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stringclasses 147
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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
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stringlengths 1
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stringlengths 73
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_dh | [119, 1] | [120, 64] | refine HasMFDerivAt.prod ?_ i.has_df' | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) (I.prod I) i.h (c, i.z') i.dh | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) (I.prod I) i.h (c, i.z') i.dh
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.has_dh | [119, 1] | [120, 64] | apply hasMFDerivAt_fst | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HasMFDerivAt (I.prod I) I (fun y => y.1) (c, i.z') dc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | have h := ContinuousLinearMap.ext_iff.mp
(extChartAt_mderiv_right_inverse' (mem_extChartAt_source I z)) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I z)) z).comp (mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z))) t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I z) z))) t
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de | [142, 1] | [146, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h : (mfderiv I I (↑(extChartAt I z)) z) ((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) t) = t
⊢ i.dei (i.de t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I (f c z)), i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I (f c z)), i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_left_inverse
(mem_extChartAt_source I (f c z))) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))).comp
(mfderiv I I (↑(extChartAt I (f c z))) (f c z)))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dei_de' | [148, 1] | [152, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I (f c z)
h :
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z)))
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I (f c z))) t
⊢ i.dei' (i.de' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I z), i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : TangentSpace I z), i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_left_inverse
(mem_extChartAt_source I z)) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
((mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)).comp (mfderiv I I (↑(extChartAt I z)) z)) t =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei | [154, 1] | [158, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : TangentSpace I z
h :
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) z)) ((mfderiv I I (↑(extChartAt I z)) z) t) =
(ContinuousLinearMap.id ℂ (TangentSpace I z)) t
⊢ i.de (i.dei t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | intro t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ), i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | have h := ContinuousLinearMap.ext_iff.mp (extChartAt_mderiv_right_inverse'
(mem_extChartAt_source I (f c z))) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply] at h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
((mfderiv I I (↑(extChartAt I (f c z))) (f c z)).comp
(mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))))
t =
(ContinuousLinearMap.id ℂ (TangentSpace I (↑(extChartAt I (f c z)) (f c z)))) t
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.de_dei' | [160, 1] | [164, 89] | exact h | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : ℂ
h :
(mfderiv I I (↑(extChartAt I (f c z))) (f c z))
((mfderiv I I (↑(extChartAt I (f c z)).symm) (↑(extChartAt I (f c z)) (f c z))) t) =
t
⊢ i.de' (i.dei' t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dhi_dh | [166, 1] | [172, 54] | intro ⟨u, v⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dhi (i.dh t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dhi (i.dh t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dhi_dh | [166, 1] | [172, 54] | simp only [Cinv.dh, Cinv.dhi, dc, dz, Cinv.dfi', Cinv.df', Cinv.df, i.dei_de', i.dei_de,
i.dfzi_dfz, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
ContinuousLinearMap.sub_apply, ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd',
ContinuousLinearMap.add_apply, ContinuousLinearMap.map_add, ContinuousLinearMap.map_sub,
add_sub_cancel_left, ContinuousLinearMap.coe_snd] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dhi (i.dh (u, v)) = (u, v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dh_dhi | [174, 1] | [180, 100] | intro ⟨u, v⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dh (i.dhi t) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ (t : ℂ × ℂ), i.dh (i.dhi t) = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.dh_dhi | [174, 1] | [180, 100] | simp only [Cinv.dh, Cinv.dhi, dc, dz, Cinv.dfi', Cinv.df', Cinv.df, i.de_dei', i.de_dei,
i.dfz_dfzi, ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply,
ContinuousLinearMap.sub_apply, ContinuousLinearMap.coe_fst', ContinuousLinearMap.coe_snd',
ContinuousLinearMap.add_apply, ContinuousLinearMap.map_add, ContinuousLinearMap.map_sub,
add_sub_cancel_left, ← add_sub_assoc, ContinuousLinearMap.coe_snd, ContinuousLinearMap.coe_fst] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
u v : ℂ
⊢ i.dh (i.dhi (u, v)) = (u, v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | have a := ContDiffAt.localInverse_apply_image i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | have e : ContDiffAt.localInverse i.ha.contDiffAt i.has_dhe le_top = i.he.symm := rfl | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | rw [e] at a | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ⋯.localInverse ⋯ ⋯ (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | clear e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
e : ⋯.localInverse ⋯ ⋯ = ↑i.he.symm
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | simp only [Cinv.z', Cinv.h, Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] at a | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.z')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_at | [194, 1] | [200, 9] | rw [a] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
a : ↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z)) = (c, ↑(extChartAt I z) z)
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).2 = ↑(extChartAt I z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | generalize ht :
((extChartAt II (c, z)).source ∩ extChartAt II (c, z) ⁻¹' i.he.source : Set (ℂ × S)) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have o : IsOpen t := by
rw [← ht]
exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_source | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have m : (c, z) ∈ t := by
simp only [mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht]
exact ContDiffAt.mem_toPartialHomeomorph_source i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | apply Filter.eventuallyEq_of_mem (o.mem_nhds m) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), i.g x.1 (f x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | intro x m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m : (c, z) ∈ t
⊢ EqOn (fun x => i.g x.1 (f x.1 x.2)) (fun x => x.2) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [mem_inter_iff, mem_preimage, extChartAt_prod, extChartAt_eq_refl, ← ht,
PartialEquiv.prod_source, PartialEquiv.refl_source, mem_prod_eq, mem_univ, true_and_iff,
PartialEquiv.prod_coe, PartialEquiv.refl_coe, id] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x ∈ t
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | have inv := i.he.left_inv m.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [Cinv.g] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ (fun x => i.g x.1 (f x.1 x.2)) x = (fun x => x.2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | generalize hq : i.he.symm = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [hq] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
inv : ↑i.he.symm (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (↑i.he (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [Cinv.h, Cinv.f', PartialEquiv.left_inv _ m.1] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
inv : ↑q (i.h (x.1, ↑(extChartAt I z) x.2)) = (x.1, ↑(extChartAt I z) x.2)
hq : i.he.symm = q
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [inv, PartialEquiv.left_inv _ m.1] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
m✝ : (c, z) ∈ t
x : ℂ × S
m : x.2 ∈ (extChartAt I z).source ∧ (x.1, ↑(extChartAt I z) x.2) ∈ i.he.source
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
inv : ↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2)) = (x.1, ↑(extChartAt I z) x.2)
⊢ ↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) (f x.1 x.2))).2 = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | rw [← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_source | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
⊢ IsOpen ((extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | simp only [mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ (c, z) ∈ t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ (c, z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.left_inv | [207, 1] | [226, 47] | exact ContDiffAt.mem_toPartialHomeomorph_source i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.he.source = t
o : IsOpen t
⊢ ↑(extChartAt (I.prod I) (c, z)) (c, z) ∈ i.he.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | intro x m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ x ∈ i.he.target, (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ x ∈ i.he.target, (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | have e : i.he (i.he.symm x) = x := i.he.right_inv m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
⊢ (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | generalize hq : i.he.symm x = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
⊢ (↑i.he.symm x).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [hq] at e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
e : ↑i.he (↑i.he.symm x) = x
q : ℂ × ℂ
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top, Cinv.h] at e | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : ↑i.he q = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.inv_fst | [229, 1] | [234, 11] | rw [← e] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
x : ℂ × ℂ
m : x ∈ i.he.target
q : ℂ × ℂ
e : (q.1, i.f' q) = x
hq : ↑i.he.symm x = q
⊢ q.1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | generalize ht : ((extChartAt II (c, f c z)).source ∩ extChartAt II (c, f c z) ⁻¹' i.he.target
: Set (ℂ × T)) = t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have o : IsOpen t := by
rw [← ht]
exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m' : (c, extChartAt I (f c z) (f c z)) ∈ i.he.toPartialEquiv.target := by
have m := ContDiffAt.image_mem_toPartialHomeomorph_target i.ha.contDiffAt i.has_dhe le_top
have e : i.h (c, i.z') = (c, extChartAt I (f c z) (f c z)) := by
simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)]
rw [e] at m; exact m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m : (c, f c z) ∈ t := by
simp only [m', mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht,
extChartAt_prod, PartialEquiv.prod_coe, extChartAt_eq_refl, PartialEquiv.refl_coe, id,
PartialEquiv.prod_source, prod_mk_mem_set_prod_eq, PartialEquiv.refl_source, mem_univ] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine fm.mp (Filter.eventually_of_mem (o.mem_nhds m) ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | intro x m mf | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ ∀ x ∈ t,
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source →
f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [mem_inter_iff, mem_preimage, extChartAt_prod, extChartAt_eq_refl,
PartialEquiv.prod_source, PartialEquiv.refl_source, mem_prod_eq, mem_univ, true_and_iff,
PartialEquiv.prod_coe, PartialEquiv.refl_coe, id, ← ht] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x ∈ t
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have inv := i.he.right_inv m.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.g] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (i.g x.1 x.2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | generalize hq : i.he.symm = q | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
⊢ f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [hq] at inv mf | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
mf : f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
inv : ↑i.he (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [Cinv.he, ContDiffAt.toPartialHomeomorph_coe i.ha.contDiffAt i.has_dhe le_top] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : ↑i.he (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have q1 : (q (x.1, extChartAt I (f c z) x.2)).1 = x.1 := by simp only [← hq, i.inv_fst _ m.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.h, Cinv.f', Prod.eq_iff_fst_eq_snd_eq, q1] at inv | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | nth_rw 2 [← PartialEquiv.left_inv _ m.1] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) = x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | nth_rw 2 [← inv.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2))) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm (↑(extChartAt I (f c z)) x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine (PartialEquiv.left_inv _ mf).symm | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
hq : i.he.symm = q
q1 : (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
inv :
True ∧
↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)) =
↑(extChartAt I (f c z)) x.2
⊢ f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) =
↑(extChartAt I (f c z)).symm
(↑(extChartAt I (f c z)) (f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [← ht] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen t | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _)
i.he.open_target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
⊢ IsOpen ((extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have m := ContDiffAt.image_mem_toPartialHomeomorph_target i.ha.contDiffAt i.has_dhe le_top | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | have e : i.h (c, i.z') = (c, extChartAt I (f c z) (f c z)) := by
simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rw [e] at m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact m | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
e : i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
⊢ (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [Cinv.h, Cinv.z', Cinv.f', PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m : i.h (c, i.z') ∈ (ContDiffAt.toPartialHomeomorph i.h ⋯ ⋯ ⋯).target
⊢ i.h (c, i.z') = (c, ↑(extChartAt I (f c z)) (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [m', mem_inter_iff, mem_preimage, mem_extChartAt_source, true_and_iff, ← ht,
extChartAt_prod, PartialEquiv.prod_coe, extChartAt_eq_refl, PartialEquiv.refl_coe, id,
PartialEquiv.prod_source, prod_mk_mem_set_prod_eq, PartialEquiv.refl_source, mem_univ] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ (c, f c z) ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
⊢ (c, f c z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine ContinuousAt.eventually_mem ?_ (extChartAt_source_mem_nhds' I ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source | case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z)
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply i.fa.continuousAt.comp₂_of_eq continuousAt_fst | case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z) | case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine ContinuousAt.comp ?_ ?_ | case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (extChartAt I z).symm.1 (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at] | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact continuousAt_extChartAt_symm I _ | case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (extChartAt I z).invFun (↑(extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply continuousAt_snd.comp | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z) | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | refine (PartialHomeomorph.continuousAt i.he.symm ?_).comp ?_ | case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [m', (he i).symm_source] | case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) ∈ i.he.symm.source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply continuousAt_fst.prod | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z) | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply (continuousAt_extChartAt I _).comp_of_eq | case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z) | case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z)
case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | exact continuousAt_snd | case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ContinuousAt (fun x => x.2) (c, f c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | rfl | case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hh.refine_2.refine_2.hy
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ (c, f c z).2 = f c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at, PartialEquiv.left_inv _ (mem_extChartAt_source _ _),
PartialEquiv.invFun_as_coe] | case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ ((c, f c z).1, (extChartAt I z).symm.1 (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) = (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [i.inv_at, PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f (c, f c z).1 (↑(extChartAt I z).symm (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2) ∈
(extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | apply mem_extChartAt_source | case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m : (c, f c z) ∈ t
⊢ f c z ∈ (extChartAt I (f c z)).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.right_inv | [237, 1] | [284, 43] | simp only [← hq, i.inv_fst _ m.2] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
t : Set (ℂ × T)
ht : (extChartAt (I.prod I) (c, f c z)).source ∩ ↑(extChartAt (I.prod I) (c, f c z)) ⁻¹' i.he.target = t
o : IsOpen t
m' : (c, ↑(extChartAt I (f c z)) (f c z)) ∈ i.he.target
m✝ : (c, f c z) ∈ t
fm :
∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z),
f x.1 (↑(extChartAt I z).symm (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
x : ℂ × T
m : x.2 ∈ (extChartAt I (f c z)).source ∧ (x.1, ↑(extChartAt I (f c z)) x.2) ∈ i.he.target
q : PartialHomeomorph (ℂ × ℂ) (ℂ × ℂ)
mf : f x.1 (↑(extChartAt I z).symm (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).2) ∈ (extChartAt I (f c z)).source
inv : i.h (↑q (x.1, ↑(extChartAt I (f c z)) x.2)) = (x.1, ↑(extChartAt I (f c z)) x.2)
hq : i.he.symm = q
⊢ (↑q (x.1, ↑(extChartAt I (f c z)) x.2)).1 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | apply AnalyticAt.holomorphicAt | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | have d : ContDiffAt ℂ ⊤ i.he.symm _ :=
ContDiffAt.to_localInverse i.ha.contDiffAt i.has_dhe le_top | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | have e : i.h (c, i.z') = (c, i.fz') := by
simp only [Cinv.h, Cinv.fz', Cinv.f']
simp only [Cinv.z', (extChartAt I z).left_inv (mem_extChartAt_source _ _)] | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | rw [e] at d | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | exact (contDiffAt_iff_analytic_at2 le_top).mp d | case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ ↑i.he.symm (c, i.fz')
e : i.h (c, i.z') = (c, i.fz')
⊢ AnalyticAt ℂ ↑i.he.symm (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | simp only [Cinv.h, Cinv.fz', Cinv.f'] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ i.h (c, i.z') = (c, i.fz') | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ i.h (c, i.z') = (c, i.fz')
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.he_symm_holomorphic | [286, 1] | [293, 63] | simp only [Cinv.z', (extChartAt I z).left_inv (mem_extChartAt_source _ _)] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
d : ContDiffAt ℂ ⊤ (↑i.he.symm) (i.h (c, i.z'))
⊢ (c, ↑(extChartAt I (f c z)) (f c (↑(extChartAt I z).symm i.z'))) = (c, ↑(extChartAt I (f c z)) (f c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/Inverse.lean | ComplexInverseFun.Cinv.ga | [296, 1] | [303, 19] | apply (HolomorphicAt.extChartAt_symm (mem_extChartAt_target I z)).comp_of_eq | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (uncurry i.g) (c, f c z) | case gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
case e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 = ↑(extChartAt I z) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (uncurry i.g) (c, f c z)
TACTIC:
|
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