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 | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have lo : ∀ a, a ≤ t → f (p.extend a) ∈ s := by
intro a h; contrapose h; simp only [not_le]
replace h : ∀ᶠ b in 𝓝 a, f (p.extend b) ∉ s :=
(fc.comp p.continuous_extend).continuousAt.eventually_mem (sc.isOpen_compl.mem_nhds h)
simp only [← hu, mem_inter_iff, mem_iInter₂, mem_Iic] at tu ⊢
rcases ((frequently... | 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] | by_cases t1 : t = 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 pos
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 ... | 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] | replace t1 : t < 1 := Ne.lt_of_le t1 m.2 | case neg
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine ((pc.joinedIn _ ft b fb).mono (preimage_mono sc.frontier_subset)).symm.trans
(JoinedIn.symm ?_) | case neg
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | generalize hq : (fun a : unitInterval ↦ p.extend (min a t)) = q | case neg
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have qc : Continuous q := by
rw [← hq]; exact p.continuous_extend.comp (continuous_subtype_val.min continuous_const) | case neg
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine ⟨⟨⟨q,qc⟩,?_,?_⟩,?_⟩ | case neg
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 ... | case neg.refine_1
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 : IsPathConn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [← hu, bddAbove_def] | 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] | use 1 | 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 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:
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] | intro t ⟨m, _⟩ | 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] | exact m.2 | 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... | no goals | 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] | rw [← hu] | 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] | use 0, left_mem_Icc.mpr zero_le_one | 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 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:
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] | simp only [mem_iInter₂, mem_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] | intro a m | 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] | contrapose m | 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] | 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 (... | no goals | 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] | rw [← hu] | 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] | apply isClosed_Icc.inter | 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] | apply isClosed_iInter | 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 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:
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] | intro a | 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] | apply isClosed_iInter | 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.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 ... | 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 _ | case h.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 ... | case h.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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact isClosed_Iic | case h.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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [← uc.closure_eq, ← ht] | 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] | exact csSup_mem_closure un bdd | 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... | no goals | 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] | rw [← hu] at tu | 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] | exact tu.1 | 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... | no goals | 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] | intro a 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' 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] | contrapose 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' 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] | simp only [not_le] | 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] | replace h : ∀ᶠ b in 𝓝 a, f (p.extend b) ∉ s :=
(fc.comp p.continuous_extend).continuousAt.eventually_mem (sc.isOpen_compl.mem_nhds 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' 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] | simp only [← hu, mem_inter_iff, mem_iInter₂, mem_Iic] at tu ⊢ | 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 ((frequently_lt_nhds a).and_eventually h).exists with ⟨c, ca, cs⟩ | 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] | exact lt_of_le_of_lt (tu.2 c cs) ca | 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... | no goals | 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] | use p.symm | case pos
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 ... | 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 pos
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro a | 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] | simp only [p.symm_apply, Function.comp, mem_preimage] | 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] | rw [← Path.extend_extends'] | 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] | apply lo | 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.a
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 ... | 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] | rw [t1] | case h.a
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 ... | case h.a
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | unit_interval | case h.a
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [frontier, mem_diff, sc.closure_eq] | 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] | constructor | 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 left
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... | 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] | convert lo t (le_refl _) | case left
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... | case h.e'_4.h.e'_1
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 : IsPathCon... | Please generate a tactic in lean4 to solve the state.
STATE:
case left
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✝ : Pat... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [ge_iff_le, zero_le_one, not_true, gt_iff_lt, mem_Icc, Path.extend_extends _ m] | case h.e'_4.h.e'_1
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 : IsPathCon... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_1
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
in... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have e : p ⟨t, m⟩ = p.extend t := by
simp only [Path.extend, IccExtend_apply, min_eq_right m.2, max_eq_right m.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [e] | 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] | clear e | 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] | simp only [← @mem_preimage _ _ (f.comp p.extend), ← ht] | 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] | by_contra h | 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] | rw [ht] at h | 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 o : IsOpen (f ∘ p.extend ⁻¹' interior s) :=
isOpen_interior.preimage (fc.comp p.continuous_extend) | 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] | rcases (nhds_basis_Ioo t).mem_iff.mp (o.mem_nhds h) with ⟨⟨x, y⟩, ⟨xt, ty⟩, h⟩ | 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.intro.mk.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
p... | 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] | simp only [subset_def, mem_Ioo, and_imp, mem_preimage, Function.comp] at xt ty h | case right.intro.mk.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
p... | case right.intro.mk.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
p... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.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 : Topologi... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases exists_between (lt_min ty t1) with ⟨z, tz, zy1⟩ | case right.intro.mk.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
p... | case right.intro.mk.intro.intro.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... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.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 : Topologi... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases lt_min_iff.mp zy1 with ⟨zy, z1⟩ | case right.intro.mk.intro.intro.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... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.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✝... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | suffices h : z ∈ u by linarith [le_csSup bdd h] | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rw [← hu] | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine ⟨⟨_root_.trans m.1 tz.le, z1.le⟩, ?_⟩ | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_iInter₂, mem_Iic] | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro w ws | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | contrapose ws | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [not_not, not_le] at ws ⊢ | case right.intro.mk.intro.intro.intro.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 :... | case right.intro.mk.intro.intro.intro.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 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | by_cases xw : x < w | case right.intro.mk.intro.intro.intro.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 :... | case pos
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.intro.mk.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | refine interior_subset (h _ xw (_root_.trans ws zy)) | case pos
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [not_lt] at xw | case neg
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 ... | case neg
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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact lo _ (_root_.trans xw xt.le) | case neg
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 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : Path... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Path.extend, IccExtend_apply, min_eq_right m.2, max_eq_right m.1] | 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... | no goals | 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] | linarith [le_csSup bdd 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' front... | no goals | 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] | rw [← hq] | 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] | exact p.continuous_extend.comp (continuous_subtype_val.min continuous_const) | 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... | no goals | 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] | simp only [← hq] | case neg.refine_1
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 : IsPathConn... | case neg.refine_1
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 : IsPathConn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Icc.coe_zero, min_eq_left m.1, p.extend_zero] | case neg.refine_1
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 : IsPathConn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [← hq] | case neg.refine_2
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 : IsPathConn... | case neg.refine_2
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 : IsPathConn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [Icc.coe_one, min_eq_right m.2, Path.extend, IccExtend_apply, max_eq_right m.1] | case neg.refine_2
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 : IsPathConn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro ⟨a, n⟩ | case neg.refine_3
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 : IsPathConn... | case neg.refine_3
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 : IsPathConn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_preimage, Path.coe_mk_mk, ← hq, Subtype.coe_mk] | case neg.refine_3
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 : IsPathConn... | case neg.refine_3
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 : IsPathConn... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | exact lo _ (min_le_right _ _) | case neg.refine_3
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 : IsPathConn... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_3
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
ins... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_iff_differentiableOn | [31, 1] | [36, 56] | constructor | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f s
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_iff_differentiableOn | [31, 1] | [36, 56] | exact AnalyticOn.differentiableOn | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ AnalyticOn ℂ f s → DifferentiableOn ℂ f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_iff_differentiableOn | [31, 1] | [36, 56] | intro d z zs | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ DifferentiableOn ℂ f s → AnalyticOn ℂ f s | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
d : DifferentiableOn ℂ f s
z : ℂ
zs : z ∈ s
⊢ AnalyticAt ℂ f z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
⊢ DifferentiableOn ℂ f s → AnalyticOn ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_iff_differentiableOn | [31, 1] | [36, 56] | exact DifferentiableOn.analyticAt d (o.mem_nhds zs) | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
d : DifferentiableOn ℂ f s
z : ℂ
zs : z ∈ s
⊢ AnalyticAt ℂ f z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
s : Set ℂ
o : IsOpen s
d : DifferentiableOn ℂ f s
z : ℂ
zs : z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_univ_iff_differentiable | [39, 1] | [42, 52] | simp only [← differentiableOn_univ] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
⊢ AnalyticOn ℂ f univ ↔ Differentiable ℂ f | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
⊢ AnalyticOn ℂ f univ ↔ DifferentiableOn ℂ f univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
⊢ AnalyticOn ℂ f univ ↔ Differentiable ℂ f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticOn_univ_iff_differentiable | [39, 1] | [42, 52] | exact analyticOn_iff_differentiableOn isOpen_univ | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
⊢ AnalyticOn ℂ f univ ↔ DifferentiableOn ℂ f univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
⊢ AnalyticOn ℂ f univ ↔ DifferentiableOn ℂ f univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | constructor | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ AnalyticAt ℂ f c ↔ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ AnalyticAt ℂ f c → ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ AnalyticAt ℂ f c ↔ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | intro fa | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ AnalyticAt ℂ f c → ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa : AnalyticAt ℂ f c
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ AnalyticAt ℂ f c → ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | rcases fa.exists_ball_analyticOn with ⟨r, rp, fa⟩ | case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa : AnalyticAt ℂ f c
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z | case mp.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, Differentiable... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa : AnalyticAt ℂ f c
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, Differentiable... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | exact fa.differentiableOn.eventually_differentiableAt (Metric.ball_mem_nhds _ rp) | case mp.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, Differentiable... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : A... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | intro d | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ (∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z) → AnalyticAt ℂ f c | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
⊢ AnalyticAt ℂ f c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
⊢ (∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z) → Analytic... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | rcases Metric.eventually_nhds_iff.mp d with ⟨r, rp, d⟩ | case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
⊢ AnalyticAt ℂ f c | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
⊢ Analytic... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | have dr : DifferentiableOn ℂ f (ball c r) := by
intro z zs; simp only [Metric.mem_ball] at zs; exact (d zs).differentiableWithinAt | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | rw [← analyticOn_iff_differentiableOn isOpen_ball] at dr | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | exact dr _ (Metric.mem_ball_self rp) | case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → Differentia... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | intro z zs | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → DifferentiableAt ℂ f y
⊢ Differe... | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → DifferentiableAt ℂ f y
z : ℂ
zs ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | simp only [Metric.mem_ball] at zs | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → DifferentiableAt ℂ f y
z : ℂ
zs ... | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → DifferentiableAt ℂ f y
z : ℂ
zs ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_iff_eventually_differentiableAt | [45, 1] | [54, 41] | exact (d zs).differentiableWithinAt | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d : ∀ ⦃y : ℂ⦄, dist y c < r → DifferentiableAt ℂ f y
z : ℂ
zs ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → E
c : ℂ
d✝ : ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ f z
r : ℝ
rp : r > 0
d... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticOn.exp | [57, 1] | [58, 76] | rw [analyticOn_univ_iff_differentiable] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
⊢ AnalyticOn ℂ Complex.exp univ | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
⊢ Differentiable ℂ Complex.exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
⊢ AnalyticOn ℂ Complex.exp univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | AnalyticOn.exp | [57, 1] | [58, 76] | exact Complex.differentiable_exp | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
⊢ Differentiable ℂ Complex.exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
⊢ Differentiable ℂ Complex.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/HolomorphicUpstream.lean | analyticAt_log | [65, 1] | [69, 35] | rw [analyticAt_iff_eventually_differentiableAt] | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ AnalyticAt ℂ log c | E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ ∀ᶠ (z : ℂ) in 𝓝 c, DifferentiableAt ℂ log z | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
inst✝³ : CompleteSpace E
F : Type
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
c : ℂ
m : c ∈ Complex.slitPlane
⊢ AnalyticAt ℂ log c
TACTIC:
|
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