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stringlengths 6
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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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