url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | apply inter_subset_left | case 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 u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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 u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | rw [isPreconnected_iff_subset_of_fully_disjoint_closed sc] | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ IsPreconnected s ↔ ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ IsPreconnected s ↔ ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | constructor | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro h u v uo vo suv uv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have suc : IsClosed (s ∩ u) := isClosed_closed_inter sc vo uv suv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have svc : IsClosed (s ∩ v) := isClosed_closed_inter sc uo uv.symm ((union_comm u v).subst suv) | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have h0 : s ⊆ s ∩ u ∪ s ∩ v := by
simp only [←inter_union_distrib_left]; exact subset_inter (subset_refl _) suv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have h1 : Disjoint (s ∩ u) (s ∩ v) := Disjoint.inter_left' _ (Disjoint.inter_right' _ uv) | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' h (s ∩ u) (s ∩ v) suc svc h0 h1 with su sv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v
case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | simp only [←inter_union_distrib_left] | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ u ∪ s ∩ v | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ u ∪ s ∩ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact subset_inter (subset_refl _) suv | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | left | case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v | case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact (subset_inter_iff.mp su).2 | case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | right | case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact (subset_inter_iff.mp sv).2 | case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro h u v uc vc suv uv | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | rcases NormalSpace.normal u v uc vc uv with ⟨u', v', uo, vo, uu, vv, uv'⟩ | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' h u' v' uo vo (_root_.trans suv (union_subset_union uu vv)) uv' with h h | case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v
case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | left | case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro x m | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' (mem_union _ _ _).mp (suv m) with mu mv | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact mu | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exfalso | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact disjoint_left.mp uv' (h m) (vv mv) | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | right | case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro x m | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' (mem_union _ _ _).mp (suv m) with mu mv | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exfalso | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact disjoint_right.mp uv' (h m) (uu mu) | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact mv | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | contrapose p | 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
p : ∀ (a : I), IsPreconnected (s a)
c : ∀ (a : I), IsCompact (s a)
⊢ IsPreconnected (⋂ a, s a) | 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)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
p : ∀ (a : I), IsPreconnected (s a)
c : ∀ (a : I), IsCompact (s a)
⊢ IsPreconnected (⋂ a, s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | have ci : IsClosed (⋂ a, s a) := isClosed_iInter fun i ↦ (c i).isClosed | 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)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | 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)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [isPreconnected_iff_subset_of_fully_disjoint_open ci, not_forall] at p | 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)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [isPreconnected_iff_subset_of_fully_disjoint_open (c _).isClosed, not_forall] | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a) | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases p with ⟨u, v, uo, vo, suv, uv, no⟩ | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case 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✝
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)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | 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)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | have e : ∃ a, s a ⊆ u ∪ v := by
by_contra h; simp only [not_exists, Set.not_subset] at h
suffices n : (⋂ a, s a \ (u ∪ v)).Nonempty by
rcases n with ⟨x, n⟩; simp only [mem_iInter, mem_diff, forall_and, forall_const] at n
rw [← mem_iInter] at n; simp only [suv n.1, not_true, imp_false] at n; exact n.2
apply IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
intro a b; rcases d a b with ⟨c, ac, bc⟩
use c, diff_subset_diff_left ac, diff_subset_diff_left bc
intro a; rcases h a with ⟨x, xa, xuv⟩; exact ⟨x, mem_diff_of_mem xa xuv⟩
intro a; exact (c a).diff (uo.union vo)
intro a; exact ((c a).diff (uo.union vo)).isClosed | case 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✝
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)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case 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✝
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)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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✝
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)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases e with ⟨a, auv⟩ | case 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✝
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)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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✝
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)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | use a, u, v, uo, vo, auv, uv | case 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | contrapose no | 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v) | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v) | 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✝
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)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [not_not] at no ⊢ | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v) | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | cases' no with su sv | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inl
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | 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✝
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
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | left | case h.inl
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inl.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✝
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact _root_.trans (iInter_subset _ _) su | case h.inl.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✝
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl.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✝
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
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | right | case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inr.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✝
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact _root_.trans (iInter_subset _ _) sv | case h.inr.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✝
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.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✝
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
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | by_contra 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✝
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)
⊢ ∃ a, s a ⊆ u ∪ v | 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 : ¬∃ 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)
⊢ ∃ a, s a ⊆ u ∪ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [not_exists, Set.not_subset] at 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✝
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 : ¬∃ a, s a ⊆ u ∪ v
⊢ False | 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
⊢ 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 : ¬∃ 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] | suffices n : (⋂ a, s a \ (u ∪ v)).Nonempty by
rcases n with ⟨x, n⟩; simp only [mem_iInter, mem_diff, forall_and, forall_const] at n
rw [← mem_iInter] at n; simp only [suv n.1, not_true, imp_false] at n; exact n.2 | 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
⊢ False | 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, s a \ (u ∪ v)).Nonempty | 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
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | apply IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed | 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, s a \ (u ∪ v)).Nonempty | case htd
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
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
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), (s i \ (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)) | 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
⊢ (⋂ a, s a \ (u ∪ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a b | case htd
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
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
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), (s i \ (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 htd
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 b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd
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
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
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), (s i \ (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] | rcases d a b with ⟨c, ac, bc⟩ | case htd
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 b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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 htd.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 b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd
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 b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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] | use c, diff_subset_diff_left ac, diff_subset_diff_left bc | case htd.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 b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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 htn
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), (s i \ (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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd.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 b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
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), (s i \ (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 htn
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), (s i \ (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 htn
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
⊢ (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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htn
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), (s i \ (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] | rcases h a with ⟨x, xa, xuv⟩ | case htn
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
⊢ (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 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)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htn
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
⊢ (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] | 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:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.