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/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | intro ⟨b, y⟩ m' | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | simp only [mem_prod_eq, mem_diff, mem_univ, true_and_iff] at m' ⊢ | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | refine ⟨?_, (cs1 m'.2).2⟩ | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | apply uu | case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a,... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | use cs0 m'.1, (cs1 m'.2).1 | case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.a
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | rw [e, nhdsWithin_prod_eq, nhdsWithin_univ] | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact Filter.prod_mem_prod (co0.mem_nhds cm0) cn1 | case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.univ_prod | [33, 1] | [48, 40] | exact cc0.isPreconnected.prod cp1 | case h.refine_3
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t)ᶜ = univ ×ˢ tᶜ
a : X
x : Y
u : Set (X × Y)
un : u ∈ 𝓝 (a, x... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
X : Type
inst✝⁵ : TopologicalSpace X
Y : Type
inst✝⁴ : TopologicalSpace Y
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
inst✝ : LocallyConnectedSpace X
t : Set Y
n : Nonseparating t
e : (univ ×ˢ t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [dense_iff_inter_open] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ Dense tᶜ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (U : Set S), IsOpen U → U.Nonempty → (U ∩ tᶜ).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro u uo ⟨z, m⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (U : Set S), IsOpen U → U.Nonempty → (U ∩ tᶜ).Nonempty | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
⊢ (u ∩ tᶜ).Nonempt... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | by_cases zt : z ∉ t | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
⊢ (u ∩ tᶜ).Nonempt... | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∉ ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use z, m, zt | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∉ ... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : ¬z ∉... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [not_not] at zt | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : ¬z ∉... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | generalize hv : (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' u = v | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have vo : IsOpen v := by
rw [← hv]
exact (continuousOn_extChartAt_symm I z).isOpen_inter_preimage
(isOpen_extChartAt_target I z) uo | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have vn : v.Nonempty := by
use extChartAt I z z
simp only [mem_inter_iff, mem_extChartAt_target, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), m, ← hv] | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases dense_iff_inter_open.mp (h z).dense v vo vn with ⟨y, m⟩ | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ ... | case neg.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use(extChartAt I z).symm y | case neg.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage, mem_compl_iff, not_and, ← hv] at m | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases m with ⟨⟨ym, yu⟩, yt⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m✝ : z ∈ u
zt : z ∈ t... | case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, ym, yu, true_and_iff, mem_compl_iff] | case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact yt ym | case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [← hv] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact (continuousOn_extChartAt_symm I z).isOpen_inter_preimage
(isOpen_extChartAt_target I z) uo | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use extChartAt I z z | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
v : Set... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_extChartAt_target, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), m, ← hv] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
u : Set S
uo : IsOpen u
z : S
m : z ∈ u
zt : z ∈ t
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro z u zt un | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
⊢ ∀ (x : S) (u : Set S), x ∈ t → u ∈ 𝓝 x → ∃ c ⊆ u \ t, c... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ t, c ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have m : extChartAt I z z ∈ (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' t := by
simp only [mem_inter_iff, mem_extChartAt_target I z, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), zt] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ∃ c ⊆ u \ t, c ... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have n : (extChartAt I z).target ∩ (extChartAt I z).symm ⁻¹' u ∈ 𝓝 (extChartAt I z z) := by
apply Filter.inter_mem
exact (isOpen_extChartAt_target I z).mem_nhds (mem_extChartAt_target I z)
exact extChartAt_preimage_mem_nhds _ un | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases (h z).loc _ _ m n with ⟨c, cs, cn, cp⟩ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | case intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use(extChartAt I z).source ∩ extChartAt I z ⁻¹' c | case intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(ex... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | refine ⟨?_, ?_, ?_⟩ | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_extChartAt_target I z, true_and_iff, mem_preimage,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), zt] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
⊢ ↑(extChartAt I ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply Filter.inter_mem | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | case hs
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact (isOpen_extChartAt_target I z).mem_nhds (mem_extChartAt_target I z) | case hs
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | case ht
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact extChartAt_preimage_mem_nhds _ un | case ht
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(ext... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ht
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply Set.ext | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extChartAt ... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro x | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage, mem_image] | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | constructor | case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(extC... | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).sy... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨xz, xc⟩ | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | refine ⟨_, xc, ?_⟩ | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [PartialEquiv.left_inv _ xz] | case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(e... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨y, yc, yx⟩ | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [← yx] | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have xc := cs yc | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_diff, mem_inter_iff, mem_preimage] at xc | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have yz := xc.1.1 | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | use PartialEquiv.map_target _ yz | case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [PartialEquiv.right_inv _ yz, yc] | case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑(... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro x xm | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_preimage] at xm | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rcases xm with ⟨xz, xc⟩ | case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | replace xc := cs xc | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(ext... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_diff, mem_inter_iff, mem_preimage, PartialEquiv.map_source _ xz, true_and_iff,
PartialEquiv.left_inv _ xz] at xc | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(ext... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact xc | case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(ext... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [e] | case h.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | convert Filter.image_mem_map cn | case h.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | have ee : ⇑(extChartAt I z).symm = (extChartAt' I z).symm := rfl | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [ee, (extChartAt' I z).symm.map_nhdsWithin_eq (mem_extChartAt_target I z), ← ee] | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [extChartAt', PartialHomeomorph.symm_source,
PartialEquiv.left_inv _ (mem_extChartAt_source I z), compl_inter, inter_union_distrib_left,
inter_compl_self, empty_union, image_inter] | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply nhdsWithin_eq_nhdsWithin (mem_extChartAt_source I z) (isOpen_extChartAt_source I z) | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply Set.ext | case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m : ↑... | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro x | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [mem_inter_iff, mem_compl_iff, mem_image, mem_preimage] | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | constructor | case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m :... | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨xt, xz⟩ | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChart... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | refine ⟨⟨extChartAt I z x, ?_⟩, xz⟩ | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChart... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [PartialEquiv.left_inv _ xz, xt, PartialEquiv.map_source _ xz, not_false_iff,
and_self_iff, eq_self_iff_true] | case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.mp
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChart... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | intro ⟨⟨y, ⟨⟨yz, yt⟩, yx⟩⟩, _⟩ | case h.e'_5.h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z... | case h.e'_5.h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChar... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | simp only [← yx, yt, PartialEquiv.map_target _ yz, not_false_iff, true_and_iff] | case h.e'_5.h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_5.h.mpr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChar... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | rw [e] | case h.refine_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply cp.image | case h.refine_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 z
m... | case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | apply (continuousOn_extChartAt_symm I z).mono | case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 ... | case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | Nonseparating.complexManifold | [51, 1] | [110, 89] | exact _root_.trans cs (_root_.trans (diff_subset _ _) (inter_subset_left _ _)) | case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extChartAt I z).symm ⁻¹' t)
z : S
u : Set S
zt : z ∈ t
un : u ∈ 𝓝 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3.hf
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
t : Set S
h : ∀ (z : S), Nonseparating ((extChartAt I z).target ∩ ↑(extCha... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [isPreconnected_iff_subset_of_disjoint] at sc ⊢ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : IsPreconnected s
so : IsOpen s
ts : Nonseparating t
⊢ IsPreconnected (s \ t) | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
⊢ ∀ (u v : Set X), ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : IsPreconnected s
so : IsOpen s
ts : Nonseparating t
⊢ IsPreconnected (s \ t)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro u v uo vo suv duv | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
⊢ ∀ (u v : Set X), ... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | generalize hf : (fun u : Set X ↦ u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ y in 𝓝[tᶜ] x, y ∈ u}) = f | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have mono : ∀ u, u ⊆ f u := by rw [← hf]; exact fun _ ↦ subset_union_left _ _ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have mem : ∀ {x u c}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u := by
intro x u c m xt cn cu; rw [← hf]; right; use m, xt
simp only [Filter.eventually_iff, setOf_mem_eq]; exact Filter.mem_of_superset cn cu | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have cover : s ⊆ f u ∪ f v := by
intro x m
by_cases xt : x ∉ t; exact union_subset_union (mono _) (mono _) (suv (mem_diff_of_mem m xt))
simp only [not_not] at xt
rcases ts.loc x s xt (so.mem_nhds m) with ⟨c, cst, cn, cp⟩
have d := inter_subset_inter_left (u ∩ v) cst; rw [duv, subset_empty_iff] at d
cases' i... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have fdiff : ∀ {u}, f u \ t ⊆ u := by
intro u x m; simp only [mem_diff, mem_union, mem_setOf, ← hf] at m
simp only [m.2, false_and_iff, and_false_iff, or_false_iff, not_false_iff, and_true_iff] at m
exact m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have fnon : ∀ {x u}, IsOpen u → x ∈ f u → ∀ᶠ y in 𝓝[tᶜ] x, y ∈ u := by
intro x u o m; simp only [mem_union, mem_setOf, ← hf] at m
cases' m with xu m; exact (o.eventually_mem xu).filter_mono nhdsWithin_le_nhds; exact m.2.2 | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have disj : s ∩ (f u ∩ f v) = ∅ := by
contrapose duv; simp only [← ne_eq, ← nonempty_iff_ne_empty] at duv ⊢
rcases duv with ⟨x, m⟩; simp only [mem_inter_iff] at m
have b := ((so.eventually_mem m.1).filter_mono nhdsWithin_le_nhds).and
((fnon uo m.2.1).and (fnon vo m.2.2))
simp only [eventually_nhdsWithin_iff... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | cases' sc (f u) (f v) (fopen uo) (fopen vo) cover disj with su sv | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set ... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | left | case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set ... | case inl.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Se... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact _root_.trans (diff_subset_diff_left su) fdiff | case inl.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Se... | case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set ... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | right | case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set ... | case inr.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Se... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact _root_.trans (diff_subset_diff_left sv) fdiff | case inr.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Se... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [← hf] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | exact fun _ ↦ subset_union_left _ _ | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro u o | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : Is... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | rw [isOpen_iff_eventually] | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | intro x m | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | by_cases xu : x ∈ u | X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : I... | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | by_cases xt : x ∉ t | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | case pos
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [not_not] at xt | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | have n := m | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | simp only [mem_union, xt, xu, false_or_iff, true_and_iff, mem_setOf,
eventually_nhdsWithin_iff, ← hf] at n | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Nonseparating.lean | IsPreconnected.open_diff | [115, 1] | [169, 61] | refine (so.eventually_mem n.1).mp (n.2.eventually_nhds.mp (eventually_of_forall fun y n m ↦ ?_)) | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ ... |
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