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/OneDimension.lean
|
critical_iter
|
[271, 1]
|
[280, 43]
|
rw [Function.iterate_succ', Critical,
mfderiv_comp z (fa _).mdifferentiableAt (fa.iter _ _).mdifferentiableAt,
mderiv_comp_eq_zero_iff] at c
|
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : Critical f^[n + 1] z
⊢ Precritical f z
|
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0 ∨ mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : Critical f^[n + 1] z
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
critical_iter
|
[271, 1]
|
[280, 43]
|
cases' c with c c
|
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0 ∨ mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
case succ.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0
⊢ Precritical f z
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0 ∨ mfderiv I I f^[n] z = 0
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
critical_iter
|
[271, 1]
|
[280, 43]
|
use n, c
|
case succ.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0
⊢ Precritical f z
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f (f^[n] z) = 0
⊢ Precritical f z
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
critical_iter
|
[271, 1]
|
[280, 43]
|
exact h c
|
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → S
z : S
fa : Holomorphic I I f
n : ℕ
h : Critical f^[n] z → Precritical f z
c : mfderiv I I f^[n] z = 0
⊢ Precritical f z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
apply HolomorphicAt.analyticAt II I
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ AnalyticAt ℂ (uncurry (_root_.inChart f c z)) (c, ↑(_root_.extChartAt I z) z)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (uncurry (_root_.inChart f c z)) (c, ↑(_root_.extChartAt I z) z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ AnalyticAt ℂ (uncurry (_root_.inChart f c z)) (c, ↑(_root_.extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
apply (HolomorphicAt.extChartAt (mem_extChartAt_source I (f c z))).comp_of_eq
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (uncurry (_root_.inChart f c z)) (c, ↑(_root_.extChartAt I z) z)
|
case gh
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => f x.1 (↑(_root_.extChartAt I z).symm x.2)) (c, ↑(_root_.extChartAt I z) z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (uncurry (_root_.inChart f c z)) (c, ↑(_root_.extChartAt I z) z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
apply fa.comp₂_of_eq holomorphicAt_fst
|
case gh
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => f x.1 (↑(_root_.extChartAt I z).symm x.2)) (c, ↑(_root_.extChartAt I z) z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(_root_.extChartAt I z).symm x.2) (c, ↑(_root_.extChartAt I z) z)
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case gh
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => f x.1 (↑(_root_.extChartAt I z).symm x.2)) (c, ↑(_root_.extChartAt I z) z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
apply (HolomorphicAt.extChartAt_symm (mem_extChartAt_target I z)).comp_of_eq holomorphicAt_snd
|
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(_root_.extChartAt I z).symm x.2) (c, ↑(_root_.extChartAt I z) z)
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, ↑(_root_.extChartAt I z) z).2 = ↑(_root_.extChartAt I z) z
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(_root_.extChartAt I z).symm x.2) (c, ↑(_root_.extChartAt I z) z)
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
repeat' simp only [PartialEquiv.left_inv _ (mem_extChartAt_source I z)]
|
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, ↑(_root_.extChartAt I z) z).2 = ↑(_root_.extChartAt I z) z
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case gh.ga
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ (c, ↑(_root_.extChartAt I z) z).2 = ↑(_root_.extChartAt I z) z
case gh.e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ((c, ↑(_root_.extChartAt I z) z).1, ↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = (c, z)
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
HolomorphicAt.inChart
|
[295, 1]
|
[302, 74]
|
simp only [PartialEquiv.left_inv _ (mem_extChartAt_source I z)]
|
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case e
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ f (c, ↑(_root_.extChartAt I z) z).1 (↑(_root_.extChartAt I z).symm (c, ↑(_root_.extChartAt I z) z).2) = f c z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
apply (fa.continuousAt.eventually_mem ((isOpen_extChartAt_source I (f c z)).mem_nhds
(mem_extChartAt_source I (f c z)))).mp
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
apply ((isOpen_extChartAt_source II (c, z)).eventually_mem (mem_extChartAt_source _ _)).mp
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
refine fa.eventually.mp (eventually_of_forall ?_)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ (x : ℂ × S),
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, z),
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
intro ⟨e, w⟩ fa m fm
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ (x : ℂ × S),
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
m : (e, w) ∈ (extChartAt (I.prod I) (c, z)).source
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
⊢ ∀ (x : ℂ × S),
HolomorphicAt (I.prod I) I (uncurry f) x →
x ∈ (extChartAt (I.prod I) (c, z)).source →
uncurry f x ∈ (extChartAt I (f c z)).source →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c z x.1) (↑(extChartAt I z) x.2) = 0)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [extChartAt_prod, PartialEquiv.prod_source, extChartAt_eq_refl, PartialEquiv.refl_source,
mem_prod, mem_univ, true_and_iff] at m
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
m : (e, w) ∈ (extChartAt (I.prod I) (c, z)).source
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
m : (e, w) ∈ (extChartAt (I.prod I) (c, z)).source
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [uncurry] at fm
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : uncurry f (e, w) ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
have m' := PartialEquiv.map_source _ m
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [← mfderiv_eq_zero_iff_deriv_eq_zero]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f (e, w).1) (e, w).2 = 0 ↔ deriv (inChart f c z (e, w).1) (↑(extChartAt I z) (e, w).2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
have cd : HolomorphicAt I I (extChartAt I (f c z)) (f e w) := HolomorphicAt.extChartAt fm
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
have fd : HolomorphicAt I I (f e ∘ (extChartAt I z).symm) (extChartAt I z w) := by
simp only [Function.comp]
exact HolomorphicAt.comp_of_eq fa.along_snd (HolomorphicAt.extChartAt_symm m')
(PartialEquiv.right_inv _ m)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
have ce : inChart f c z e = extChartAt I (f c z) ∘ f e ∘ (extChartAt I z).symm := rfl
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
rw [ce, mfderiv_comp_of_eq cd.mdifferentiableAt fd.mdifferentiableAt ?blah,
mfderiv_comp_of_eq fa.along_snd.mdifferentiableAt
(HolomorphicAt.extChartAt_symm m').mdifferentiableAt]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
(mfderiv I I (↑(extChartAt I (f c z))) ((f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w))).comp
((mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w))).comp
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w))) =
0
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ ↑(extChartAt I z).symm (↑(extChartAt I z) w) = w
case blah
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w) = f e w
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔ mfderiv I I (inChart f c z e) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [Function.comp]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I z).symm x)) (↑(extChartAt I z) w)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
exact HolomorphicAt.comp_of_eq fa.along_snd (HolomorphicAt.extChartAt_symm m')
(PartialEquiv.right_inv _ m)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I z).symm x)) (↑(extChartAt I z) w)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I z).symm x)) (↑(extChartAt I z) w)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [mderiv_comp_eq_zero_iff, Function.comp]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
(mfderiv I I (↑(extChartAt I (f c z))) ((f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w))).comp
((mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w))).comp
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w))) =
0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e (↑(extChartAt I z).symm (↑(extChartAt I z) w))) = 0 ∨
mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w)) = 0 ∨
mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
(mfderiv I I (↑(extChartAt I (f c z))) ((f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w))).comp
((mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w))).comp
(mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w))) =
0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
rw [(extChartAt I z).left_inv m]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e (↑(extChartAt I z).symm (↑(extChartAt I z) w))) = 0 ∨
mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w)) = 0 ∨
mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e w) = 0 ∨
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e (↑(extChartAt I z).symm (↑(extChartAt I z) w))) = 0 ∨
mfderiv I I (fun y => f e y) (↑(extChartAt I z).symm (↑(extChartAt I z) w)) = 0 ∨
mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [extChartAt_mderiv_ne_zero' fm, false_or]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e w) = 0 ∨
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (↑(extChartAt I (f c z))) (f e w) = 0 ∨
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
constructor
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 →
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0 →
mfderiv I I (f e) w = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 ↔
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
intro h
|
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 →
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (f e) w = 0 →
mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
left
|
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
|
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
exact h
|
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (f e) w = 0
⊢ mfderiv I I (fun y => f e y) w = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
intro h
|
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0 →
mfderiv I I (f e) w = 0
|
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0 →
mfderiv I I (f e) w = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
cases' h with h h
|
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0
⊢ mfderiv I I (f e) w = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0 ∨ mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
exact h
|
case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0
⊢ mfderiv I I (f e) w = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inl
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (fun y => f e y) w = 0
⊢ mfderiv I I (f e) w = 0
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simpa only using extChartAt_symm_mderiv_ne_zero' m' h
|
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.inr
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
h : mfderiv I I (↑(extChartAt I z).symm) (↑(extChartAt I z) w) = 0
⊢ mfderiv I I (f e) w = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
exact PartialEquiv.left_inv _ m
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ ↑(extChartAt I z).symm (↑(extChartAt I z) w) = w
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ ↑(extChartAt I z).symm (↑(extChartAt I z) w) = w
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
inChart_critical
|
[305, 1]
|
[334, 57]
|
simp only [Function.comp, PartialEquiv.left_inv _ m]
|
case blah
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w) = f e w
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case blah
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa✝ : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
e : ℂ
w : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (e, w)
fm : f e w ∈ (extChartAt I (f c z)).source
m : w ∈ (extChartAt I z).source
m' : ↑(extChartAt I z) w ∈ (extChartAt I z).target
cd : HolomorphicAt I I (↑(extChartAt I (f c z))) (f e w)
fd : HolomorphicAt I I (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w)
ce : inChart f c z e = ↑(extChartAt I (f c z)) ∘ f e ∘ ↑(extChartAt I z).symm
⊢ (f e ∘ ↑(extChartAt I z).symm) (↑(extChartAt I z) w) = f e w
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
set g := inChart f c z
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
have g0 := inChart_critical fa
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
refine g0.mp (g0n.mp (eventually_of_forall fun w g0 e ↦ ?_))
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ mfderiv I I (f w.1) w.2 ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
rw [Ne, e]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ mfderiv I I (f w.1) w.2 ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ ¬deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ mfderiv I I (f w.1) w.2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
exact g0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ ¬deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
g0n : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
w : ℂ × S
g0 : deriv (g w.1) (↑(extChartAt I z) w.2) ≠ 0
e : mfderiv I I (f w.1) w.2 = 0 ↔ deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
⊢ ¬deriv (inChart f c z w.1) (↑(extChartAt I z) w.2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
refine ContinuousAt.eventually_ne ?_ ?_
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), deriv (g p.1) (↑(extChartAt I z) p.2) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
have e : (fun p : ℂ × S ↦ deriv (g p.1) (extChartAt I z p.2)) =
(fun p : ℂ × ℂ ↦ deriv (g p.1) p.2) ∘ fun p : ℂ × S ↦ (p.1, extChartAt I z p.2) := rfl
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
rw [e]
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt ((fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)) (c, z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt (fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
exact fa.inChart.deriv2.continuousAt.comp_of_eq
(continuousAt_fst.prod ((continuousAt_extChartAt I z).comp_of_eq continuousAt_snd rfl))
rfl
|
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt ((fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)) (c, z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
e :
(fun p => deriv (g p.1) (↑(extChartAt I z) p.2)) =
(fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)
⊢ ContinuousAt ((fun p => deriv (g p.1) p.2) ∘ fun p => (p.1, ↑(extChartAt I z) p.2)) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
contrapose f0
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : ¬deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
⊢ ¬mfderiv I I (f c) z ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
f0 : mfderiv I I (f c) z ≠ 0
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
⊢ deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
simp only [not_not, Function.comp] at f0 ⊢
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : ¬deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
⊢ ¬mfderiv I I (f c) z ≠ 0
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ mfderiv I I (f c) z = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : ¬deriv (g (c, z).1) (↑(extChartAt I z) (c, z).2) ≠ 0
⊢ ¬mfderiv I I (f c) z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
rw [g0.self_of_nhds]
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ mfderiv I I (f c) z = 0
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ deriv (inChart f c z (c, z).1) (↑(extChartAt I z) (c, z).2) = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ mfderiv I I (f c) z = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually'
|
[337, 1]
|
[352, 23]
|
exact f0
|
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ deriv (inChart f c z (c, z).1) (↑(extChartAt I z) (c, z).2) = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
g : ℂ → ℂ → ℂ := inChart f c z
g0 : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), mfderiv I I (f p.1) p.2 = 0 ↔ deriv (inChart f c z p.1) (↑(extChartAt I z) p.2) = 0
f0 : deriv (g c) (↑(extChartAt I z) z) = 0
⊢ deriv (inChart f c z (c, z).1) (↑(extChartAt I z) (c, z).2) = 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
set c : ℂ := 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
set g : ℂ → S → T := fun _ z ↦ f z
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
have ga : HolomorphicAt II I (uncurry g) (c, z) := by
have e : uncurry g = f ∘ fun p ↦ p.2 := rfl; rw [e]
apply HolomorphicAt.comp_of_eq fa holomorphicAt_snd; simp only
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
have pc : Tendsto (fun z ↦ (c, z)) (𝓝 z) (𝓝 (c, z)) := continuousAt_const.prod continuousAt_id
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
exact pc.eventually (mfderiv_ne_zero_eventually' ga f0)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, z)
pc : Tendsto (fun z => (c, z)) (𝓝 z) (𝓝 (c, z))
⊢ ∀ᶠ (w : S) in 𝓝 z, mfderiv I I f w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
have e : uncurry g = f ∘ fun p ↦ p.2 := rfl
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
rw [e]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (uncurry g) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
apply HolomorphicAt.comp_of_eq fa holomorphicAt_snd
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ HolomorphicAt (I.prod I) I (f ∘ fun p => p.2) (c, z)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
mfderiv_ne_zero_eventually
|
[355, 1]
|
[363, 58]
|
simp only
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S → T
z : S
fa : HolomorphicAt I I f z
f0 : mfderiv I I f z ≠ 0
c : ℂ := 0
g : ℂ → S → T := fun x z => f z
e : uncurry g = f ∘ fun p => p.2
⊢ (c, z).2 = z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isOpen_noncritical
|
[366, 1]
|
[368, 89]
|
rw [isOpen_iff_eventually]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsOpen {p | ¬Critical (f p.1) p.2}
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsOpen {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isOpen_noncritical
|
[366, 1]
|
[368, 89]
|
intro ⟨c, z⟩ m
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2}
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ ∀ x ∈ {p | ¬Critical (f p.1) p.2}, ∀ᶠ (y : ℂ × S) in 𝓝 x, y ∈ {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isOpen_noncritical
|
[366, 1]
|
[368, 89]
|
exact mfderiv_ne_zero_eventually' (fa _) m
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : ℂ
z : S
m : (c, z) ∈ {p | ¬Critical (f p.1) p.2}
⊢ ∀ᶠ (y : ℂ × S) in 𝓝 (c, z), y ∈ {p | ¬Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isClosed_critical
|
[371, 1]
|
[374, 49]
|
have c := (isOpen_noncritical fa).isClosed_compl
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsClosed {p | Critical (f p.1) p.2}
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2}
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isClosed_critical
|
[371, 1]
|
[374, 49]
|
simp only [compl_setOf, not_not] at c
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2}
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2}
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {p | ¬Critical (f p.1) p.2}ᶜ
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
isClosed_critical
|
[371, 1]
|
[374, 49]
|
exact c
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : ℂ → S → T
fa : Holomorphic (I.prod I) I (uncurry f)
c : IsClosed {a | Critical (f a.1) a.2}
⊢ IsClosed {p | Critical (f p.1) p.2}
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
rw [holomorphic_iff]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Holomorphic (I.prod I) I f
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Holomorphic (I.prod I) I f
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
use fc
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
|
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ Continuous f ∧
∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
intro p
|
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
|
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
⊢ ∀ (x : S × T),
AnalyticAt ℂ (↑(extChartAt I (f x)) ∘ f ∘ ↑(extChartAt (I.prod I) x).symm) (↑(extChartAt (I.prod I) x) x)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
apply osgood_at'
|
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p)
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ AnalyticAt ℂ (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
have fm : ∀ᶠ q in 𝓝 (extChartAt II p p),
f ((extChartAt II p).symm q) ∈ (extChartAt I (f p)).source := by
refine (fc.continuousAt.comp (continuousAt_extChartAt_symm II p)).eventually_mem
((isOpen_extChartAt_source I (f p)).mem_nhds ?_)
simp only [Function.comp, (extChartAt II p).left_inv (mem_extChartAt_source _ _)]
apply mem_extChartAt_source
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
apply ((isOpen_extChartAt_target II p).eventually_mem (mem_extChartAt_target II p)).mp
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
refine fm.mp (eventually_of_forall fun q fm m ↦ ⟨?_, ?_, ?_⟩)
|
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
|
case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
⊢ ∀ᶠ (x : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
x ∈ (extChartAt (I.prod I) p).target →
ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) x ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, x.2)) x.1 ∧
AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (x.1, z)) x.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
refine (fc.continuousAt.comp (continuousAt_extChartAt_symm II p)).eventually_mem
((isOpen_extChartAt_source I (f p)).mem_nhds ?_)
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ ∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
simp only [Function.comp, (extChartAt II p).left_inv (mem_extChartAt_source _ _)]
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ (f ∘ ↑(extChartAt (I.prod I) p).symm) (↑(extChartAt (I.prod I) p) p) ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
apply mem_extChartAt_source
|
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
⊢ f p ∈ (extChartAt I (f p)).source
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
exact (continuousAt_extChartAt' I fm).comp_of_eq
(fc.continuousAt.comp (continuousAt_extChartAt_symm'' _ m)) rfl
|
case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ ContinuousAt (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) q
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
apply HolomorphicAt.analyticAt I I
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
refine (HolomorphicAt.extChartAt fm).comp_of_eq ?_ rfl
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
rw [extChartAt_prod] at m
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
simp only [Function.comp, extChartAt_prod, PartialEquiv.prod_symm, PartialEquiv.prod_coe,
PartialEquiv.prod_target, mem_prod_eq] at m ⊢
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (z, q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
exact (f0 _ _).comp (HolomorphicAt.extChartAt_symm m.1)
|
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm z, ↑(extChartAt I p.2).symm q.2)) q.1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
apply HolomorphicAt.analyticAt I I
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ AnalyticAt ℂ (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
refine (HolomorphicAt.extChartAt fm).comp_of_eq ?_ rfl
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (↑(extChartAt I (f p)) ∘ f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
rw [extChartAt_prod] at m
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ (extChartAt (I.prod I) p).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
simp only [Function.comp, extChartAt_prod, PartialEquiv.prod_symm, PartialEquiv.prod_coe,
PartialEquiv.prod_target, mem_prod_eq] at m ⊢
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q ∈ ((extChartAt I p.1).prod (extChartAt I p.2)).target
⊢ HolomorphicAt I I (fun z => (f ∘ ↑(extChartAt (I.prod I) p).symm) (q.1, z)) q.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/OneDimension.lean
|
osgoodManifold
|
[379, 1]
|
[404, 60]
|
exact (f1 _ _).comp (HolomorphicAt.extChartAt_symm m.2)
|
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
f : S × T → U
fc : Continuous f
f0 : ∀ (x : S) (y : T), HolomorphicAt I I (fun x => f (x, y)) x
f1 : ∀ (x : S) (y : T), HolomorphicAt I I (fun y => f (x, y)) y
p : S × T
fm✝ :
∀ᶠ (q : ℂ × ℂ) in 𝓝 (↑(extChartAt (I.prod I) p) p),
f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
q : ℂ × ℂ
fm : f (↑(extChartAt (I.prod I) p).symm q) ∈ (extChartAt I (f p)).source
m : q.1 ∈ (extChartAt I p.1).target ∧ q.2 ∈ (extChartAt I p.2).target
⊢ HolomorphicAt I I (fun z => f (↑(extChartAt I p.1).symm q.1, ↑(extChartAt I p.2).symm z)) q.2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
intro z zs
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
⊢ ∀ c ∈ interior s, ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f c) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap c s t))
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
⊢ ∀ c ∈ interior s, ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f c) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap c s t))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
rcases Metric.isOpen_iff.mp isOpen_interior z zs with ⟨r, rp, rh⟩
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
exists r, rp
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∃ r, 0 < r ∧ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
intro t tp tr
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
⊢ ∀ (s : ℝ), 0 < s → s < r → g (f z) ≤ ⨍ (t : ℝ) in itau, g (f (circleMap z s t))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
have cs : closedBall z t ⊆ s :=
_root_.trans (Metric.closedBall_subset_ball tr) (_root_.trans rh interior_subset)
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
simp only [fh.mean z t tp cs]
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (f z) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
have n := NiceVolume.itau
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
apply ConvexOn.map_set_average_le gc c.continuousOn isClosed_univ n.ne_zero n.ne_top
|
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
|
case intro.intro.hfs
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict itau, f (circleMap z t x) ∈ univ
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ g (⨍ (t_1 : ℝ) in itau, f (circleMap z t t_1)) ≤ ⨍ (t_1 : ℝ) in itau, g (f (circleMap z t t_1))
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
simp only [Set.mem_univ, Filter.eventually_true]
|
case intro.intro.hfs
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict itau, f (circleMap z t x) ∈ univ
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.hfs
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict itau, f (circleMap z t x) ∈ univ
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
exact (fh.cont.mono cs).integrableOn_sphere tp
|
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.hfi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (fun x => f (circleMap z t x)) itau volume
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.convex
|
[88, 1]
|
[102, 77]
|
exact ((c.comp_continuousOn fh.cont).mono cs).integrableOn_sphere tp
|
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.hgi
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → E
s : Set ℂ
g : E → ℝ
fh : HarmonicOn f s
c : Continuous g
gc : ConvexOn ℝ univ g
z : ℂ
zs : z ∈ interior s
r : ℝ
rp : r > 0
rh : ball z r ⊆ interior s
t : ℝ
tp : 0 < t
tr : t < r
cs : closedBall z t ⊆ s
n : NiceVolume itau
⊢ IntegrableOn (g ∘ fun x => f (circleMap z t x)) itau volume
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.subharmonicOn
|
[105, 1]
|
[108, 65]
|
have e : (fun z ↦ f z) = fun z ↦ (fun x ↦ x) (f z) := rfl
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
⊢ SubharmonicOn (fun z => f z) s
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => f z) s
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
⊢ SubharmonicOn (fun z => f z) s
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.subharmonicOn
|
[105, 1]
|
[108, 65]
|
rw [e]
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => f z) s
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => (fun x => x) (f z)) s
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => f z) s
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Hartogs/Subharmonic.lean
|
HarmonicOn.subharmonicOn
|
[105, 1]
|
[108, 65]
|
exact h.convex continuous_id (convexOn_id convex_univ)
|
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => (fun x => x) (f z)) s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝¹⁵ : RCLike S
inst✝¹⁴ : SMulCommClass ℝ S S
T : Type
inst✝¹³ : RCLike T
inst✝¹² : SMulCommClass ℝ T T
E : Type
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : CompleteSpace E
inst✝⁹ : NormedSpace ℝ E
F : Type
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : CompleteSpace F
inst✝⁶ : NormedSpace ℝ F
H : Type
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : CompleteSpace H
inst✝³ : NormedSpace ℂ H
inst✝² : SecondCountableTopology E
inst✝¹ : SecondCountableTopology F
inst✝ : SecondCountableTopology H
f : ℂ → ℝ
s : Set ℂ
h : HarmonicOn f s
e : (fun z => f z) = fun z => (fun x => x) (f z)
⊢ SubharmonicOn (fun z => (fun x => x) (f z)) s
TACTIC:
|
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