url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [e] | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact IsPreconnected.directed_iInter d p c | case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Ioc_eq_empty ab, nhdsWithin_empty, MapClusterPt, Filter.map_bot, ClusterPt.bot,
setOf_false, isPreconnected_empty] | case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : ¬a < b
⊢ IsPreconnected {x | MapClusterPt x (𝓝... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro a b ab | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | refine closure_mono (monotone_image ?_) | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact Ioc_subset_Ioc (le_refl _) (Subtype.coe_le_coe.mpr ab) | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ b✝)
ab✝ : a✝ < b✝
s : ↑(Ioc a✝ b✝) → Set X
hs : (fun t => clo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a✝ b✝ : ℝ
rc : ContinuousOn r (Ioc a✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro ⟨t, m⟩ | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | refine (isPreconnected_Ioc.image _ (rc.mono ?_)).closure | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] at m | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Subtype.coe_mk, Ioc_subset_Ioc_iff m.1, m.2, le_refl, true_and_iff] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro t | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rw [← hs] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact isClosed_closure.isCompact | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply Set.ext | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro x | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_setOf, mem_iInter, mapClusterPt_iff, mem_closure_iff_nhds, Set.Nonempty,
@forall_comm _ (Set X), ← hs] | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply forall_congr' | case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro u | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [@forall_comm _ (u ∈ 𝓝 x)] | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply forall_congr' | case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => clo... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro _ | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_inter_iff, Filter.frequently_iff, nhdsWithin_Ioc_eq_nhdsWithin_Ioi ab] | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | constructor | case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro h ⟨t, m⟩ | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have tm : Ioc a t ∈ 𝓝[Ioi a] a := by
apply Ioc_mem_nhdsWithin_Ioi
simp only [mem_Ioc] at m; simp only [mem_Ico]; use le_refl _, m.1 | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases h tm with ⟨v, vm, vu⟩ | case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t =... | case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
h... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | exact ⟨r v, vu, mem_image_of_mem _ vm⟩ | case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mp.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | apply Ioc_mem_nhdsWithin_Ioi | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] at m | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ico] | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use le_refl _, m.1 | case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closu... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Io... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | intro h v vm | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : Continuous... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases mem_nhdsWithin_Ioi_iff_exists_Ioc_subset.mp vm with ⟨w, wa, wv⟩ | case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t ... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : Continuous... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioi] at wa | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | have m : min w b ∈ Ioc a b := by simp only [mem_Ioc]; use lt_min wa ab, min_le_right _ _ | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases h ⟨_, m⟩ with ⟨x, xu, rx⟩ | case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
... | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [Subtype.coe_mk, mem_image, mem_Ioc, le_min_iff] at rx | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rcases rx with ⟨c, ⟨ac, cw, _⟩, cx⟩ | case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a... | case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use c, wv (mem_Ioc.mpr ⟨ac, cw⟩) | case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b... | case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.mpr.intro.intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | rwa [cx] | case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | simp only [mem_Ioc] | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.limits_Ioc | [129, 1] | [167, 53] | use lt_min wa ab, min_le_right _ _ | X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
ab : a < b
s : ↑(Ioc a b) → Set X
hs : (fun t => closure (r '... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
inst✝¹ : CompactSpace X
inst✝ : T4Space X
r : ℝ → X
a b : ℝ
rc : ContinuousOn r (Ioc a b)
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | generalize hu : (fun x : s ↦ (x : X)) ⁻¹' t = u | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
⊢ s ⊆ interior t | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ s ⊆ int... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have uo : IsOpen u := by
rw [← subset_interior_iff_isOpen]; intro ⟨x, m⟩ h
simp only [mem_preimage, Subtype.coe_mk, ← hu] at h
have n := op ⟨m, h⟩
simp only [mem_interior_iff_mem_nhds, preimage_coe_mem_nhds_subtype, Subtype.coe_mk,
← hu] at n ⊢
exact nhdsWithin_le_nhds n | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ s ⊆ int... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have uc : IsClosed u := by
rw [← closure_eq_iff_isClosed]; refine subset_antisymm ?_ subset_closure
rw [← hu]
refine _root_.trans (continuous_subtype_val.closure_preimage_subset _) ?_
intro ⟨x, m⟩ h; exact cl ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have p : IsPreconnected (univ : Set s) := (Subtype.preconnectedSpace sp).isPreconnected_univ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | cases' disjoint_or_subset_of_isClopen p ⟨uc, uo⟩ with h h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← subset_interior_iff_isOpen] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ IsOpen ... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ int... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | intro ⟨x, m⟩ h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
⊢ u ⊆ int... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [mem_preimage, Subtype.coe_mk, ← hu] at h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | have n := op ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [mem_interior_iff_mem_nhds, preimage_coe_mem_nhds_subtype, Subtype.coe_mk,
← hu] at n ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact nhdsWithin_le_nhds n | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← closure_eq_iff_isClosed] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | refine subset_antisymm ?_ subset_closure | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← hu] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | refine _root_.trans (continuous_subtype_val.closure_preimage_subset _) ?_ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | intro ⟨x, m⟩ h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact cl ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [univ_disjoint, preimage_eq_empty_iff, Subtype.range_coe, ← hu] at h | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exfalso | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact ne.not_disjoint h.symm | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← Subtype.coe_preimage_self, ← hu, preimage_subset_preimage_iff] at h | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact _root_.trans (subset_inter (subset_refl _) h) op | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [Subtype.range_coe, subset_refl] | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | have uc : IsPathConnected (univ : Set s) := by
convert sc.preimage_coe (subset_refl _); apply Set.ext; intro x
simp only [mem_univ, true_iff_iff, mem_preimage, Subtype.mem] | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ IsPathConnected ... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | have e : f '' s = s.restrict f '' univ := by
apply Set.ext; intro y; constructor
intro ⟨x, m, e⟩; use⟨x, m⟩, mem_univ _, e
intro ⟨⟨x, m⟩, _, e⟩; use x, m, e | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | rw [e] | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | exact uc.image (continuousOn_iff_continuous_restrict.mp fc) | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | convert sc.preimage_coe (subset_refl _) | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ IsPathConnected ... | case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ univ... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | apply Set.ext | case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ univ... | case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ ∀ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro x | case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ ∀ ... | case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | simp only [mem_univ, true_iff_iff, mem_preimage, Subtype.mem] | case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | apply Set.ext | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro y | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | constructor | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | 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
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro ⟨x, m, e⟩ | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | use⟨x, m⟩, mem_univ _, e | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro ⟨⟨x, m⟩, _, e⟩ | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | use x, m, e | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | rw [← abs_of_nonneg r0, ← image_circleMap_Ioc z r] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (sphere z r) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (sphere z r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | refine IsPathConnected.image ?_ (continuous_circleMap _ _) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π)) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | exact (convex_Ioc 0 (2 * π)).isPathConnected (nonempty_Ioc.mpr Real.two_pi_pos) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have pc' := pc | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases pc' with ⟨b, fb, j⟩ | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConne... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | use b | case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConne... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_preimage, mem_singleton_iff] at fb j ⊢ | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have bs : f b ∈ s := sc.frontier_subset fb | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | use bs | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro x fx | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have p := PathConnectedSpace.somePath x b | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | generalize hu : Icc (0 : ℝ) 1 ∩ ⋂ (a) (_ : f (p.extend a) ∉ s), Iic a = u | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have bdd : BddAbove u := by rw [← hu, bddAbove_def]; use 1; intro t ⟨m, _⟩; exact m.2 | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have un : u.Nonempty := by
rw [← hu]; use 0, left_mem_Icc.mpr zero_le_one; simp only [mem_iInter₂, mem_Iic]; intro a m
contrapose m; simp only [not_not, p.extend_of_le_zero (not_le.mp m).le, fx] | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have uc : IsClosed u := by
rw [← hu]; apply isClosed_Icc.inter; apply isClosed_iInter; intro a; apply isClosed_iInter
intro _; exact isClosed_Iic | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | generalize ht : sSup u = t | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have tu : t ∈ u := by rw [← uc.closure_eq, ← ht]; exact csSup_mem_closure un bdd | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have m : t ∈ Icc (0 : ℝ) 1 := by rw [← hu] at tu; exact tu.1 | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
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