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stringlengths 1
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stringlengths 6
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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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