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... |
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 : ... | 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), IsOp... |
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 : ... | 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) →
∀... | 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 ... |
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) →
∀... | 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 : S... | 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 → I... |
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 : S... | 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 : S... | 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 → ... |
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 : S... | 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 : S... | 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 → ... |
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 : S... | 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 : S... | 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 → ... |
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 : S... | 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 : S... | 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 → ... |
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 : S... | 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... | 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 → ... |
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 ... | 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 ... | 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... |
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 ... | 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... |
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... | 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... | 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 ... |
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... | 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), IsClose... |
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... | 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... | 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 ... |
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... | 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), IsClose... |
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... | 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 ... | 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 → Is... |
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 ... | 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 → Dis... | 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 → I... |
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 → Dis... | 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 ... | 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... |
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 ... | 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 ∪ ... | 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 : IsCl... |
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 ∪ ... | 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 ∪ ... | 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 : Is... |
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 ∪ ... | 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 ⊆ ... | 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 : Is... |
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 ⊆ ... | 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 ⊆ ... | 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 ... |
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 ⊆ ... | 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 ⊆ ... | 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 ... |
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 ⊆ ... | 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 ... |
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 ... | 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 ∪ ... | 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 : IsCl... |
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 ∪ ... | 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 ∪ ... | 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 : Is... |
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 ∪ ... | 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 ⊆ ... | 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 : Is... |
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 ⊆ ... | 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 ⊆ ... | 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 ... |
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 ⊆ ... | 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 ⊆ ... | 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 ... |
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 ⊆ ... | 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 ... |
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 ... | 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)... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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)... | 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)... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [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)... | 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [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 : 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 :
... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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 :
... | 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), IsCom... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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
appl... | 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), IsCom... | 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), IsCom... | 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
i... |
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), IsCom... | 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),... | 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
i... |
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),... | 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... | 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 : Nonemp... |
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... | 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... | 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 Supe... |
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... | 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... | 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 Supe... |
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... | 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,... | 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 Supe... |
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,... | 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 (⋂ ... | 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 ... |
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 (⋂ ... | 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,... | 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 : Directe... |
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,... | 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 (⋂ ... | 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 ... |
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 (⋂ ... | 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 : Directe... |
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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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 :... | 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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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 :... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | 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... | 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... | 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 Su... |
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... | 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 : IsC... | 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 Su... |
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 : IsC... | 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... | 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 :... |
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... | 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... | 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 Su... |
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... | case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsCl... | 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 Su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact ⟨x, mem_diff_of_mem xa xuv⟩ | case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsCl... | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | Please generate a tactic in lean4 to solve the state.
STATE:
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact (c a).diff (uo.union vo) | case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s... | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Su... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed S... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact ((c a).diff (uo.union vo)).isClosed | case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed S... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases n with ⟨x, n⟩ | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v :... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [mem_iInter, mem_diff, forall_and, forall_const] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rw [← mem_iInter] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [suv n.1, not_true, imp_false] at n | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact n.2 | case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | generalize hs : (fun a ↦ closure (r '' Ici a)) = s | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have m : Antitone s := by
intro a b ab; rw [← hs]; exact closure_mono (monotone_image (Ici_subset_Ici.mpr ab)) | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have d : Directed Superset s := by
intro a b; exact ⟨a ⊔ b, m le_sup_left, m le_sup_right⟩ | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have p : ∀ a, IsPreconnected (s a) := by
intro a; rw [← hs]; exact ((p _).image _ rc.continuousOn).closure | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have c : ∀ a, IsCompact (s a) := by
intro a; rw [← hs]; exact isClosed_closure.isCompact | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | have e : {x | MapClusterPt x atTop r} = ⋂ a, s a := by
ext x
simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm P, ← hs]
apply forall_congr'; intro t
simp only [@forall_comm P, mem_inter_iff, mem_image, mem_Ici, @and_comm (_ ∈ t),
exists_exists_and_eq_an... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [e] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact IsPreconnected.directed_iInter d p c | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a b ab | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact closure_mono (monotone_image (Ici_subset_Ici.mpr ab)) | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a b | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact ⟨a ⊔ b, m le_sup_left, m le_sup_right⟩ | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact ((p _).image _ rc.continuousOn).closure | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | rw [← hs] | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | exact isClosed_closure.isCompact | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | ext x | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsPreconne... | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm P, ← hs] | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | apply forall_congr' | case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), IsP... | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | intro t | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atTop | [95, 1] | [114, 53] | simp only [@forall_comm P, mem_inter_iff, mem_image, mem_Ici, @and_comm (_ ∈ t),
exists_exists_and_eq_and, Filter.frequently_atTop, exists_prop] | case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p✝ : ∀ (a : P), I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeSup P
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | set r' : Pᵒᵈ → X := r | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | have rc' : Continuous r' := rc | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | have p' : ∀ a : Pᵒᵈ, IsPreconnected (Ici a) := fun a ↦ p a | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_atBot | [118, 1] | [124, 43] | exact IsPreconnected.limits_atTop p' rc' | X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : TopologicalSpace P
inst✝ : Nonempty P
p : ∀ (a : P), IsPreconnec... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁹ : TopologicalSpace X
I : Type
inst✝⁸ : TopologicalSpace I
inst✝⁷ : ConditionallyCompleteLinearOrder I
inst✝⁶ : DenselyOrdered I
inst✝⁵ : OrderTopology I
inst✝⁴ : CompactSpace X
inst✝³ : T4Space X
P : Type
inst✝² : SemilatticeInf P
inst✝¹ : Top... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | by_cases ab : ¬a < b | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
⊢ IsPreconnected {x | MapClusterPt x (𝓝[Ioc a b] a) r} | case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [not_not] at ab | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬¬a < b
⊢ IsPreconnected {x | MapClusterPt x (�... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | generalize hs : (fun t : Ioc a b ↦ closure (r '' Ioc a t)) = s | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝[... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have n : Nonempty (Ioc a b) := ⟨b, right_mem_Ioc.mpr ab⟩ | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have m : Monotone s := by
intro a b ab; rw [← hs]; refine closure_mono (monotone_image ?_)
exact Ioc_subset_Ioc (le_refl _) (Subtype.coe_le_coe.mpr ab) | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have d : Directed Superset s := fun a b ↦ ⟨min a b, m (min_le_left _ _), m (min_le_right _ _)⟩ | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have p : ∀ t, IsPreconnected (s t) := by
intro ⟨t, m⟩; rw [← hs]; refine (isPreconnected_Ioc.image _ (rc.mono ?_)).closure
simp only [mem_Ioc] at m
simp only [Subtype.coe_mk, Ioc_subset_Ioc_iff m.1, m.2, le_refl, true_and_iff] | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have c : ∀ t, IsCompact (s t) := by intro t; rw [← hs]; exact isClosed_closure.isCompact | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
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