⋂ t, s t)
TACTIC:
|
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[Ioc a b] a) r} | 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 (Ioc a b)
ab : ¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r}
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
⊢ Monotone s | 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 '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ s a ≤ s b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
⊢ Monotone s
TACTIC:
|
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 '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ s a ≤ s 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✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ (fun t => closure (r '' Ioc a✝ ↑t)) a ≤ (fun t => closure (r '' Ioc a✝ ↑t)) b | 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✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ s a ≤ s b
TACTIC:
|
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 => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ (fun t => closure (r '' Ioc a✝ ↑t)) a ≤ (fun t => closure (r '' Ioc a✝ ↑t)) 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✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ Ioc a✝ ↑a ≤ Ioc a✝ ↑b | 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✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ (fun t => closure (r '' Ioc a✝ ↑t)) a ≤ (fun t => closure (r '' Ioc a✝ ↑t)) b
TACTIC:
|
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 => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ Ioc a✝ ↑a ≤ Ioc a✝ ↑b | 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✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => closure (r '' Ioc a✝ ↑t)) = s
n : Nonempty ↑(Ioc a✝ b✝)
a b : ↑(Ioc a✝ b✝)
ab : a ≤ b
⊢ Ioc a✝ ↑a ≤ Ioc a✝ ↑b
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
⊢ ∀ (t : ↑(Ioc a b)), IsPreconnected (s 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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected (s ⟨t, m⟩) | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
⊢ ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected (s ⟨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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected ((fun t => closure (r '' Ioc a ↑t)) ⟨t, m⟩) | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected (s ⟨t, m⟩)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected ((fun t => closure (r '' Ioc a ↑t)) ⟨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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ Ioc a ↑⟨t, m⟩ ⊆ Ioc a b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ IsPreconnected ((fun t => closure (r '' Ioc a ↑t)) ⟨t, m⟩)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ Ioc a ↑⟨t, m⟩ ⊆ Ioc 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝¹ : Monotone s
d : Directed Superset s
t : ℝ
m✝ : t ∈ Ioc a b
m : a < t ∧ t ≤ b
⊢ Ioc a ↑⟨t, m✝⟩ ⊆ Ioc a b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
t : ℝ
m : t ∈ Ioc a b
⊢ Ioc a ↑⟨t, m⟩ ⊆ Ioc a b
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝¹ : Monotone s
d : Directed Superset s
t : ℝ
m✝ : t ∈ Ioc a b
m : a < t ∧ t ≤ b
⊢ Ioc a ↑⟨t, m✝⟩ ⊆ Ioc a b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝¹ : Monotone s
d : Directed Superset s
t : ℝ
m✝ : t ∈ Ioc a b
m : a < t ∧ t ≤ b
⊢ Ioc a ↑⟨t, m✝⟩ ⊆ Ioc a b
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
⊢ ∀ (t : ↑(Ioc a b)), IsCompact (s 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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact (s t) | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
⊢ ∀ (t : ↑(Ioc a b)), IsCompact (s t)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact (s 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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact ((fun t => closure (r '' Ioc a ↑t)) t) | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact (s t)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact ((fun t => closure (r '' Ioc a ↑t)) t) | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
t : ↑(Ioc a b)
⊢ IsCompact ((fun t => closure (r '' Ioc a ↑t)) t)
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ ∀ (x : X), x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ {x | MapClusterPt x (𝓝[Ioc a b] a) r} = ⋂ t, s t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ ∀ (x : X), x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
⊢ ∀ (x : X), x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ s) ↔
∀ (b_1 : Set X) (a_1 : ↑(Ioc a b)), b_1 ∈ 𝓝 x → ∃ x, x ∈ b_1 ∩ r '' Ioc a ↑a_1 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ x ∈ {x | MapClusterPt x (𝓝[Ioc a b] a) r} ↔ x ∈ ⋂ t, s t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ s) ↔
∀ (b_1 : Set X) (a_1 : ↑(Ioc a b)), b_1 ∈ 𝓝 x → ∃ x, x ∈ b_1 ∩ r '' Ioc a ↑a_1 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ ∀ (a_1 : Set X),
(a_1 ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ a_1) ↔
∀ (a_2 : ↑(Ioc a b)), a_1 ∈ 𝓝 x → ∃ x, x ∈ a_1 ∩ r '' Ioc a ↑a_2 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ (∀ s ∈ 𝓝 x, ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ s) ↔
∀ (b_1 : Set X) (a_1 : ↑(Ioc a b)), b_1 ∈ 𝓝 x → ∃ x, x ∈ b_1 ∩ r '' Ioc a ↑a_1
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ ∀ (a_1 : Set X),
(a_1 ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ a_1) ↔
∀ (a_2 : ↑(Ioc a b)), a_1 ∈ 𝓝 x → ∃ x, x ∈ a_1 ∩ r '' Ioc a ↑a_2 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), u ∈ 𝓝 x → ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
⊢ ∀ (a_1 : Set X),
(a_1 ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ a_1) ↔
∀ (a_2 : ↑(Ioc a b)), a_1 ∈ 𝓝 x → ∃ x, x ∈ a_1 ∩ r '' Ioc a ↑a_2
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), u ∈ 𝓝 x → ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ u ∈ 𝓝 x → ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), u ∈ 𝓝 x → ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ u ∈ 𝓝 x → ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ u ∈ 𝓝 x → ((∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_2 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_2) | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ (u ∈ 𝓝 x → ∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ u ∈ 𝓝 x → ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ u ∈ 𝓝 x → ((∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_2 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_2) | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
⊢ u ∈ 𝓝 x → ((∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_2 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_2)
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1 | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∃ᶠ (a : ℝ) in 𝓝[Ioc a b] a, r a ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x, x ∈ u ∩ r '' Ioc a ↑a_1
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) → ∀ (a_2 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_2
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1) → ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) ↔ ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) → ∀ (a_2 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_2 | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩ | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u) → ∀ (a_2 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_2
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩ | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩ | 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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
v : ℝ
vm : v ∈ Ioc a t
vu : r v ∈ u
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩ | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩
TACTIC:
|
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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
v : ℝ
vm : v ∈ Ioc a t
vu : r v ∈ u
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩ | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
tm : Ioc a t ∈ 𝓝[>] a
v : ℝ
vm : v ∈ Ioc a t
vu : r v ∈ u
⊢ ∃ x ∈ u, x ∈ r '' Ioc a ↑⟨t, m⟩
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ Ioc a t ∈ 𝓝[>] a | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ a ∈ Ico a t | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ Ioc a t ∈ 𝓝[>] a
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ a ∈ Ico a t | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ∈ Ico a t | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : t ∈ Ioc a b
⊢ a ∈ Ico a t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ∈ Ico a t | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ≤ a ∧ a < t | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ∈ Ico a t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ≤ a ∧ a < t | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
t : ℝ
m : a < t ∧ t ≤ b
⊢ a ≤ a ∧ a < t
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1) → ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
⊢ ∃ x ∈ v, r x ∈ u | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
⊢ (∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1) → ∀ {U : Set ℝ}, U ∈ 𝓝[>] a → ∃ x ∈ U, r x ∈ u
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
⊢ ∃ x ∈ v, r x ∈ u | 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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wa : w ∈ Ioi a
wv : Ioc a w ⊆ v
⊢ ∃ x ∈ v, r x ∈ u | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wa : w ∈ Ioi a
wv : Ioc a w ⊆ v
⊢ ∃ x ∈ v, r x ∈ u | 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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ ∃ x ∈ v, r x ∈ u | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wa : w ∈ Ioi a
wv : Ioc a w ⊆ v
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ ∃ x ∈ v, r x ∈ u | 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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
⊢ ∃ x ∈ v, r x ∈ u | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
⊢ ∃ x ∈ v, r x ∈ u | 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 b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : x ∈ r '' Ioc a ↑⟨min w b, m⟩
⊢ ∃ x ∈ v, r x ∈ u | 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 : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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 b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : x ∈ r '' Ioc a ↑⟨min w b, m⟩
⊢ ∃ x ∈ v, r x ∈ u | 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 b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : ∃ x_1, (a < x_1 ∧ x_1 ≤ w ∧ x_1 ≤ b) ∧ r x_1 = x
⊢ ∃ x ∈ v, r x ∈ u | 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
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : x ∈ r '' Ioc a ↑⟨min w b, m⟩
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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 b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : ∃ x_1, (a < x_1 ∧ x_1 ≤ w ∧ x_1 ≤ b) ∧ r x_1 = x
⊢ ∃ x ∈ v, r x ∈ u | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ ∃ x ∈ v, r x ∈ u | 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
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
rx : ∃ x_1, (a < x_1 ∧ x_1 ≤ w ∧ x_1 ≤ b) ∧ r x_1 = x
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ ∃ x ∈ v, r x ∈ u | 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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ r c ∈ u | 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
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ ∃ x ∈ v, r x ∈ u
TACTIC:
|
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 => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ r c ∈ u | 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 (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m✝ : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c✝ : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x✝ : X
u : Set X
a✝ : u ∈ 𝓝 x✝
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
m : min w b ∈ Ioc a b
x : X
xu : x ∈ u
c : ℝ
cx : r c = x
ac : a < c
cw : c ≤ w
right✝ : c ≤ b
⊢ r c ∈ u
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ min w b ∈ Ioc 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ a < min w b ∧ min w b ≤ b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ min w b ∈ Ioc a b
TACTIC:
|
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 '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ a < min w b ∧ min w b ≤ b | 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)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '' Ioc a ↑t)) = s
n : Nonempty ↑(Ioc a b)
m : Monotone s
d : Directed Superset s
p : ∀ (t : ↑(Ioc a b)), IsPreconnected (s t)
c : ∀ (t : ↑(Ioc a b)), IsCompact (s t)
x : X
u : Set X
a✝ : u ∈ 𝓝 x
h : ∀ (a_1 : ↑(Ioc a b)), ∃ x ∈ u, x ∈ r '' Ioc a ↑a_1
v : Set ℝ
vm : v ∈ 𝓝[>] a
w : ℝ
wv : Ioc a w ⊆ v
wa : a < w
⊢ a < min w b ∧ min w b ≤ b
TACTIC:
|
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 ⊆ interior t | 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 ∩ closure t ⊆ t
⊢ s ⊆ interior t
TACTIC:
|
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 ⊆ 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
uo : IsOpen u
⊢ s ⊆ interior t | 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ s ⊆ interior t
TACTIC:
|
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 : IsOpen u
⊢ 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
uo : IsOpen u
uc : IsClosed u
⊢ s ⊆ interior t | 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOpen u
⊢ s ⊆ interior t
TACTIC:
|
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 : IsOpen u
uc : IsClosed u
⊢ 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
uo : IsOpen u
uc : IsClosed u
p : IsPreconnected univ
⊢ s ⊆ interior t | 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOpen u
uc : IsClosed u
⊢ s ⊆ interior t
TACTIC:
|
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 : IsOpen u
uc : IsClosed u
p : IsPreconnected univ
⊢ s ⊆ interior t | 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
uo : IsOpen u
uc : IsClosed u
p : IsPreconnected univ
h : Disjoint univ u
⊢ s ⊆ interior t
case inr
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 : IsOpen u
uc : IsClosed u
p : IsPreconnected univ
h : univ ⊆ u
⊢ s ⊆ interior t | 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOpen u
uc : IsClosed u
p : IsPreconnected univ
⊢ s ⊆ interior t
TACTIC:
|
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 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
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ interior 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ IsOpen u
TACTIC:
|
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 ⊆ interior 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
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : ⟨x, m⟩ ∈ u
⊢ ⟨x, m⟩ ∈ interior 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ interior u
TACTIC:
|
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 ∈ s
h : ⟨x, m⟩ ∈ u
⊢ ⟨x, m⟩ ∈ interior 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
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : x ∈ t
⊢ ⟨x, m⟩ ∈ interior 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : ⟨x, m⟩ ∈ u
⊢ ⟨x, m⟩ ∈ interior u
TACTIC:
|
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 ∈ s
h : x ∈ t
⊢ ⟨x, m⟩ ∈ interior 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
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : x ∈ t
n : x ∈ interior t
⊢ ⟨x, m⟩ ∈ interior 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : x ∈ t
⊢ ⟨x, m⟩ ∈ interior u
TACTIC:
|
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 ∈ s
h : x ∈ t
n : x ∈ interior t
⊢ ⟨x, m⟩ ∈ interior 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
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : x ∈ t
n : t ∈ 𝓝 x
⊢ t ∈ 𝓝[s] x | 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 ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m : x ∈ s
h : x ∈ t
n : x ∈ interior t
⊢ ⟨x, m⟩ ∈ interior u
TACTIC:
|
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