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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
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stringlengths 6
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stringlengths 1
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2.09M
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_a | [81, 1] | [83, 52] | induction' n with n h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ (f c)^[n] a = a | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (f c)^[0] a = a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ (f c)^[n] a = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_a | [81, 1] | [83, 52] | simp only [Function.iterate_zero_apply] | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (f c)^[0] a = a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (f c)^[0] a = a
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_a | [81, 1] | [83, 52] | simp only [Function.iterate_succ_apply', h, s.f0] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : (f c)^[n] a = a
⊢ (f c)^[n + 1] a = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | rw [@analyticAt_iff_holomorphicAt _ _ (ℂ × ℂ) (ModelProd ℂ ℂ) _ _ _ ℂ ℂ _ _ _ II I] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ AnalyticAt ℂ (uncurry s.fl) (c, 0) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (uncurry s.fl) (c, 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ AnalyticAt ℂ (uncurry s.fl) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | refine (((analyticAt_id _ _).sub analyticAt_const).holomorphicAt I I).comp ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (uncurry s.fl) (c, 0) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I
(fun x => (↑(extChartAt I a) ∘ f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (uncurry s.fl) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | refine (HolomorphicAt.extChartAt ?_).comp ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I
(fun x => (↑(extChartAt I a) ∘ f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (f (c, 0).1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) (c, 0).2) ∈ (extChartAt I a).source
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I
(fun x => (↑(extChartAt I a) ∘ f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | simp only [s.f0, extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans,
ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.coe_coe, Function.comp_apply, zero_add,
PartialEquiv.coe_trans_symm, PartialHomeomorph.coe_coe_symm, ModelWithCorners.toPartialEquiv_coe_symm,
ModelWithCorners.left_inv, PartialHomeomorph.left_inv, mem_chart_source, PartialEquiv.trans_source,
ModelWithCorners.source_eq, Set.preimage_univ, Set.inter_univ] | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (f (c, 0).1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) (c, 0).2) ∈ (extChartAt I a).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (f (c, 0).1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) (c, 0).2) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | refine (s.fa _).comp₂ holomorphicAt_fst ?_ | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a).symm ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1 ∘ ↑(extChartAt I a).symm) ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | refine (HolomorphicAt.extChartAt_symm ?_).comp ?_ | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a).symm ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0) | case refine_2.refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (fun z => z + ↑(extChartAt I a) a) (c, 0).2 ∈ (extChartAt I a).target
case refine_2.refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (fun z => z + ↑(extChartAt I a) a) x.2) (c, 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a).symm ((fun z => z + ↑(extChartAt I a) a) x.2)) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | simp only [extChartAt, PartialHomeomorph.extend, PartialEquiv.coe_trans,
ModelWithCorners.toPartialEquiv_coe, PartialHomeomorph.coe_coe, Function.comp_apply, zero_add,
PartialEquiv.trans_target, ModelWithCorners.target_eq, ModelWithCorners.toPartialEquiv_coe_symm,
Set.mem_inter_iff, Set.mem_range_self, Set.mem_preimage, ModelWithCorners.left_inv,
PartialHomeomorph.map_source, mem_chart_source, and_self_iff] | case refine_2.refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (fun z => z + ↑(extChartAt I a) a) (c, 0).2 ∈ (extChartAt I a).target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (fun z => z + ↑(extChartAt I a) a) (c, 0).2 ∈ (extChartAt I a).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fla | [86, 1] | [102, 72] | exact ((analyticAt_snd _).add analyticAt_const).holomorphicAt _ _ | case refine_2.refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (fun z => z + ↑(extChartAt I a) a) x.2) (c, 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ HolomorphicAt (I.prod I) I (fun x => (fun z => z + ↑(extChartAt I a) a) x.2) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphicAt_iter | [105, 1] | [110, 62] | induction' n with n h | S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
n : ℕ
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h p)) p | case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[0] (h p)) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
n : ℕ
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h p)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphicAt_iter | [105, 1] | [110, 62] | simp only [Function.iterate_zero, id] | case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[0] (h p)) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => h p) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[0] (h p)) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphicAt_iter | [105, 1] | [110, 62] | exact ha | case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => h p) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h p
⊢ HolomorphicAt (I.prod I) I (fun p => h p) p
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphicAt_iter | [105, 1] | [110, 62] | simp_rw [Function.iterate_succ'] | case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p | case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p) ∘ (f (g p))^[n]) (h✝ p)) p | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n + 1] (h✝ p)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphicAt_iter | [105, 1] | [110, 62] | exact (s.fa _).comp₂ ga h | case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p) ∘ (f (g p))^[n]) (h✝ p)) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁶ : TopologicalSpace S
inst✝⁵ : CompactSpace S
inst✝⁴ : T3Space S
inst✝³ : ChartedSpace ℂ S
inst✝² : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
g : ℂ × T → ℂ
h✝ : ℂ × T → S
p : ℂ × T
ga : HolomorphicAt (I.prod I) I g p
ha : HolomorphicAt (I.prod I) I h✝ p
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f (g p))^[n] (h✝ p)) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f (g p) ∘ (f (g p))^[n]) (h✝ p)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuous_iter | [113, 1] | [117, 78] | induction' n with n h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
n : ℕ
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => (f (g x))^[n] (h x) | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => (f (g x))^[0] (h x)
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
n : ℕ
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => (f (g x))^[n] (h x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuous_iter | [113, 1] | [117, 78] | simp only [Function.iterate_zero, id] | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => (f (g x))^[0] (h x)
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => h x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => (f (g x))^[0] (h x)
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuous_iter | [113, 1] | [117, 78] | exact hc | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => h x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
gc : Continuous g
hc : Continuous h
⊢ Continuous fun x => h x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuous_iter | [113, 1] | [117, 78] | simp_rw [Function.iterate_succ'] | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x))^[n + 1] (h✝ x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuous_iter | [113, 1] | [117, 78] | exact s.fa.continuous.comp (gc.prod_mk h) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
gc : Continuous g
hc : Continuous h✝
n : ℕ
h : Continuous fun x => (f (g x))^[n] (h✝ x)
⊢ Continuous fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousOn_iter | [120, 1] | [124, 88] | induction' n with n h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
n : ℕ
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => (f (g x))^[n] (h x)) t | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => (f (g x))^[0] (h x)) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
n : ℕ
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => (f (g x))^[n] (h x)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousOn_iter | [120, 1] | [124, 88] | simp only [Function.iterate_zero, id] | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => (f (g x))^[0] (h x)) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => h x) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => (f (g x))^[0] (h x)) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousOn_iter | [120, 1] | [124, 88] | exact hc | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => h x) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h t
⊢ ContinuousOn (fun x => h x) t
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousOn_iter | [120, 1] | [124, 88] | simp_rw [Function.iterate_succ'] | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) t | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x))^[n + 1] (h✝ x)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousOn_iter | [120, 1] | [124, 88] | exact s.fa.continuous.comp_continuousOn (gc.prod h) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
t : Set T
gc : ContinuousOn g t
hc : ContinuousOn h✝ t
n : ℕ
h : ContinuousOn (fun x => (f (g x))^[n] (h✝ x)) t
⊢ ContinuousOn (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousAt_iter | [127, 1] | [131, 81] | induction' n with n h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
n : ℕ
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => (f (g x))^[n] (h x)) x | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => (f (g x))^[0] (h x)) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
n : ℕ
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => (f (g x))^[n] (h x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousAt_iter | [127, 1] | [131, 81] | simp only [Function.iterate_zero, id] | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => (f (g x))^[0] (h x)) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => h x) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => (f (g x))^[0] (h x)) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousAt_iter | [127, 1] | [131, 81] | exact hc | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => h x) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h x
⊢ ContinuousAt (fun x => h x) x
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousAt_iter | [127, 1] | [131, 81] | simp_rw [Function.iterate_succ'] | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x))^[n + 1] (h✝ x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.continuousAt_iter | [127, 1] | [131, 81] | exact (s.fa _).continuousAt.comp (gc.prod h) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
T : Type
inst✝ : TopologicalSpace T
g : T → ℂ
h✝ : T → S
x : T
gc : ContinuousAt g x
hc : ContinuousAt h✝ x
n : ℕ
h : ContinuousAt (fun x => (f (g x))^[n] (h✝ x)) x
⊢ ContinuousAt (fun x => (f (g x) ∘ (f (g x))^[n]) (h✝ x)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphic_prod_iter | [139, 1] | [141, 66] | intro p | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) (I.prod I) fun p => (p.1, (f p.1)^[n] p.2) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) (I.prod I) fun p => (p.1, (f p.1)^[n] p.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphic_prod_iter | [139, 1] | [141, 66] | apply holomorphicAt_fst.prod | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.holomorphic_prod_iter | [139, 1] | [141, 66] | apply s.holomorphic_iter | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fl0 | [144, 1] | [146, 37] | simp only [Super.fl, _root_.fl, s.f0, Function.comp_apply, zero_add, PartialEquiv.left_inv,
mem_extChartAt_source, sub_self] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ s.fl c 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ s.fl c 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [Critical, mfderiv_eq_fderiv, Super.fl] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ Critical (s.fl c) 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ Critical (s.fl c) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | have p := (s.fla c).along_snd.leading_approx | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p :
(fun z =>
uncurry s.fl ((c, 0).1, z) -
(z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 •
leadingCoeff (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2) =o[𝓝 (c, 0).2]
fun z => (z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [sub_zero, Algebra.id.smul_eq_mul, Super.fl, s.fd, s.fc, mul_one, uncurry] at p | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p :
(fun z =>
uncurry s.fl ((c, 0).1, z) -
(z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 •
leadingCoeff (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2) =o[𝓝 (c, 0).2]
fun z => (z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p :
(fun z =>
uncurry s.fl ((c, 0).1, z) -
(z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 •
leadingCoeff (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2) =o[𝓝 (c, 0).2]
fun z => (z - (c, 0).2) ^ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | generalize hg : _root_.fl f a c = g | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
g : ℂ → ℂ
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
⊢ fderiv ℂ (_root_.fl f a c) 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | rw [hg] at p | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
g : ℂ → ℂ
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
p : (fun z => _root_.fl f a c z - z ^ d) =o[𝓝 0] fun z => z ^ d
g : ℂ → ℂ
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | have g0 : g 0 = 0 := by rw [← hg]; exact s.fl0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ fderiv ℂ g 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ fderiv ℂ g 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | apply HasFDerivAt.fderiv | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ fderiv ℂ g 0 = 0 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ HasFDerivAt g 0 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ fderiv ℂ g 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [hasFDerivAt_iff_isLittleO_nhds_zero, ContinuousLinearMap.zero_apply, sub_zero,
zero_add, g0] | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ HasFDerivAt g 0 0 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ HasFDerivAt g 0 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | have od : (fun z : ℂ ↦ z ^ d) =o[𝓝 0] (fun z ↦ z) := by
rw [Asymptotics.isLittleO_iff]; intro e ep
apply ((@Metric.isOpen_ball ℂ _ 0 (min 1 e)).eventually_mem (mem_ball_self (by bound))).mp
refine eventually_of_forall fun z b ↦ ?_
simp only at b; rw [mem_ball_zero_iff, Complex.norm_eq_abs, lt_min_iff] at b
simp only [Complex.norm_eq_abs, Complex.abs.map_pow]
rw [← Nat.sub_add_cancel s.d2, pow_add, pow_two]
calc abs z ^ (d - 2) * (abs z * abs z)
_ ≤ (1:ℝ) ^ (d - 2) * (abs z * abs z) := by bound
_ = abs z * abs z := by simp only [one_pow, one_mul]
_ ≤ e * abs z := by bound | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | have p' := (p.trans od).add od | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x - x ^ d + x ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [sub_add_cancel] at p' | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x - x ^ d + x ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x - x ^ d + x ^ d) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | refine p'.congr_left ?_ | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ ∀ (x : ℂ), g x = g x - 0 x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ (fun h => g h - 0 h) =o[𝓝 0] fun h => h
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | intro z | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ ∀ (x : ℂ), g x = g x - 0 x | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
z : ℂ
⊢ g z = g z - 0 z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
⊢ ∀ (x : ℂ), g x = g x - 0 x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | exact (sub_zero _).symm | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
z : ℂ
⊢ g z = g z - 0 z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
od : (fun z => z ^ d) =o[𝓝 0] fun z => z
p' : (fun x => g x) =o[𝓝 0] fun z => z
z : ℂ
⊢ g z = g z - 0 z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | rw [← hg] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ g 0 = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ _root_.fl f a c 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ g 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | exact s.fl0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ _root_.fl f a c 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
⊢ _root_.fl f a c 0 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | rw [Asymptotics.isLittleO_iff] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ (fun z => z ^ d) =o[𝓝 0] fun z => z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ c * ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ (fun z => z ^ d) =o[𝓝 0] fun z => z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | intro e ep | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ c * ‖x‖ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ e * ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ c * ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | apply ((@Metric.isOpen_ball ℂ _ 0 (min 1 e)).eventually_mem (mem_ball_self (by bound))).mp | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ e * ‖x‖ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ ball 0 (min 1 e) → ‖x ^ d‖ ≤ e * ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖x ^ d‖ ≤ e * ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | refine eventually_of_forall fun z b ↦ ?_ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ ball 0 (min 1 e) → ‖x ^ d‖ ≤ e * ‖x‖ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : z ∈ ball 0 (min 1 e)
⊢ ‖z ^ d‖ ≤ e * ‖z‖ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, x ∈ ball 0 (min 1 e) → ‖x ^ d‖ ≤ e * ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | rw [mem_ball_zero_iff, Complex.norm_eq_abs, lt_min_iff] at b | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : z ∈ ball 0 (min 1 e)
⊢ ‖z ^ d‖ ≤ e * ‖z‖ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ ‖z ^ d‖ ≤ e * ‖z‖ | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : z ∈ ball 0 (min 1 e)
⊢ ‖z ^ d‖ ≤ e * ‖z‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [Complex.norm_eq_abs, Complex.abs.map_pow] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ ‖z ^ d‖ ≤ e * ‖z‖ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ d ≤ e * Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ ‖z ^ d‖ ≤ e * ‖z‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | rw [← Nat.sub_add_cancel s.d2, pow_add, pow_two] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ d ≤ e * Complex.abs z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ (d - 2) * (Complex.abs z * Complex.abs z) ≤ e * Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ d ≤ e * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | calc abs z ^ (d - 2) * (abs z * abs z)
_ ≤ (1:ℝ) ^ (d - 2) * (abs z * abs z) := by bound
_ = abs z * abs z := by simp only [one_pow, one_mul]
_ ≤ e * abs z := by bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ (d - 2) * (Complex.abs z * Complex.abs z) ≤ e * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ (d - 2) * (Complex.abs z * Complex.abs z) ≤ e * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ 0 < min 1 e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
⊢ 0 < min 1 e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ (d - 2) * (Complex.abs z * Complex.abs z) ≤ 1 ^ (d - 2) * (Complex.abs z * Complex.abs z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z ^ (d - 2) * (Complex.abs z * Complex.abs z) ≤ 1 ^ (d - 2) * (Complex.abs z * Complex.abs z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | simp only [one_pow, one_mul] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ 1 ^ (d - 2) * (Complex.abs z * Complex.abs z) = Complex.abs z * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ 1 ^ (d - 2) * (Complex.abs z * Complex.abs z) = Complex.abs z * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_0 | [149, 1] | [172, 35] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z * Complex.abs z ≤ e * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
g : ℂ → ℂ
p : (fun z => g z - z ^ d) =o[𝓝 0] fun z => z ^ d
hg : _root_.fl f a c = g
g0 : g 0 = 0
e : ℝ
ep : 0 < e
z : ℂ
b : Complex.abs z < 1 ∧ Complex.abs z < e
⊢ Complex.abs z * Complex.abs z ≤ e * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | have h := s.critical_0 c | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ Critical (f c) a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
⊢ Critical (f c) a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ Critical (f c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | have e := PartialEquiv.left_inv _ (mem_extChartAt_source I a) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
⊢ Critical (f c) a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
⊢ Critical (f c) a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
⊢ Critical (f c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | contrapose h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
⊢ Critical (f c) a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : ¬Critical (f c) a
⊢ ¬Critical (s.fl c) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
h : Critical (s.fl c) 0
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
⊢ Critical (f c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | simp only [Critical, Super.fl, fl, ← ne_eq] at h ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : ¬Critical (f c) a
⊢ ¬Critical (s.fl c) 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (_root_.fl f a c) 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : ¬Critical (f c) a
⊢ ¬Critical (s.fl c) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | simp only [mfderiv_eq_fderiv, _root_.fl, Function.comp] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (_root_.fl f a c) 0 ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ fderiv ℂ (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) - ↑(extChartAt I a) a)
0 ≠
0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (_root_.fl f a c) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | rw [fderiv_sub_const, ←mfderiv_eq_fderiv] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ fderiv ℂ (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) - ↑(extChartAt I a) a)
0 ≠
0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)))) 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ fderiv ℂ (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) - ↑(extChartAt I a) a)
0 ≠
0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | apply mderiv_comp_ne_zero' (extChartAt_mderiv_ne_zero' ?_) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)))) 0 ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) 0 ≠ 0
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ f c (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a) (f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)))) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | apply mderiv_comp_ne_zero' (f := f c) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) 0 ≠ 0 | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ≠ 0
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a))) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | rw [zero_add, e] | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ≠ 0 | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) a ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | exact h | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) a ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (f c) a ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | apply mderiv_comp_ne_zero' (extChartAt_symm_mderiv_ne_zero' ?_) | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 ≠ 0 | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x + ↑(extChartAt I a) a) 0 ≠ 0
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | rw [mfderiv_eq_fderiv, fderiv_add_const, ←mfderiv_eq_fderiv] | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x + ↑(extChartAt I a) a) 0 ≠ 0 | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x) 0 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x + ↑(extChartAt I a) a) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | exact id_mderiv_ne_zero | case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x) 0 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ mfderiv I I (fun x => x) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | rw [zero_add] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | apply mem_extChartAt_target | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | simp only [zero_add, e, s.f0] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ f c (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ∈ (extChartAt I a).source | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ a ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ f c (↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a)) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.critical_a | [175, 1] | [188, 32] | apply mem_extChartAt_source | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ a ∈ (extChartAt I a).source | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
h : mfderiv I I (f c) a ≠ 0
⊢ a ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | refine ⟨(s.fa _).along_snd, ?_⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ NontrivialHolomorphicAt (f c) a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ f c a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ NontrivialHolomorphicAt (f c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [s.f0] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ f c a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ f c a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | contrapose n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ a | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ¬∃ᶠ (w : S) in 𝓝 a, f c w ≠ a
⊢ ¬∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
⊢ ∃ᶠ (w : S) in 𝓝 a, f c w ≠ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [Filter.not_frequently, not_not, Super.fl, fl, Function.comp, sub_eq_zero] at n ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ¬∃ᶠ (w : S) in 𝓝 a, f c w ≠ a
⊢ ¬∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ¬∃ᶠ (w : S) in 𝓝 a, f c w ≠ a
⊢ ¬∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | have gc : ContinuousAt (fun x ↦ (extChartAt I a).symm (x + extChartAt I a a)) 0 := by
refine (continuousAt_extChartAt_symm I a).comp_of_eq ?_ (by simp only [zero_add])
exact continuousAt_id.add continuousAt_const | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [ContinuousAt, zero_add, PartialEquiv.left_inv _ (mem_extChartAt_source _ _)] at gc | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | refine (gc.eventually n).mp (eventually_of_forall ?_) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ (x : ℂ), f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a → _root_.fl f a c x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, _root_.fl f a c x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | intro x h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ (x : ℂ), f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a → _root_.fl f a c x = 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
x : ℂ
h : f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a
⊢ _root_.fl f a c x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
⊢ ∀ (x : ℂ), f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a → _root_.fl f a c x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [_root_.fl, Function.comp, h, sub_self] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
x : ℂ
h : f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a
⊢ _root_.fl f a c x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
gc : Tendsto (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) (𝓝 0) (𝓝 a)
x : ℂ
h : f c (↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = a
⊢ _root_.fl f a c x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | have e := (nontrivialHolomorphicAt_of_order (s.fla c).along_snd ?_).nonconst | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 (c, 0).2, uncurry s.fl ((c, 0).1, w) ≠ uncurry s.fl ((c, 0).1, (c, 0).2)
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [s.fl0, uncurry] at e | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 (c, 0).2, uncurry s.fl ((c, 0).1, w) ≠ uncurry s.fl ((c, 0).1, (c, 0).2)
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 (c, 0).2, uncurry s.fl ((c, 0).1, w) ≠ uncurry s.fl ((c, 0).1, (c, 0).2)
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | exact e | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
e : ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
⊢ ∃ᶠ (w : ℂ) in 𝓝 0, s.fl c w ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [Super.fl, s.fd, uncurry] | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 ≠ 0 | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ d ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ orderAt (fun y => uncurry s.fl ((c, 0).1, y)) (c, 0).2 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | exact s.d0 | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ d ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ d ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | refine (continuousAt_extChartAt_symm I a).comp_of_eq ?_ (by simp only [zero_add]) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ContinuousAt (fun x => x + ↑(extChartAt I a) a) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | exact continuousAt_id.add continuousAt_const | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ContinuousAt (fun x => x + ↑(extChartAt I a) a) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ ContinuousAt (fun x => x + ↑(extChartAt I a) a) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_nontrivial | [191, 1] | [204, 63] | simp only [zero_add] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ 0 + ↑(extChartAt I a) a = ↑(extChartAt I a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
c : ℂ
n : ∀ᶠ (x : S) in 𝓝 a, f c x = a
⊢ 0 + ↑(extChartAt I a) a = ↑(extChartAt I a) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_in_chart | [207, 1] | [211, 98] | apply ContinuousAt.eventually_mem_nhd | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source | case fc
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ContinuousAt (fun y => f y.1 y.2) (c, a)
case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_in_chart | [207, 1] | [211, 98] | exact (s.fa.continuous.comp continuous_id).continuousAt | case fc
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ContinuousAt (fun y => f y.1 y.2) (c, a)
case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2) | case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2) | Please generate a tactic in lean4 to solve the state.
STATE:
case fc
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ContinuousAt (fun y => f y.1 y.2) (c, a)
case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_in_chart | [207, 1] | [211, 98] | simp only [s.f0, Function.comp_id, Function.uncurry_apply_pair, extChartAt_source_mem_nhds I a] | case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case m
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (extChartAt I a).source ∈ 𝓝 (f (c, a).1 (c, a).2)
TACTIC:
|
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