url
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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_iff_attracts | [419, 1] | [423, 29] | exact ⟨n, h _ (le_refl _)⟩ | case mpr.intro
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
h✝ : Attracts (f c) z a
n : ℕ
h : ∀ (n_1 : ℕ), n ≤ n_1 → (f c)^[n_1] z ∈ {z | (c, z) ∈ s.near}
⊢ (c, z) ∈ s.basin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
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
h✝ : Attracts (f c) z a
n : ℕ
h : ∀ (n_1 : ℕ), n ≤ n_1 → (f c)^[n_1] z ∈ {z | (c, z) ∈ s.near}
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp1 | [433, 1] | [436, 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
p : ℂ × S
⊢ (s.fp^[n] p).1 = p.1 | 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
p : ℂ × S
⊢ (s.fp^[0] p).1 = p.1
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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).1 = p.1 | 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
⊢ (s.fp^[n] p).1 = p.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp1 | [433, 1] | [436, 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
p : ℂ × S
⊢ (s.fp^[0] p).1 = p.1 | no goals | 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
p : ℂ × S
⊢ (s.fp^[0] p).1 = p.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp1 | [433, 1] | [436, 52] | simp only [Function.iterate_succ_apply', h, fp] | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).1 = p.1 | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).1 = p.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp2 | [438, 1] | [442, 55] | 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
p : ℂ × S
⊢ (s.fp^[n] p).2 = (f p.1)^[n] p.2 | 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
p : ℂ × S
⊢ (s.fp^[0] p).2 = (f p.1)^[0] p.2
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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.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
n : ℕ
p : ℂ × S
⊢ (s.fp^[n] p).2 = (f p.1)^[n] p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp2 | [438, 1] | [442, 55] | 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
p : ℂ × S
⊢ (s.fp^[0] p).2 = (f p.1)^[0] p.2 | no goals | 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
p : ℂ × S
⊢ (s.fp^[0] p).2 = (f p.1)^[0] p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp2 | [438, 1] | [442, 55] | have c := s.fp1 n p | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.2 | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
c : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.2 | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fp2 | [438, 1] | [442, 55] | simp only [Function.iterate_succ_apply', c, h, fp] | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
c : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.2 | 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
p : ℂ × S
n : ℕ
h : (s.fp^[n] p).2 = (f p.1)^[n] p.2
c : (s.fp^[n] p).1 = p.1
⊢ (s.fp^[n + 1] p).2 = (f p.1)^[n + 1] p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | intro p 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
⊢ HolomorphicOn (I.prod I) I (uncurry s.bottcherNear) s.near | 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) 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
⊢ HolomorphicOn (I.prod I) I (uncurry s.bottcherNear) s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | have e : uncurry s.bottcherNear =
(fun p : ℂ × ℂ ↦ _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p : ℂ × S ↦
(p.1, extChartAt I a p.2 - extChartAt I a a) :=
rfl | 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | rw [e] | 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (uncurry s.bottcherNear) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | clear e | 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
e :
uncurry s.bottcherNear =
(fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | have h1 := (bottcherNear_analytic s.superNearC _ (s.mem_near_to_near' m)).holomorphicAt 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | have h2 : HolomorphicAt II II (fun p : ℂ × S ↦
(p.1, extChartAt I a p.2 - extChartAt I a a)) p := by
apply holomorphicAt_fst.prod; apply HolomorphicAt.sub
exact (HolomorphicAt.extChartAt (s.near_subset_chart m)).comp holomorphicAt_snd
exact holomorphicAt_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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | refine h1.comp_of_eq h2 ?_ | 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p = (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ HolomorphicAt (I.prod I) I
((fun p => _root_.bottcherNear (s.fl p.1) d p.2) ∘ fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | simp only [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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p = (p.1, ↑(extChartAt I a) p.2 - ↑(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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
h2 : HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
⊢ (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p = (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | apply HolomorphicAt.sub | 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a) p | case fa
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2) p
case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) 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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | exact (HolomorphicAt.extChartAt (s.near_subset_chart m)).comp holomorphicAt_snd | case fa
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2) p
case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) p | case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) p | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) x.2) p
case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_holomorphic | [453, 1] | [468, 23] | exact holomorphicAt_const | case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ga
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
p : ℂ × S
m : p ∈ s.near
h1 :
HolomorphicAt (I.prod I) I (fun p => _root_.bottcherNear (s.fl p.1) d p.2)
(p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I a) a) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_holomorphic | [474, 1] | [478, 10] | refine (s.bottcherNear_holomorphic _ ?_).comp₂ holomorphicAt_fst (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
r : (c, (f c)^[n] z) ∈ s.near
⊢ HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near | 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
r : (c, (f c)^[n] z) ∈ s.near
⊢ HolomorphicAt (I.prod I) I (uncurry (s.bottcherNearIter n)) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearIter_holomorphic | [474, 1] | [478, 10] | exact r | 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
r : (c, (f c)^[n] z) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near | 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
r : (c, (f c)^[n] z) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn | [481, 1] | [488, 92] | simp only [Super.bottcherNear] | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c (f c z) = s.bottcherNear c z ^ d | 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c (f c z) = s.bottcherNear c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn | [481, 1] | [488, 92] | have e : extChartAt I a (f c z) - extChartAt I a a =
s.fl c (extChartAt I a z - extChartAt I a a) := by
simp only [Function.comp, Super.fl, _root_.fl, sub_add_cancel,
PartialEquiv.left_inv _ (s.near_subset_chart 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d | 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
m : (c, z) ∈ s.near
e : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a = s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d | 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn | [481, 1] | [488, 92] | rw [e, _root_.bottcherNear_eqn (s.superNearC.s (Set.mem_univ c)) (s.mem_near_to_near' 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
m : (c, z) ∈ s.near
e : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a = s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d | 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
m : (c, z) ∈ s.near
e : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a = s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) =
_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn | [481, 1] | [488, 92] | simp only [Function.comp, Super.fl, _root_.fl, sub_add_cancel,
PartialEquiv.left_inv _ (s.near_subset_chart 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
m : (c, z) ∈ s.near
⊢ ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a = s.fl c (↑(extChartAt I a) z - ↑(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
m : (c, z) ∈ s.near
⊢ ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a = s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn_iter | [491, 1] | [495, 25] | 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
m : (c, z) ∈ s.near
n : ℕ
⊢ s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c ((f c)^[0] z) = s.bottcherNear c z ^ d ^ 0
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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1) | 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
m : (c, z) ∈ s.near
n : ℕ
⊢ s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn_iter | [491, 1] | [495, 25] | simp only [Function.iterate_zero_apply, pow_zero, pow_one] | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c ((f c)^[0] z) = s.bottcherNear c z ^ d ^ 0
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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1) | 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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1) | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c ((f c)^[0] z) = s.bottcherNear c z ^ d ^ 0
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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eqn_iter | [491, 1] | [495, 25] | simp only [Function.iterate_succ_apply', s.bottcherNear_eqn (s.iter_stays_near m n), h, ←
pow_mul, ← pow_succ] | 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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1) | 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
m : (c, z) ∈ s.near
n : ℕ
h : s.bottcherNear c ((f c)^[n] z) = s.bottcherNear c z ^ d ^ n
⊢ s.bottcherNear c ((f c)^[n + 1] z) = s.bottcherNear c z ^ d ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearp_eqn | [498, 1] | [500, 51] | rcases p with ⟨c, 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
p : ℂ × S
m : p ∈ s.near
⊢ s.bottcherNearp (s.fp p) = s.bottcherNearp p ^ d | case mk
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
z : S
m : (c, z) ∈ s.near
⊢ s.bottcherNearp (s.fp (c, z)) = s.bottcherNearp (c, z) ^ d | 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
p : ℂ × S
m : p ∈ s.near
⊢ s.bottcherNearp (s.fp p) = s.bottcherNearp p ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNearp_eqn | [498, 1] | [500, 51] | exact s.bottcherNear_eqn m | case mk
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
z : S
m : (c, z) ∈ s.near
⊢ s.bottcherNearp (s.fp (c, z)) = s.bottcherNearp (c, z) ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
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
z : S
m : (c, z) ∈ s.near
⊢ s.bottcherNearp (s.fp (c, z)) = s.bottcherNearp (c, z) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_lt_one | [503, 1] | [506, 93] | simp only [Super.bottcherNear, mem_setOf] | 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
m : (c, z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c z) < 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
m : (c, z) ∈ s.near
⊢ Complex.abs (_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a)) < 1 | 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
m : (c, z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c z) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_lt_one | [503, 1] | [506, 93] | exact _root_.bottcherNear_lt_one (s.superNearC.s (Set.mem_univ c)) (s.mem_near_to_near' 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
m : (c, z) ∈ s.near
⊢ Complex.abs (_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a)) < 1 | 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
m : (c, z) ∈ s.near
⊢ Complex.abs (_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a)) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | simp only [Super.bottcherNear] | 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c z = 0 ↔ z = 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = 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
m : (c, z) ∈ s.near
⊢ s.bottcherNear c z = 0 ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | constructor | 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a | case 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 → z = a
case mpr
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
m : (c, z) ∈ s.near
⊢ z = a → _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | intro za | case 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 → z = a | case 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
m : (c, z) ∈ s.near
za : _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
⊢ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
m : (c, z) ∈ s.near
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 → z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | contrapose za | case 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
m : (c, z) ∈ s.near
za : _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
⊢ z = a | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ¬_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
m : (c, z) ∈ s.near
za : _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
⊢ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | apply bottcherNear_ne_zero (s.superNearC.s (Set.mem_univ _)) (s.mem_near_to_near' m) | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ¬_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) (c, z).2 - ↑(extChartAt I a) a ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ¬_root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | simp only [sub_ne_zero] | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) (c, z).2 - ↑(extChartAt I a) a ≠ 0 | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) z ≠ ↑(extChartAt I a) a | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) (c, z).2 - ↑(extChartAt I a) a ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | exact (extChartAt I a).injOn.ne (s.near_subset_chart m) (mem_extChartAt_source I a) za | case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) z ≠ ↑(extChartAt I a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
m : (c, z) ∈ s.near
za : ¬z = a
⊢ ↑(extChartAt I a) z ≠ ↑(extChartAt I a) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | intro za | case mpr
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
m : (c, z) ∈ s.near
⊢ z = a → _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case mpr
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
m : (c, z) ∈ s.near
za : z = a
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
m : (c, z) ∈ s.near
⊢ z = a → _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_eq_zero | [509, 1] | [516, 58] | simp only [za, sub_self, bottcherNear_zero] | case mpr
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
m : (c, z) ∈ s.near
za : z = a
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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
m : (c, z) ∈ s.near
za : z = a
⊢ _root_.bottcherNear (s.fl c) d (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_a | [519, 1] | [520, 62] | simp only [Super.bottcherNear, sub_self, bottcherNear_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
⊢ s.bottcherNear c 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
⊢ s.bottcherNear c a = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | apply mderiv_comp_ne_zero' (f := _root_.bottcherNear (s.fl c) d) | 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
⊢ mfderiv I I (s.bottcherNear c) 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
⊢ mfderiv I I (_root_.bottcherNear (s.fl c) d) (↑(extChartAt I a) a - ↑(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
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) 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
⊢ mfderiv I I (s.bottcherNear c) a ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | simp only [sub_self, mfderiv_eq_fderiv,
(_root_.bottcherNear_monic (s.superNearC.s (Set.mem_univ c))).hasFDerivAt.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
⊢ mfderiv I I (_root_.bottcherNear (s.fl c) d) (↑(extChartAt I a) a - ↑(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
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 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
⊢ mfderiv I I (_root_.bottcherNear (s.fl c) d) (↑(extChartAt I a) a - ↑(extChartAt I a) a) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | exact ContinuousLinearMap.smulRight_ne_zero ContinuousLinearMap.one_ne_zero (by norm_num) | 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
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 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
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | norm_num | 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
⊢ 1 ≠ 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
⊢ 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | have u : (fun z : S ↦ extChartAt I a z - extChartAt I a a) =
extChartAt I a - fun _ : S ↦ extChartAt I a a := rfl | 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
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) 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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) a) 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
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) a) a ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | rw [u, mfderiv_sub, mfderiv_const, sub_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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) 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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (↑(extChartAt I a)) a ≠ 0
case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | 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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (fun x => ↑(extChartAt I a) x - ↑(extChartAt I a) a) a ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | exact extChartAt_mderiv_ne_zero a | 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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (↑(extChartAt I a)) a ≠ 0
case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | 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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ mfderiv I I (↑(extChartAt I a)) a ≠ 0
case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | exact (HolomorphicAt.extChartAt (mem_extChartAt_source I a)).mdifferentiableAt | case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hf
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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (↑(extChartAt I a)) a
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_mfderiv_ne_zero | [523, 1] | [534, 34] | apply mdifferentiableAt_const | case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.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
u : (fun z => ↑(extChartAt I a) z - ↑(extChartAt I a) a) = ↑(extChartAt I a) - fun x => ↑(extChartAt I a) a
⊢ MDifferentiableAt I I (fun x => ↑(extChartAt I a) a) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_has_inv | [537, 1] | [544, 45] | have h := complex_inverse_fun (s.bottcherNear_holomorphic _ (s.mem_near c))
(s.bottcherNear_mfderiv_ne_zero 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
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = 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
c : ℂ
h :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, s.bottcherNear c a) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.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
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_has_inv | [537, 1] | [544, 45] | simp only [s.bottcherNear_a] 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
h :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, s.bottcherNear c a) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = 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
c : ℂ
h :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, 0) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.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
h :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, s.bottcherNear c a) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, s.bottcherNear c a), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.bottcherNear_has_inv | [537, 1] | [544, 45] | exact 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 :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, 0) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.2 | 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
h :
∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, 0) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, a), g x.1 (s.bottcherNear x.1 x.2) = x.2) ∧
∀ᶠ (x : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear x.1 (g x.1 x.2) = x.2
⊢ ∃ bi,
HolomorphicAt (I.prod I) I (uncurry bi) (c, 0) ∧
(∀ᶠ (p : ℂ × S) in 𝓝 (c, a), bi p.1 (s.bottcherNear p.1 p.2) = p.2) ∧
∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), s.bottcherNear p.1 (bi p.1 p.2) = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have t : ContinuousAt (fun p : ℂ × S ↦ (p.1, extChartAt I a p.2 - extChartAt I a a)) (c, a) := by
refine continuousAt_fst.prod (ContinuousAt.sub ?_ continuousAt_const)
exact (continuousAt_extChartAt I a).comp_of_eq continuousAt_snd rfl | 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), Critical (f p.1) p.2 ↔ p.2 = 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
t : ContinuousAt (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (c, a)
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = 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 (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [ContinuousAt, sub_self] at t | 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
t : ContinuousAt (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (c, a)
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = 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
t : ContinuousAt (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (c, a)
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply (inChart_critical (s.fa (c, a))).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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), Critical (f p.1) p.2 ↔ p.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply (t.eventually (df_ne_zero s.superNearC (Set.mem_univ c))).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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have am := 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have em := ((isOpen_extChartAt_source I a).eventually_mem am).prod_inr (𝓝 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 c ×ˢ 𝓝 a, x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [← nhds_prod_eq] at em | 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 c ×ˢ 𝓝 a, x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 c ×ˢ 𝓝 a, x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply em.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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have ezm : ∀ᶠ p : ℂ × S in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source := by
refine (s.fa _).continuousAt.eventually_mem (extChartAt_source_mem_nhds' I ?_)
simp only [uncurry, s.f0, 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply ezm.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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a) | case hp
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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (x : ℂ × S) in 𝓝 (c, a),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | clear t em | case hp
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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a) | case hp
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
am : a ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | intro ⟨e, z⟩ ezm zm d0 m0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a) | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f (e, z).1 (e, z).2 ∈ (extChartAt I a).source
zm : (e, z).2 ∈ (extChartAt I a).source
d0 :
deriv (s.fl ((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).1)
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 =
0 ↔
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 = 0
m0 : mfderiv I I (f (e, z).1) (e, z).2 = 0 ↔ deriv (inChart f c a (e, z).1) (↑(extChartAt I a) (e, z).2) = 0
⊢ Critical (f (e, z).1) (e, z).2 ↔ (e, z).2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
⊢ ∀ (x : ℂ × S),
f x.1 x.2 ∈ (extChartAt I a).source →
x.2 ∈ (extChartAt I a).source →
(deriv (s.fl (x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).1)
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 =
0 ↔
(x.1, ↑(extChartAt I a) x.2 - ↑(extChartAt I a) a).2 = 0) →
(mfderiv I I (f x.1) x.2 = 0 ↔ deriv (inChart f c a x.1) (↑(extChartAt I a) x.2) = 0) →
(Critical (f x.1) x.2 ↔ x.2 = a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only at ezm zm d0 m0 ⊢ | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f (e, z).1 (e, z).2 ∈ (extChartAt I a).source
zm : (e, z).2 ∈ (extChartAt I a).source
d0 :
deriv (s.fl ((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).1)
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 =
0 ↔
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 = 0
m0 : mfderiv I I (f (e, z).1) (e, z).2 = 0 ↔ deriv (inChart f c a (e, z).1) (↑(extChartAt I a) (e, z).2) = 0
⊢ Critical (f (e, z).1) (e, z).2 ↔ (e, z).2 = a | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
d0 : deriv (s.fl e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ ↑(extChartAt I a) z - ↑(extChartAt I a) a = 0
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
⊢ Critical (f e) z ↔ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f (e, z).1 (e, z).2 ∈ (extChartAt I a).source
zm : (e, z).2 ∈ (extChartAt I a).source
d0 :
deriv (s.fl ((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).1)
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 =
0 ↔
((e, z).1, ↑(extChartAt I a) (e, z).2 - ↑(extChartAt I a) a).2 = 0
m0 : mfderiv I I (f (e, z).1) (e, z).2 = 0 ↔ deriv (inChart f c a (e, z).1) (↑(extChartAt I a) (e, z).2) = 0
⊢ Critical (f (e, z).1) (e, z).2 ↔ (e, z).2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [Super.fl, fl, sub_eq_zero, (PartialEquiv.injOn _).eq_iff zm am] at d0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
d0 : deriv (s.fl e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ ↑(extChartAt I a) z - ↑(extChartAt I a) a = 0
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
⊢ Critical (f e) z ↔ z = a | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ Critical (f e) z ↔ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
d0 : deriv (s.fl e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ ↑(extChartAt I a) z - ↑(extChartAt I a) a = 0
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
⊢ Critical (f e) z ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [Critical, m0, ← d0] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ Critical (f e) z ↔ z = a | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ Critical (f e) z ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | unfold inChart | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | clear m0 d0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
m0 : mfderiv I I (f e) z = 0 ↔ deriv (inChart f c a e) (↑(extChartAt I a) z) = 0
d0 : deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔ z = a
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | generalize hg : (fun w ↦ extChartAt I (f c a) (f e ((extChartAt I a).symm w))) = g | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
⊢ deriv (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) = 0 ↔
deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have hg' : extChartAt I a ∘ f e ∘ (extChartAt I a).symm = g := by
rw [← hg]; simp only [Function.comp, s.f0] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | rw [_root_.fl, hg'] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔ deriv (_root_.fl f a e) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | clear hg' | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
hg' : ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | rw [Iff.comm] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv g (↑(extChartAt I a) z) = 0 ↔
deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have dg : DifferentiableAt ℂ g (extChartAt I a z) := by
rw [← hg]; apply AnalyticAt.differentiableAt; apply HolomorphicAt.analyticAt I I
simp only [s.f0]
apply (HolomorphicAt.extChartAt _).comp; apply (s.fa _).along_snd.comp
exact HolomorphicAt.extChartAt_symm (PartialEquiv.map_source _ zm)
simp only [PartialEquiv.left_inv _ zm, s.f0]; exact ezm | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have d0 : ∀ z, DifferentiableAt ℂ (fun z ↦ z - extChartAt I a a) z := fun z ↦
differentiableAt_id.sub (differentiableAt_const _) | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | have d1 : DifferentiableAt ℂ (g ∘ fun z : ℂ ↦ z + extChartAt I a a)
(extChartAt I a z - extChartAt I a a) := by
apply DifferentiableAt.comp; simp only [sub_add_cancel, dg]
exact differentiableAt_id.add (differentiableAt_const _) | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [deriv.comp _ (d0 _) d1, deriv_sub_const, deriv_id'', one_mul] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔
deriv g (↑(extChartAt I a) z) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv ((fun z => z - ↑(extChartAt I a) a) ∘ g ∘ fun z => z + ↑(extChartAt I a) a)
(↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | rw [deriv.comp _ _ _] | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔
deriv g (↑(extChartAt I a) z) = 0 | case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a) *
deriv (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) =
0 ↔
deriv g (↑(extChartAt I a) z) = 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
d1 : DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
⊢ deriv (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) = 0 ↔
deriv g (↑(extChartAt I a) z) = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | refine continuousAt_fst.prod (ContinuousAt.sub ?_ 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
⊢ ContinuousAt (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (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
⊢ ContinuousAt (fun p => ↑(extChartAt I a) p.2) (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
⊢ ContinuousAt (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (c, a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact (continuousAt_extChartAt I a).comp_of_eq continuousAt_snd rfl | 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 p => ↑(extChartAt I a) p.2) (c, 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
⊢ ContinuousAt (fun p => ↑(extChartAt I a) p.2) (c, a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | refine (s.fa _).continuousAt.eventually_mem (extChartAt_source_mem_nhds' 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ uncurry f (c, 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ ∀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.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [uncurry, s.f0, 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ uncurry f (c, 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
t : Tendsto (fun p => (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a)) (𝓝 (c, a)) (𝓝 (c, 0))
am : a ∈ (extChartAt I a).source
em : ∀ᶠ (x : ℂ × S) in 𝓝 (c, a), x.2 ∈ (extChartAt I a).source
⊢ uncurry f (c, a) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w)) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [Function.comp, 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w)) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ ↑(extChartAt I a) ∘ f e ∘ ↑(extChartAt I a).symm = fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ DifferentiableAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply AnalyticAt.differentiableAt | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ DifferentiableAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ AnalyticAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ DifferentiableAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply HolomorphicAt.analyticAt I I | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ AnalyticAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ AnalyticAt ℂ (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [s.f0] | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I a) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply (HolomorphicAt.extChartAt _).comp | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I a) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I a).symm x)) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun w => ↑(extChartAt I a) (f e (↑(extChartAt I a).symm w))) (↑(extChartAt I a) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply (s.fa _).along_snd.comp | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I a).symm x)) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => ↑(extChartAt I a).symm x) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => f e (↑(extChartAt I a).symm x)) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact HolomorphicAt.extChartAt_symm (PartialEquiv.map_source _ zm) | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => ↑(extChartAt I a).symm x) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ HolomorphicAt I I (fun x => ↑(extChartAt I a).symm x) (↑(extChartAt I a) 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [PartialEquiv.left_inv _ zm, 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e z ∈ (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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e (↑(extChartAt I a).symm (↑(extChartAt I a) z)) ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | exact ezm | 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e z ∈ (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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
⊢ f e z ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | apply DifferentiableAt.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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | case 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a)
case hf
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I 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
c : ℂ
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (g ∘ fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.f_noncritical_near_a | [548, 1] | [589, 61] | simp only [sub_add_cancel, dg] | case 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a)
case hf
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | case hf
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a) | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ g (↑(extChartAt I a) z - ↑(extChartAt I a) a + ↑(extChartAt I a) a)
case hf
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
am : a ∈ (extChartAt I a).source
ezm✝ : ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), f p.1 p.2 ∈ (extChartAt I a).source
e : ℂ
z : S
ezm : f e z ∈ (extChartAt I a).source
zm : z ∈ (extChartAt I a).source
g : ℂ → ℂ
hg : (fun w => ↑(extChartAt I (f c a)) (f e (↑(extChartAt I a).symm w))) = g
dg : DifferentiableAt ℂ g (↑(extChartAt I a) z)
d0 : ∀ (z : ℂ), DifferentiableAt ℂ (fun z => z - ↑(extChartAt I a) a) z
⊢ DifferentiableAt ℂ (fun z => z + ↑(extChartAt I a) a) (↑(extChartAt I a) z - ↑(extChartAt I a) a)
TACTIC:
|
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