url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact nhdsWithin_le_nhds n | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
x : X
m :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← closure_eq_iff_isClosed] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | refine subset_antisymm ?_ subset_closure | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← hu] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | refine _root_.trans (continuous_subtype_val.closure_preimage_subset _) ?_ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | intro ⟨x, m⟩ h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact cl ⟨m, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
uo : IsOp... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [univ_disjoint, preimage_eq_empty_iff, Subtype.range_coe, ← hu] at h | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exfalso | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact ne.not_disjoint h.symm | case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | rw [← Subtype.coe_preimage_self, ← hu, preimage_subset_preimage_iff] at h | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | exact _root_.trans (subset_inter (subset_refl _) h) op | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.relative_clopen | [170, 1] | [191, 47] | simp only [Subtype.range_coe, subset_refl] | case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl : s ∩ closure t ⊆ t
u : Set ↑s
hu : (fun x => ↑x) ⁻¹' t = u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s t : Set X
sp : IsPreconnected s
ne : (s ∩ t).Nonempty
op : s ∩ t ⊆ interior t
cl... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | have uc : IsPathConnected (univ : Set s) := by
convert sc.preimage_coe (subset_refl _); apply Set.ext; intro x
simp only [mem_univ, true_iff_iff, mem_preimage, Subtype.mem] | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ IsPathConnected ... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | have e : f '' s = s.restrict f '' univ := by
apply Set.ext; intro y; constructor
intro ⟨x, m, e⟩; use⟨x, m⟩, mem_univ _, e
intro ⟨⟨x, m⟩, _, e⟩; use x, m, e | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | rw [e] | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | exact uc.image (continuousOn_iff_continuous_restrict.mp fc) | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | convert sc.preimage_coe (subset_refl _) | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ IsPathConnected ... | case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ univ... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | apply Set.ext | case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ univ... | case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ ∀ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro x | case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
⊢ ∀ ... | case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
x : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | simp only [mem_univ, true_iff_iff, mem_preimage, Subtype.mem] | case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
x : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | apply Set.ext | X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPathConnect... | case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro y | case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | constructor | case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsPath... | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro ⟨x, m, e⟩ | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | use⟨x, m⟩, mem_univ _, e | case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : IsP... | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | intro ⟨⟨x, m⟩, _, e⟩ | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.image_of_continuousOn | [195, 1] | [205, 70] | use x, m, e | case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
sc : IsPathConnected s
f : X → Y
fc : ContinuousOn f s
uc : Is... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X✝ : Type
inst✝⁶ : TopologicalSpace X✝
I : Type
inst✝⁵ : TopologicalSpace I
inst✝⁴ : ConditionallyCompleteLinearOrder I
inst✝³ : DenselyOrdered I
inst✝² : OrderTopology I
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | rw [← abs_of_nonneg r0, ← image_circleMap_Ioc z r] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (sphere z r) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (sphere z r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | refine IsPathConnected.image ?_ (continuous_circleMap _ _) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π)) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (circleMap z r '' Ioc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPathConnected_sphere | [208, 1] | [211, 82] | exact (convex_Ioc 0 (2 * π)).isPathConnected (nonempty_Ioc.mpr Real.two_pi_pos) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
z : ℂ
r : ℝ
r0 : 0 ≤ r
⊢ IsPathConnected (Ioc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have pc' := pc | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | rcases pc' with ⟨b, fb, j⟩ | X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹' front... | case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConne... | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | use b | case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConne... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | simp only [mem_preimage, mem_singleton_iff] at fb j ⊢ | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have bs : f b ∈ s := sc.frontier_subset fb | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | use bs | case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (f ⁻¹... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathCo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | intro x fx | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have p := PathConnectedSpace.somePath x b | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | generalize hu : Icc (0 : ℝ) 1 ∩ ⋂ (a) (_ : f (p.extend a) ∉ s), Iic a = u | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have bdd : BddAbove u := by rw [← hu, bddAbove_def]; use 1; intro t ⟨m, _⟩; exact m.2 | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have un : u.Nonempty := by
rw [← hu]; use 0, left_mem_Icc.mpr zero_le_one; simp only [mem_iInter₂, mem_Iic]; intro a m
contrapose m; simp only [not_not, p.extend_of_le_zero (not_le.mp m).le, fx] | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have uc : IsClosed u := by
rw [← hu]; apply isClosed_Icc.inter; apply isClosed_iInter; intro a; apply isClosed_iInter
intro _; exact isClosed_Iic | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | generalize ht : sSup u = t | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have tu : t ∈ u := by rw [← uc.closure_eq, ← ht]; exact csSup_mem_closure un bdd | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPathConnected.of_frontier | [218, 1] | [281, 34] | have m : t ∈ Icc (0 : ℝ) 1 := by rw [← hu] at tu; exact tu.1 | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
s : Set Y
pc : IsPathConnected (... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X✝ : Type
inst✝⁷ : TopologicalSpace X✝
I : Type
inst✝⁶ : TopologicalSpace I
inst✝⁵ : ConditionallyCompleteLinearOrder I
inst✝⁴ : DenselyOrdered I
inst✝³ : OrderTopology I
X Y : Type
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : Pa... |
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... |
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