url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/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.... |
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... |
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) (↑(extCh... | 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) (↑(extCh... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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... | 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... |
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... | 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 ... |
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... |
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... | 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.... |
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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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)... | 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 :
((mfde... |
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)... | 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 :
(mfder... |
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... |
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 (mfder... | 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) =... |
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 (mfder... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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 ... | 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 ... |
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 ... | 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... |
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
TACTI... |
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... | 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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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... | 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... |
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... | 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 ... |
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
TAC... |
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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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)
... |
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
TAC... |
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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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)
... |
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)... | 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... |
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)... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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... | 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')) ... |
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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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, ↑(e... | 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')) ... |
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, ↑(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')
⊢ (↑i.he.symm (c, ↑(extChartAt I (f c z)) (f c z))).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
a : ↑i.he.symm (i.h (c, i.z')) = (c, i.... |
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 = ↑... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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, ↑(e... | 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.... |
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, ↑(e... | 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 ... |
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.h... | 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 ... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.h... | 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.pro... |
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.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 : Set (ℂ × S)
ht : (extChartAt (I.prod I) (c, z)).source ∩ ↑(extChartAt (I.prod I) (c, z)) ⁻¹' i.h... | 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.pro... |
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.h... | 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.pro... |
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 =... |
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.... |
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... | 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... |
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... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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... |
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... |
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... |
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))... | 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.... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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_... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝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))... | 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.pro... |
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))... | 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.pro... |
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))... | 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.pro... |
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))... | 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.pro... |
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))... | 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 ... | 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.pro... |
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 ... | 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.pr... | 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 : (ext... |
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.pr... | 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 ∩ ↑(extChar... | 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 : (... |
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 ∩ ↑(extChar... | 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 ∩ ↑(extChar... | 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 × ... |
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 ∩ ↑(extChar... | 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 × ... |
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 ∩ ↑(extChar... | 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 ∩ ↑(extChar... | 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 × ... |
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 ∩ ↑(extChar... | 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 ∩ ... | 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 × ... |
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 ∩ ... | 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 : ... |
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 ∩ ... | 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 ∩ ... | 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 : ... |
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 ∩ ... | 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... | 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 : ... |
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... | 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... |
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... | 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... |
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.pro... | 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 : (e... |
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 ... | 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 ... | 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 : (ext... |
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 ... | 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 : (ext... |
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))... | 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.pro... |
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) ↑... |
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... |
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 ... | 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)... |
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 ... | 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... | 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)... |
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... | 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 (... |
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... | 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... |
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... | 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... |
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... | 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... |
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