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/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
rw [← hg]
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
exact n.holomorphicAt.2
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
contrapose h
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : ∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : ¬𝓝 (f z) ≤ Filter.map f (𝓝 z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : ∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
clear h
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : ¬𝓝 (f z) ≤ Filter.map f (𝓝 z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : ¬𝓝 (f z) ≤ Filter.map f (𝓝 z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [Filter.not_eventually]
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∃ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ¬∀ᶠ (z_1 : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), g z_1 = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
apply n.inCharts.nonconst.mp
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∃ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z),
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)) ≠
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) →
¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∃ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z), ¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [← hg, Ne, imp_self, Filter.eventually_true]
|
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z),
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)) ≠
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) →
¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ) in 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z),
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)) ≠
↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) →
¬g x = g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [← extChartAt_map_nhds' I z, Filter.map_map] at h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map g (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map g (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) z) z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
replace h := @Filter.map_mono _ _ (extChartAt I (f z)).symm _ _ h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [← hg] at h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (g (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (Filter.map (g ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [extChartAt_symm_map_nhds' I (f z), Filter.map_map, Function.comp] at h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f z))) ≤
Filter.map (↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm)
(Filter.map
((fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) ∘ ↑(extChartAt 𝓘(ℂ, ℂ) z)) (𝓝 z))
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
have e : (fun w ↦ (extChartAt I (f z)).symm
(extChartAt I (f z) (f ((extChartAt I z).symm (extChartAt I z w))))) =ᶠ[𝓝 z] f := by
apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp
apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp
refine eventually_of_forall fun w fm m ↦ ?_
simp only [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ fm]
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
rw [Filter.map_congr e] at h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (f z) ≤ Filter.map f (𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
exact h
|
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (f z) ≤ Filter.map f (𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (f z) ≤ Filter.map f (𝓝 z)
e :
(𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
⊢ 𝓝 (f z) ≤ Filter.map f (𝓝 z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
apply ((isOpen_extChartAt_source I z).eventually_mem (mem_extChartAt_source I z)).mp
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ (𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ (𝓝 z).EventuallyEq
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
f
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
apply (n.holomorphicAt.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f z))).mp
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source →
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
refine eventually_of_forall fun w fm m ↦ ?_
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source →
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
w : S
fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source
m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ (fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
w =
f w
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
⊢ ∀ᶠ (x : S) in 𝓝 z,
f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source →
x ∈ (extChartAt 𝓘(ℂ, ℂ) z).source →
(fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
x =
f x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds
|
[201, 1]
|
[225, 42]
|
simp only [PartialEquiv.left_inv _ m, PartialEquiv.left_inv _ fm]
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
w : S
fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source
m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ (fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
w =
f w
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : S → T
z : S
n : NontrivialHolomorphicAt f z
g : ℂ → ℂ
hg : (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ g (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 (f z) ≤
Filter.map
(fun x =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x)))))
(𝓝 z)
w : S
fm : f w ∈ (extChartAt 𝓘(ℂ, ℂ) (f z)).source
m : w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ (fun w =>
↑(extChartAt 𝓘(ℂ, ℂ) (f z)).symm
(↑(extChartAt 𝓘(ℂ, ℂ) (f z)) (f (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) w)))))
w =
f w
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
refine le_antisymm ?_ (continuousAt_fst.prod fa.continuousAt)
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
⊢ 𝓝 (c, f c z) = Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
⊢ 𝓝 (c, f c z) = Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
generalize hg : (fun e x ↦ extChartAt I (f c z) (f e ((extChartAt I z).symm x))) = g
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
have ga : AnalyticAt ℂ (uncurry g) (c, extChartAt I z z) := by
rw [← hg]; exact (holomorphicAt_iff.mp fa).2
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
have gn : NontrivialHolomorphicAt (g c) (extChartAt I z z) := by rw [← hg]; exact n.inCharts
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
have h := gn.nhds_le_map_nhds_param' ga
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (c, g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
simp only [nhds_prod_eq, ← extChartAt_map_nhds' I z, Filter.map_map, Filter.prod_map_id_map_eq,
Function.comp] at h
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (c, g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤
Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h : 𝓝 (c, g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤ Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
replace h := @Filter.map_mono _ _ (fun p : ℂ × ℂ ↦ (p.1, (extChartAt I (f c z)).symm p.2)) _ _ h
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤
Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) (𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z)) ≤
Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z)
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
simp only [← hg] at h
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) (𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) (𝓝 c ×ˢ 𝓝 (g c (↑(extChartAt 𝓘(ℂ, ℂ) z) z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map (fun x => (x.1, g x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2))) (𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
rw [PartialEquiv.left_inv _ (mem_extChartAt_source I z)] at h
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) z))))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
have pe := Filter.prod_map_id_map_eq (f := 𝓝 c) (g := 𝓝 (extChartAt I (f c z) (f c z)))
(m := fun x ↦ (extChartAt I (f c z)).symm x)
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 c ×ˢ Filter.map (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm x) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
rw [extChartAt_symm_map_nhds', ←nhds_prod_eq] at pe
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 c ×ˢ Filter.map (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm x) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 c ×ˢ Filter.map (fun x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm x) (𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
refine _root_.trans (le_of_eq pe) (_root_.trans h (le_of_eq ?_))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ 𝓝 (c, f c z) ≤ Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
clear h pe
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
h :
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z))) ≤
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z))
pe :
𝓝 (c, f c z) =
Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(𝓝 c ×ˢ 𝓝 (↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f c z)))
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
rw [←nhds_prod_eq, Filter.map_map]
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map (fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2))
(Filter.map
(fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 c ×ˢ 𝓝 z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
apply Filter.map_congr
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ (𝓝 (c, z)).EventuallyEq
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
fun p => (p.1, f p.1 p.2)
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ Filter.map
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(𝓝 (c, z)) =
Filter.map (fun p => (p.1, f p.1 p.2)) (𝓝 (c, z))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
apply ((isOpen_extChartAt_source II (c, z)).eventually_mem (mem_extChartAt_source II (c, z))).mp
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ (𝓝 (c, z)).EventuallyEq
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
fun p => (p.1, f p.1 p.2)
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ (𝓝 (c, z)).EventuallyEq
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
fun p => (p.1, f p.1 p.2)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
apply (fa.continuousAt.eventually_mem (extChartAt_source_mem_nhds I (f c z))).mp
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
apply eventually_of_forall
|
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ (x : ℂ × S),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
intro ⟨e, w⟩ fm m
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ (x : ℂ × S),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : uncurry f (e, w) ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : (e, w) ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ ∀ (x : ℂ × S),
uncurry f x ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source →
x ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source →
((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
x =
(fun p => (p.1, f p.1 p.2)) x
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
simp only [Function.comp, uncurry, extChartAt_prod, PartialEquiv.prod_source, mem_prod_eq] at fm m
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : uncurry f (e, w) ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : (e, w) ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : f e w ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : e ∈ (extChartAt 𝓘(ℂ, ℂ) c).source ∧ w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : uncurry f (e, w) ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : (e, w) ∈ (extChartAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) (c, z)).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
simp only [Function.comp, PartialEquiv.left_inv _ m.2, PartialEquiv.left_inv _ fm]
|
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : f e w ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : e ∈ (extChartAt 𝓘(ℂ, ℂ) c).source ∧ w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.hp
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
gn : NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
e : ℂ
w : S
fm : f e w ∈ (extChartAt 𝓘(ℂ, ℂ) (f c z)).source
m : e ∈ (extChartAt 𝓘(ℂ, ℂ) c).source ∧ w ∈ (extChartAt 𝓘(ℂ, ℂ) z).source
⊢ ((fun p => (p.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)).symm p.2)) ∘ fun x =>
(x.1, ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f x.1 (↑(extChartAt 𝓘(ℂ, ℂ) z).symm (↑(extChartAt 𝓘(ℂ, ℂ) z) x.2)))))
(e, w) =
(fun p => (p.1, f p.1 p.2)) (e, w)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
rw [← hg]
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (uncurry fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)))
(c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
exact (holomorphicAt_iff.mp fa).2
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (uncurry fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)))
(c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
⊢ AnalyticAt ℂ (uncurry fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x)))
(c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
rw [← hg]
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ NontrivialHolomorphicAt ((fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) c)
(↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ NontrivialHolomorphicAt (g c) (↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OpenMapping.lean
|
NontrivialHolomorphicAt.nhds_eq_map_nhds_param
|
[234, 1]
|
[258, 85]
|
exact n.inCharts
|
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ NontrivialHolomorphicAt ((fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) c)
(↑(extChartAt 𝓘(ℂ, ℂ) z) z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : ChartedSpace ℂ S
cms : AnalyticManifold 𝓘(ℂ, ℂ) S
T : Type
inst✝³ : TopologicalSpace T
inst✝² : ChartedSpace ℂ T
cmt : AnalyticManifold 𝓘(ℂ, ℂ) T
U : Type
inst✝¹ : TopologicalSpace U
inst✝ : ChartedSpace ℂ U
cmu : AnalyticManifold 𝓘(ℂ, ℂ) U
f : ℂ → S → T
c : ℂ
z : S
n : NontrivialHolomorphicAt (f c) z
fa : HolomorphicAt (𝓘(ℂ, ℂ).prod 𝓘(ℂ, ℂ)) 𝓘(ℂ, ℂ) (uncurry f) (c, z)
g : ℂ → ℂ → ℂ
hg : (fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) = g
ga : AnalyticAt ℂ (uncurry g) (c, ↑(extChartAt 𝓘(ℂ, ℂ) z) z)
⊢ NontrivialHolomorphicAt ((fun e x => ↑(extChartAt 𝓘(ℂ, ℂ) (f c z)) (f e (↑(extChartAt 𝓘(ℂ, ℂ) z).symm x))) c)
(↑(extChartAt 𝓘(ℂ, ℂ) z) z)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.bind_const_none
|
[23, 1]
|
[24, 22]
|
cases x <;> simp
|
α : Type u_1
β : Type u_2
x : Option α
⊢ (Option.bind x fun x => none) = none
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
x : Option α
⊢ (Option.bind x fun x => none) = none
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.isNone_false_iff_isSome
|
[28, 1]
|
[29, 22]
|
cases x <;> simp
|
α : Type u_1
x : Option α
⊢ isNone x = false ↔ isSome x = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : Option α
⊢ isNone x = false ↔ isSome x = true
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Fin.tuple_sequence₁
|
[56, 1]
|
[58, 41]
|
simp [Fin.tupleSequence, functor_norm]
|
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 1 → Type u
x : (i : Fin 1) → m (α i)
⊢ tupleSequence x = do
let r₀ ← x 0
pure (cons r₀ default)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 1 → Type u
x : (i : Fin 1) → m (α i)
⊢ tupleSequence x = do
let r₀ ← x 0
pure (cons r₀ default)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Fin.tuple_sequence₂
|
[62, 1]
|
[66, 6]
|
simp [Fin.tupleSequence, functor_norm]
|
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 2 → Type u
x : (i : Fin 2) → m (α i)
⊢ tupleSequence x = do
let r₀ ← x 0
let r₁ ← x 1
pure (cons r₀ (cons r₁ default))
|
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 2 → Type u
x : (i : Fin 2) → m (α i)
⊢ (do
let r ← x 0
let x ← tail x 0
pure (cons r (cons x default))) =
do
let r₀ ← x 0
let r₁ ← x 1
pure (cons r₀ (cons r₁ default))
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 2 → Type u
x : (i : Fin 2) → m (α i)
⊢ tupleSequence x = do
let r₀ ← x 0
let r₁ ← x 1
pure (cons r₀ (cons r₁ default))
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Fin.tuple_sequence₂
|
[62, 1]
|
[66, 6]
|
rfl
|
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 2 → Type u
x : (i : Fin 2) → m (α i)
⊢ (do
let r ← x 0
let x ← tail x 0
pure (cons r (cons x default))) =
do
let r₀ ← x 0
let r₁ ← x 1
pure (cons r₀ (cons r₁ default))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
inst✝ : LawfulMonad m
α : Fin 2 → Type u
x : (i : Fin 2) → m (α i)
⊢ (do
let r ← x 0
let x ← tail x 0
pure (cons r (cons x default))) =
do
let r₀ ← x 0
let r₁ ← x 1
pure (cons r₀ (cons r₁ default))
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.bind_isSome
|
[74, 1]
|
[75, 84]
|
cases x <;> simp
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
x : Option α
y : α → Option β
⊢ isSome (Option.bind x y) = true ↔ ∃ (h : isSome x = true), isSome (y (get x h)) = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
x : Option α
y : α → Option β
⊢ isSome (Option.bind x y) = true ↔ ∃ (h : isSome x = true), isSome (y (get x h)) = true
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.map_isSome
|
[79, 1]
|
[80, 19]
|
cases x <;> simp
|
m : Type u → Type v
inst✝ : Monad m
α β : Type u_1
x : Option α
y : α → β
⊢ isSome (y <$> x) = isSome x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α β : Type u_1
x : Option α
y : α → β
⊢ isSome (y <$> x) = isSome x
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.not_isSome'
|
[84, 1]
|
[84, 92]
|
cases x <;> simp
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
⊢ (!decide (isSome x = isNone x)) = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
⊢ (!decide (isSome x = isNone x)) = true
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.guardProp_isSome
|
[92, 1]
|
[95, 22]
|
dsimp only [Option.guardProp]
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
p : Prop
inst✝ : Decidable p
x : α
⊢ isSome (guardProp p x) = true ↔ p
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
p : Prop
inst✝ : Decidable p
x : α
⊢ isSome (if p then some x else none) = true ↔ p
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
p : Prop
inst✝ : Decidable p
x : α
⊢ isSome (guardProp p x) = true ↔ p
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.guardProp_isSome
|
[92, 1]
|
[95, 22]
|
split_ifs <;> simpa
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
p : Prop
inst✝ : Decidable p
x : α
⊢ isSome (if p then some x else none) = true ↔ p
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
p : Prop
inst✝ : Decidable p
x : α
⊢ isSome (if p then some x else none) = true ↔ p
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.coe_part_dom
|
[99, 1]
|
[99, 100]
|
cases x <;> simp
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
⊢ (↑x).Dom ↔ isSome x = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
⊢ (↑x).Dom ↔ isSome x = true
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.coe_part_eq_some
|
[103, 1]
|
[104, 74]
|
simp [Part.eq_some_iff]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
y : α
⊢ ↑x = Part.some y ↔ x = some y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : Option α
y : α
⊢ ↑x = Part.some y ↔ x = some y
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.get?_isSome_iff
|
[108, 1]
|
[110, 35]
|
rw [← not_iff_not]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : List α
n : ℕ
⊢ Option.isSome (get? x n) = true ↔ n < length x
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : List α
n : ℕ
⊢ ¬Option.isSome (get? x n) = true ↔ ¬n < length x
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : List α
n : ℕ
⊢ Option.isSome (get? x n) = true ↔ n < length x
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.get?_isSome_iff
|
[108, 1]
|
[110, 35]
|
simp [Option.isNone_iff_eq_none]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : List α
n : ℕ
⊢ ¬Option.isSome (get? x n) = true ↔ ¬n < length x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
x : List α
n : ℕ
⊢ ¬Option.isSome (get? x n) = true ↔ ¬n < length x
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Option.map_is_some'
|
[114, 1]
|
[115, 19]
|
cases x <;> simp
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
x : Option α
f : α → β
⊢ isSome (Option.map f x) = isSome x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
x : Option α
f : α → β
⊢ isSome (Option.map f x) = isSome x
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.zipWith_fst
|
[118, 1]
|
[121, 11]
|
erw [← List.map_uncurry_zip_eq_zipWith, List.map_fst_zip]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₁ ≤ length l₂
⊢ zipWith (fun a b => a) l₁ l₂ = l₁
|
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₁ ≤ length l₂
⊢ length l₁ ≤ length l₂
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₁ ≤ length l₂
⊢ zipWith (fun a b => a) l₁ l₂ = l₁
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.zipWith_fst
|
[118, 1]
|
[121, 11]
|
exact hl
|
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₁ ≤ length l₂
⊢ length l₁ ≤ length l₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₁ ≤ length l₂
⊢ length l₁ ≤ length l₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.zipWith_snd
|
[124, 1]
|
[127, 11]
|
erw [← List.map_uncurry_zip_eq_zipWith, List.map_snd_zip]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₂ ≤ length l₁
⊢ zipWith (fun a b => b) l₁ l₂ = l₂
|
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₂ ≤ length l₁
⊢ length l₂ ≤ length l₁
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₂ ≤ length l₁
⊢ zipWith (fun a b => b) l₁ l₂ = l₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
List.zipWith_snd
|
[124, 1]
|
[127, 11]
|
exact hl
|
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₂ ≤ length l₁
⊢ length l₂ ≤ length l₁
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
hl : length l₂ ≤ length l₁
⊢ length l₂ ≤ length l₁
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Multiset.map_get
|
[134, 1]
|
[136, 36]
|
simp [Finset.univ, Fintype.elems]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
l : List α
⊢ map (List.get l) Finset.univ.val = ↑l
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
l : List α
⊢ map (List.get l) Finset.univ.val = ↑l
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Multiset.get_zero
|
[142, 1]
|
[142, 88]
|
simp [Multiset.get]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
⊢ get 0 = none
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
⊢ get 0 = none
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Multiset.get_singleton
|
[144, 1]
|
[144, 105]
|
simp [Multiset.get]
|
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
a : α
⊢ get {a} = some a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
α : Type u_1
a : α
⊢ get {a} = some a
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
le_false_iff
|
[147, 1]
|
[147, 72]
|
decide
|
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, b ≤ false ↔ b = false
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, b ≤ false ↔ b = false
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
lt_true_iff
|
[151, 1]
|
[151, 70]
|
decide
|
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, b < true ↔ b = false
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, b < true ↔ b = false
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
false_lt_iff
|
[155, 1]
|
[155, 71]
|
decide
|
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, false < b ↔ b = true
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝ : Monad m
⊢ ∀ {b : Bool}, false < b ↔ b = true
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
ne_min_of_ne_and_ne
|
[158, 1]
|
[159, 81]
|
rcases min_choice x y with h | h <;> rw [h] <;> assumption
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a x y : ι
hx : a ≠ x
hy : a ≠ y
⊢ a ≠ min x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a x y : ι
hx : a ≠ x
hy : a ≠ y
⊢ a ≠ min x y
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff
|
[163, 1]
|
[167, 31]
|
rw [max_def]
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬a = max a b ↔ a < b
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ (¬a = if a ≤ b then b else a) ↔ a < b
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬a = max a b ↔ a < b
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff
|
[163, 1]
|
[167, 31]
|
split_ifs with h
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ (¬a = if a ≤ b then b else a) ↔ a < b
|
case pos
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : a ≤ b
⊢ ¬a = b ↔ a < b
case neg
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : ¬a ≤ b
⊢ ¬a = a ↔ a < b
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ (¬a = if a ≤ b then b else a) ↔ a < b
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff
|
[163, 1]
|
[167, 31]
|
simpa using h
|
case pos
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : a ≤ b
⊢ ¬a = b ↔ a < b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : a ≤ b
⊢ ¬a = b ↔ a < b
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff
|
[163, 1]
|
[167, 31]
|
simpa using le_of_not_ge h
|
case neg
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : ¬a ≤ b
⊢ ¬a = a ↔ a < b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
h : ¬a ≤ b
⊢ ¬a = a ↔ a < b
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff'
|
[171, 1]
|
[172, 22]
|
rw [max_comm]
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬b = max a b ↔ b < a
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬b = max b a ↔ b < a
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬b = max a b ↔ b < a
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
max_ne_self_iff'
|
[171, 1]
|
[172, 22]
|
simp
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬b = max b a ↔ b < a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type u_1
inst✝ : LinearOrder ι
a b : ι
⊢ ¬b = max b a ↔ b < a
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
WithTop.isSome_iff_lt_top
|
[180, 1]
|
[183, 6]
|
rw [← not_iff_not, Bool.eq_false_eq_not_eq_true, Option.not_isSome, Option.isNone_iff_eq_none,
lt_top_iff_ne_top, Ne, Classical.not_not]
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type
inst✝ : PartialOrder ι
x : WithTop ι
⊢ Option.isSome x = true ↔ x < ⊤
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type
inst✝ : PartialOrder ι
x : WithTop ι
⊢ x = none ↔ x = ⊤
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type
inst✝ : PartialOrder ι
x : WithTop ι
⊢ Option.isSome x = true ↔ x < ⊤
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
WithTop.isSome_iff_lt_top
|
[180, 1]
|
[183, 6]
|
rfl
|
m : Type u → Type v
inst✝¹ : Monad m
ι : Type
inst✝ : PartialOrder ι
x : WithTop ι
⊢ x = none ↔ x = ⊤
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
ι : Type
inst✝ : PartialOrder ι
x : WithTop ι
⊢ x = none ↔ x = ⊤
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.le_iff''
|
[233, 1]
|
[237, 8]
|
rw [Prod.Lex.le_iff', le_iff_lt_or_eq]
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x ≤ y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 ≤ y.2)
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x ≤ y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 ≤ y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.le_iff''
|
[233, 1]
|
[237, 8]
|
have := @ne_of_lt _ _ x.1 y.1
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → x.1 ≠ y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.le_iff''
|
[233, 1]
|
[237, 8]
|
tauto
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → x.1 ≠ y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → x.1 ≠ y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 ≤ y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.lt_iff''
|
[244, 1]
|
[248, 8]
|
rw [lt_iff', le_iff_lt_or_eq]
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x < y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 < y.2)
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x < y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 < y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.lt_iff''
|
[244, 1]
|
[248, 8]
|
have : x.1 < y.1 → ¬x.1 = y.1 := ne_of_lt
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → ¬x.1 = y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.lt_iff''
|
[244, 1]
|
[248, 8]
|
tauto
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → ¬x.1 = y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x y : Lex (α × β)
this : x.1 < y.1 → ¬x.1 = y.1
⊢ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2 ↔ (x.1 < y.1 ∨ x.1 = y.1) ∧ (x.1 = y.1 → x.2 < y.2)
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_le_of_le
|
[251, 1]
|
[255, 26]
|
rw [Prod.Lex.le_iff'] at h
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x ≤ y
⊢ x.1 ≤ y.1
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2
⊢ x.1 ≤ y.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x ≤ y
⊢ x.1 ≤ y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_le_of_le
|
[251, 1]
|
[255, 26]
|
cases h with
| inl h => exact h.le
| inr h => exact h.1.le
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2
⊢ x.1 ≤ y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2
⊢ x.1 ≤ y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_le_of_le
|
[251, 1]
|
[255, 26]
|
exact h.le
|
case inl
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 < y.1
⊢ x.1 ≤ y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 < y.1
⊢ x.1 ≤ y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_le_of_le
|
[251, 1]
|
[255, 26]
|
exact h.1.le
|
case inr
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 = y.1 ∧ x.2 ≤ y.2
⊢ x.1 ≤ y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x y : Lex (α × β)
h : x.1 = y.1 ∧ x.2 ≤ y.2
⊢ x.1 ≤ y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_lt_of_lt_of_le
|
[258, 1]
|
[263, 33]
|
rw [Prod.Lex.lt_iff'] at h
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h : x < y
h' : y.2 ≤ x.2
⊢ x.1 < y.1
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2
h' : y.2 ≤ x.2
⊢ x.1 < y.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h : x < y
h' : y.2 ≤ x.2
⊢ x.1 < y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_lt_of_lt_of_le
|
[258, 1]
|
[263, 33]
|
cases h with
| inl h => exact h
| inr h => cases h.2.not_le h'
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2
h' : y.2 ≤ x.2
⊢ x.1 < y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h : x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2
h' : y.2 ≤ x.2
⊢ x.1 < y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_lt_of_lt_of_le
|
[258, 1]
|
[263, 33]
|
exact h
|
case inl
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h' : y.2 ≤ x.2
h : x.1 < y.1
⊢ x.1 < y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h' : y.2 ≤ x.2
h : x.1 < y.1
⊢ x.1 < y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.fst_lt_of_lt_of_le
|
[258, 1]
|
[263, 33]
|
cases h.2.not_le h'
|
case inr
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h' : y.2 ≤ x.2
h : x.1 = y.1 ∧ x.2 < y.2
⊢ x.1 < y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : PartialOrder β
x y : Lex (α × β)
h' : y.2 ≤ x.2
h : x.1 = y.1 ∧ x.2 < y.2
⊢ x.1 < y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_fst_mono_iff
|
[275, 1]
|
[276, 56]
|
simp [le_iff']
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x : α
y₁ y₂ : β
⊢ (x, y₁) ≤ (x, y₂) ↔ y₁ ≤ y₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x : α
y₁ y₂ : β
⊢ (x, y₁) ≤ (x, y₂) ↔ y₁ ≤ y₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_fst_mono_lt_iff
|
[280, 1]
|
[281, 56]
|
simp [lt_iff']
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x : α
y₁ y₂ : β
⊢ (x, y₁) < (x, y₂) ↔ y₁ < y₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x : α
y₁ y₂ : β
⊢ (x, y₁) < (x, y₂) ↔ y₁ < y₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_snd_mono_le_iff
|
[285, 1]
|
[286, 57]
|
simp [le_iff'']
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x₁ x₂ : α
y : β
⊢ (x₁, y) ≤ (x₂, y) ↔ x₁ ≤ x₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : PartialOrder α
inst✝ : Preorder β
x₁ x₂ : α
y : β
⊢ (x₁, y) ≤ (x₂, y) ↔ x₁ ≤ x₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_snd_mono_lt_iff
|
[290, 1]
|
[291, 56]
|
simp [lt_iff']
|
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x₁ x₂ : α
y : β
⊢ (x₁, y) < (x₂, y) ↔ x₁ < x₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝² : Monad m
α : Type u_1
β : Type u_2
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝¹ : Preorder α
inst✝ : Preorder β
x₁ x₂ : α
y : β
⊢ (x₁, y) < (x₂, y) ↔ x₁ < x₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_false_lt_mk_true_iff
|
[295, 1]
|
[296, 96]
|
simp [lt_iff', le_iff_lt_or_eq]
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.49210
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x₁ x₂ : α
⊢ (x₁, false) < (x₂, true) ↔ x₁ ≤ x₂
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.49210
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x₁ x₂ : α
⊢ (x₁, false) < (x₂, true) ↔ x₁ ≤ x₂
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_true_le_mk_false_iff_lt
|
[300, 1]
|
[301, 75]
|
simp [le_iff']
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.50081
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x y : α
⊢ (x, true) ≤ (y, false) ↔ x < y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.50081
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x y : α
⊢ (x, true) ≤ (y, false) ↔ x < y
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.mk_true_lt_iff_lt
|
[305, 1]
|
[306, 68]
|
simp [lt_iff']
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.51649
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x : α
y : Lex (α × Bool)
⊢ (x, true) < y ↔ x < y.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.51649
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x : α
y : Lex (α × Bool)
⊢ (x, true) < y ↔ x < y.1
TACTIC:
|
https://github.com/kovach/etch.git
|
b9e66fe99c33dc1edd926626e598ba00d5d78627
|
Etch/Verification/Misc.lean
|
Prod.Lex.lt_mk_true_iff
|
[308, 1]
|
[310, 26]
|
simp [lt_iff', le_iff']
|
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.53384
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x : Lex (α × Bool)
y : α
⊢ x < (y, true) ↔ x ≤ (y, false)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u → Type v
inst✝¹ : Monad m
α : Type u_1
β : Type ?u.53384
r₁ : α → α → Prop
r₂ : β → β → Prop
inst✝ : PartialOrder α
x : Lex (α × Bool)
y : α
⊢ x < (y, true) ↔ x ≤ (y, false)
TACTIC:
|
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