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stringlengths 1
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stringlengths 6
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stringlengths 6
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | rcases(s.fla c).exists_ball_analyticOn with ⟨r0, r0p, fla⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source | case intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (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
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | rcases eventually_nhds_iff.mp (s.stays_in_chart c) with ⟨t, tp, ot, ta⟩ | case intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | set ch := extChartAt II (c, a) | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ (extChartAt (I.prod I) (c, a)).source,
↑(extChartAt (I.prod I) (c, a)) p ∈ ball (↑(extChartAt (I.prod I) (c, a)) (c, a)) r →
f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | set s := ch.target ∩ ch.symm ⁻¹' t | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | have os : IsOpen s :=
(continuousOn_extChartAt_symm II (c, a)).isOpen_inter_preimage
(isOpen_extChartAt_target II (c, a)) ot | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | have m : ch (c, a) ∈ s := by
apply Set.mem_inter (mem_extChartAt_target _ _)
rw [Set.mem_preimage, ch.left_inv (mem_extChartAt_source _ _)]
exact ta | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | rcases Metric.isOpen_iff.mp os (ch (c, a)) m with ⟨r1, r1p, rs⟩ | case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | case intro.intro.intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | apply Set.mem_inter (mem_extChartAt_target _ _) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ↑ch (c, a) ∈ s | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ↑(extChartAt (I.prod I) (c, a)) (c, a) ∈ ↑ch.symm ⁻¹' t | 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ↑ch (c, a) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | rw [Set.mem_preimage, ch.left_inv (mem_extChartAt_source _ _)] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ↑(extChartAt (I.prod I) (c, a)) (c, a) ∈ ↑ch.symm ⁻¹' 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ (c, a) ∈ t | 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ ↑(extChartAt (I.prod I) (c, a)) (c, a) ∈ ↑ch.symm ⁻¹' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | exact ta | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ (c, a) ∈ t | 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
⊢ (c, a) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | use min r0 r1, by bound | case intro.intro.intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) (min r0 r1)) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.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
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∃ r > 0,
AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) r → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | use fla.mono (Metric.ball_subset_ball (by bound)) | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) (min r0 r1)) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) (min r0 r1)) ∧
∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | intro p ps pr | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ f p.1 p.2 ∈ (extChartAt I a).source | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ ∀ p ∈ ch.source, ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1) → f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | apply tp p | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ f p.1 p.2 ∈ (extChartAt I a).source | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ p ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ f p.1 p.2 ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | rw [← ch.left_inv ps, ← Set.mem_preimage] | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ p ∈ t | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ ↑ch p ∈ ↑ch.symm ⁻¹' t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ p ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | exact Set.mem_of_mem_inter_right (rs (Metric.ball_subset_ball (by bound) pr)) | case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ ↑ch p ∈ ↑ch.symm ⁻¹' t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ ↑ch p ∈ ↑ch.symm ⁻¹' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ min r0 r1 > 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ min r0 r1 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ min r0 r1 ≤ r0 | 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
⊢ min r0 r1 ≤ r0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.fr_prop | [214, 1] | [235, 82] | bound | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
c : ℂ
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ min r0 r1 ≤ r1 | 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
r0 : ℝ
r0p : 0 < r0
fla : AnalyticOn ℂ (uncurry s✝.fl) (ball (c, 0) r0)
t : Set (ℂ × S)
tp : ∀ x ∈ t, f x.1 x.2 ∈ (extChartAt I a).source
ot : IsOpen t
ta : (c, a) ∈ t
ch : PartialEquiv (ℂ × S) (ℂ × ℂ) := extChartAt (I.prod I) (c, a)
s : Set (ℂ × ℂ) := ch.target ∩ ↑ch.symm ⁻¹' t
os : IsOpen s
m : ↑ch (c, a) ∈ s
r1 : ℝ
r1p : r1 > 0
rs : ball (↑ch (c, a)) r1 ⊆ s
p : ℂ × S
ps : p ∈ ch.source
pr : ↑ch p ∈ ball (↑ch (c, a)) (min r0 r1)
⊢ min r0 r1 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | prod_zero_mem_ball | [262, 1] | [264, 87] | simp only [Metric.mem_ball] at 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
c b : ℂ
r : ℝ
m : b ∈ ball c r
⊢ (b, 0) ∈ ball (c, 0) 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
c b : ℂ
r : ℝ
m : dist b c < r
⊢ (b, 0) ∈ ball (c, 0) r | 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
c b : ℂ
r : ℝ
m : b ∈ ball c r
⊢ (b, 0) ∈ ball (c, 0) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | prod_zero_mem_ball | [262, 1] | [264, 87] | simpa only [Metric.mem_ball, dist_prod_same_right] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
c b : ℂ
r : ℝ
m : dist b c < r
⊢ (b, 0) ∈ ball (c, 0) r | 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
c b : ℂ
r : ℝ
m : dist b c < r
⊢ (b, 0) ∈ ball (c, 0) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.exists_superNearC | [277, 1] | [279, 71] | refine s.superAtC.superNearC' s.fls_open fun 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
⊢ ∃ t ⊆ s.fls, SuperNearC s.fl d univ 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
x✝ : c ∈ univ
⊢ (c, 0) ∈ s.fls | 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
⊢ ∃ t ⊆ s.fls, SuperNearC s.fl d univ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.exists_superNearC | [277, 1] | [279, 71] | rw [Super.fls, Set.mem_iUnion] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
x✝ : c ∈ univ
⊢ (c, 0) ∈ s.fls | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
x✝ : c ∈ univ
⊢ ∃ i, (c, 0) ∈ ball (i, 0) (s.fr i) | 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
x✝ : c ∈ univ
⊢ (c, 0) ∈ s.fls
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.exists_superNearC | [277, 1] | [279, 71] | use 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
x✝ : c ∈ univ
⊢ ∃ i, (c, 0) ∈ ball (i, 0) (s.fr i) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
x✝ : c ∈ univ
⊢ (c, 0) ∈ ball (c, 0) (s.fr c) | 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
x✝ : c ∈ univ
⊢ ∃ i, (c, 0) ∈ ball (i, 0) (s.fr i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.exists_superNearC | [277, 1] | [279, 71] | exact mem_ball_self (s.frp c) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
x✝ : c ∈ univ
⊢ (c, 0) ∈ ball (c, 0) (s.fr c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
x✝ : c ∈ univ
⊢ (c, 0) ∈ ball (c, 0) (s.fr c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isOpen_near | [297, 1] | [300, 19] | apply (continuousOn_extChartAt _ _).isOpen_inter_preimage (isOpen_extChartAt_source _ _) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen 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
⊢ IsOpen {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ 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
⊢ IsOpen s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isOpen_near | [297, 1] | [300, 19] | exact IsOpen.preimage (continuous_fst.prod_mk (continuous_snd.sub continuous_const))
s.superNearC.o | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ 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
⊢ IsOpen {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.mem_near | [303, 1] | [307, 45] | simp only [Super.near, extChartAt_prod, PartialEquiv.prod_source, Set.mem_prod, Set.mem_inter_iff,
mem_extChartAt_source, extChartAt_eq_refl, PartialEquiv.refl_source, Set.mem_univ, true_and_iff,
Set.mem_preimage, PartialEquiv.prod_coe, PartialEquiv.refl_coe, id, Set.mem_setOf_eq, sub_self] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (c, a) ∈ 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
c : ℂ
⊢ (c, 0) ∈ 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
c : ℂ
⊢ (c, a) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.mem_near | [303, 1] | [307, 45] | exact (s.superNearC.s (Set.mem_univ _)).t0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
c : ℂ
⊢ (c, 0) ∈ 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
c : ℂ
⊢ (c, 0) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.near_subset_chart | [310, 1] | [314, 12] | have h := Set.mem_of_mem_inter_left m | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ 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
z : S
m : (c, z) ∈ s.near
h : (c, z) ∈ (extChartAt (I.prod I) (0, a)).source
⊢ 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
z : S
m : (c, z) ∈ s.near
⊢ z ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.near_subset_chart | [310, 1] | [314, 12] | simp only [extChartAt_prod, PartialEquiv.prod_source, Set.mem_prod_eq] at h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.near
h : (c, z) ∈ (extChartAt (I.prod I) (0, a)).source
⊢ 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
z : S
m : (c, z) ∈ s.near
h : c ∈ (extChartAt I 0).source ∧ z ∈ (extChartAt I a).source
⊢ 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
z : S
m : (c, z) ∈ s.near
h : (c, z) ∈ (extChartAt (I.prod I) (0, a)).source
⊢ z ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.near_subset_chart | [310, 1] | [314, 12] | exact h.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
z : S
m : (c, z) ∈ s.near
h : c ∈ (extChartAt I 0).source ∧ z ∈ (extChartAt I a).source
⊢ 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
z : S
m : (c, z) ∈ s.near
h : c ∈ (extChartAt I 0).source ∧ z ∈ (extChartAt I a).source
⊢ z ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.mem_near_to_near' | [316, 1] | [321, 10] | have h := Set.mem_of_mem_inter_right 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
p : ℂ × S
m : p ∈ s.near
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ 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
h : p ∈ ↑(extChartAt (I.prod I) (0, a)) ⁻¹' {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ 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
p : ℂ × S
m : p ∈ s.near
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.mem_near_to_near' | [316, 1] | [321, 10] | simp only [Set.mem_preimage, extChartAt_prod, PartialEquiv.prod_coe, extChartAt_eq_refl,
PartialEquiv.refl_coe, id] 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
p : ℂ × S
m : p ∈ s.near
h : p ∈ ↑(extChartAt (I.prod I) (0, a)) ⁻¹' {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ 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
h : (p.1, ↑(extChartAt I a) p.2) ∈ {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ 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
p : ℂ × S
m : p ∈ s.near
h : p ∈ ↑(extChartAt (I.prod I) (0, a)) ⁻¹' {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.mem_near_to_near' | [316, 1] | [321, 10] | 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
p : ℂ × S
m : p ∈ s.near
h : (p.1, ↑(extChartAt I a) p.2) ∈ {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ 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
p : ℂ × S
m : p ∈ s.near
h : (p.1, ↑(extChartAt I a) p.2) ∈ {p | (p.1, p.2 - ↑(extChartAt I a) a) ∈ s.near'}
⊢ (p.1, ↑(extChartAt I a) p.2 - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | simp only [Super.near, extChartAt_prod, PartialEquiv.prod_source, Set.mem_prod, Set.mem_inter_iff,
mem_extChartAt_source, extChartAt_eq_refl, PartialEquiv.refl_source, Set.mem_univ, true_and_iff,
Set.mem_preimage, PartialEquiv.prod_coe, PartialEquiv.refl_coe, id, Set.mem_setOf_eq,
sub_self] at m ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, f c z) ∈ 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ 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
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, f c z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | rcases mem_iUnion.mp (s.near_subset' m.2) with ⟨b, mb⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ ball (b, 0) (s.fr b)
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | simp only [mem_ball_iff_norm, Prod.norm_def, max_lt_iff, Prod.fst_sub, Prod.snd_sub,
sub_zero] at mb | case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ ball (b, 0) (s.fr b)
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ ball (b, 0) (s.fr b)
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | constructor | case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | case intro.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source
case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
c : ℂ
z : S
m : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | apply s.fr_stays b (c, z) | case intro.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source | case intro.left.ps
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, z) ∈ (extChartAt (I.prod I) (b, a)).source
case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ f c z ∈ (extChartAt I a).source
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | simp only [m.1, Super.near, extChartAt_prod, PartialEquiv.prod_source, Set.mem_prod,
Set.mem_inter_iff, mem_extChartAt_source, extChartAt_eq_refl, PartialEquiv.refl_source,
Set.mem_univ, true_and_iff, Set.mem_preimage, PartialEquiv.prod_coe, PartialEquiv.refl_coe, id,
Set.mem_setOf_eq, sub_self] | case intro.left.ps
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, z) ∈ (extChartAt (I.prod I) (b, a)).source
case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b) | case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left.ps
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, z) ∈ (extChartAt (I.prod I) (b, a)).source
case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | simp only [m.1, mb.1, mb.2, Super.near, extChartAt_prod, PartialEquiv.prod_source, Set.mem_prod,
Set.mem_inter_iff, mem_extChartAt_source, extChartAt_eq_refl, PartialEquiv.refl_source,
Set.mem_univ, true_and_iff, Set.mem_preimage, PartialEquiv.prod_coe, PartialEquiv.refl_coe, id,
Set.mem_setOf_eq, sub_self, mem_ball_iff_norm, Prod.norm_def, max_lt_iff, Prod.fst_sub,
Prod.snd_sub, sub_zero] | case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.left.pr
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ ↑(extChartAt (I.prod I) (b, a)) (c, z) ∈ ball (↑(extChartAt (I.prod I) (b, a)) (b, a)) (s.fr b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | have h := (s.superNearC.s (Set.mem_univ c)).ft m.2 | case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | simp only [Super.fl, _root_.fl, Function.comp, sub_add_cancel, PartialEquiv.left_inv _ m.1] at h | case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : s.fl c (↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.stays_near | [324, 1] | [346, 12] | exact h | case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.right
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ 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 : z ∈ (extChartAt I a).source ∧ (c, ↑(extChartAt I a) z - ↑(extChartAt I a) a) ∈ s.near'
b : ℂ
mb : ‖c - b‖ < s.fr b ∧ ‖↑(extChartAt I a) z - ↑(extChartAt I a) a‖ < s.fr b
h : ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a ∈ {z | (c, z) ∈ s.near'}
⊢ (c, ↑(extChartAt I a) (f c z) - ↑(extChartAt I a) a) ∈ s.near'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_stays_near | [349, 1] | [352, 82] | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
⊢ (c, (f c)^[n] z) ∈ s.near | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, (f c)^[0] z) ∈ s.near
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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ 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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
⊢ (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_stays_near | [349, 1] | [352, 82] | simp only [Function.iterate_zero, id, m] | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, (f c)^[0] z) ∈ s.near
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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, (f c)^[0] z) ∈ s.near
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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_stays_near | [349, 1] | [352, 82] | simp only [Nat.add_succ, Function.iterate_succ', s.stays_near h, Function.comp] | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near | 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
c : ℂ
z : S
m : (c, z) ∈ s.near
n : ℕ
h : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_stays_near' | [355, 1] | [357, 88] | rw [← Nat.sub_add_cancel ab, Function.iterate_add_apply] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a b : ℕ
m : (c, (f c)^[a] z) ∈ s.near
ab : a ≤ b
⊢ (c, (f c)^[b] z) ∈ 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✝
a b : ℕ
m : (c, (f c)^[a] z) ∈ s.near
ab : a ≤ b
⊢ (c, (f c)^[b - a] ((f c)^[a] z)) ∈ 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✝
a b : ℕ
m : (c, (f c)^[a] z) ∈ s.near
ab : a ≤ b
⊢ (c, (f c)^[b] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.iter_stays_near' | [355, 1] | [357, 88] | exact s.iter_stays_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✝
a b : ℕ
m : (c, (f c)^[a] z) ∈ s.near
ab : a ≤ b
⊢ (c, (f c)^[b - a] ((f c)^[a] z)) ∈ 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✝
a b : ℕ
m : (c, (f c)^[a] z) ∈ s.near
ab : a ≤ b
⊢ (c, (f c)^[b - a] ((f c)^[a] z)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | rw [Attracts, Filter.tendsto_iff_forall_eventually_mem] at 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✝
z : S
a : Attracts (f c) z a✝
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ 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✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ 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✝
z : S
a : Attracts (f c) z a✝
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | have e := a {z | (c, z) ∈ 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✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
e : ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ {z | (c, z) ∈ s.near}
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 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✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | exact e | case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
e : ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ {z | (c, z) ∈ s.near}
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 a✝ | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
e : ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ {z | (c, z) ∈ s.near}
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 a✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | apply IsOpen.mem_nhds | case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 a✝ | case refine_1.hs
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ IsOpen {z | (c, z) ∈ s.near}
case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ {z | (c, z) ∈ s.near} ∈ 𝓝 a✝
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | apply IsOpen.snd_preimage s.isOpen_near | case refine_1.hs
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ IsOpen {z | (c, z) ∈ s.near}
case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near} | case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near} | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hs
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ IsOpen {z | (c, z) ∈ s.near}
case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.reaches_near | [360, 1] | [364, 85] | exact s.mem_near c | case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hx
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a✝ z✝ : S
d n : ℕ
s : Super f d a✝
z : S
a : ∀ s ∈ 𝓝 a✝, ∀ᶠ (x : ℕ) in atTop, (f c)^[x] z ∈ s
⊢ a✝ ∈ {z | (c, z) ∈ s.near}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | have m := s.mem_near_to_near' 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
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | have t := iterates_tendsto (s.superNearC.s (Set.mem_univ c)) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | generalize hg : (fun x : ℂ ↦ (extChartAt I a).symm (x + extChartAt I a a)) = 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | have gc : ContinuousAt g 0 := by
rw [← hg]
refine (continuousAt_extChartAt_symm'' I ?_).comp
(continuous_id.add continuous_const).continuousAt
simp only [zero_add]; exact mem_extChartAt_target 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | have g0 : g 0 = a := by
simp only [← hg]; simp only [zero_add]; exact PartialEquiv.left_inv _ (mem_extChartAt_source _ _) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | have h := gc.tendsto.comp 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | clear t gc 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [Function.comp, g0] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (g ∘ fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 (g 0))
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | rw [← attracts_shift n] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | refine Filter.Tendsto.congr ?_ 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
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ Attracts (f c) ((f c)^[n] z) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | clear 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
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] 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
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
h : Tendsto (fun x => g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a))) atTop (𝓝 a)
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | intro k | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] 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
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ g ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[k] ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ∀ (x : ℕ), g ((s.fl c)^[x] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[x] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [← 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ g ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[k] ((f c)^[n] 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
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ ↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ g ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) = (f c)^[k] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | induction' k with k 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
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ ↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm ((s.fl c)^[0] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[0] ((f c)^[n] z)
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
⊢ ↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm ((s.fl c)^[0] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[0] ((f c)^[n] z)
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a + ↑(extChartAt I a) a) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm ((s.fl c)^[0] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[0] ((f c)^[n] z)
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | rw [sub_add_cancel] | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a + ↑(extChartAt I a) a) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z)) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a + ↑(extChartAt I a) a) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | exact PartialEquiv.left_inv _ (s.near_subset_chart r) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z)) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[n] z)) = (f c)^[n] z
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [Function.iterate_succ_apply'] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
(s.fl c ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) + ↑(extChartAt I a) a) =
f c ((f c)^[k] ((f c)^[n] z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
((s.fl c)^[k + 1] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k + 1] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | generalize hx : (s.fl c)^[k] (extChartAt I a ((f c)^[n] z) - extChartAt I a a) = x | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
(s.fl c ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) + ↑(extChartAt I a) a) =
f c ((f c)^[k] ((f c)^[n] z)) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
x : ℂ
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
⊢ ↑(extChartAt I a).symm
(s.fl c ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a)) + ↑(extChartAt I a) a) =
f c ((f c)^[k] ((f c)^[n] z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | rw [hx] at h | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
x : ℂ
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z)) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
h :
↑(extChartAt I a).symm ((s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) + ↑(extChartAt I a) a) =
(f c)^[k] ((f c)^[n] z)
x : ℂ
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [Super.fl, _root_.fl, Function.comp, sub_add_cancel, h,
←Function.iterate_succ_apply' (f c)] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z)) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[k.succ] ((f c)^[n] z))) = (f c)^[k.succ] ((f c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (s.fl c x + ↑(extChartAt I a) a) = f c ((f c)^[k] ((f c)^[n] z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | apply PartialEquiv.left_inv _ (s.near_subset_chart (s.iter_stays_near r _)) | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[k.succ] ((f c)^[n] z))) = (f c)^[k.succ] ((f c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
g0 : g 0 = a
k : ℕ
x : ℂ
h : ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a) = (f c)^[k] ((f c)^[n] z)
hx : (s.fl c)^[k] (↑(extChartAt I a) ((f c)^[n] z) - ↑(extChartAt I a) a) = x
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) ((f c)^[k.succ] ((f c)^[n] z))) = (f c)^[k.succ] ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ContinuousAt g 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ContinuousAt g 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | refine (continuousAt_extChartAt_symm'' I ?_).comp
(continuous_id.add continuous_const).continuousAt | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ContinuousAt (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [zero_add] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ 0 + ↑(extChartAt I a) a ∈ (extChartAt I a).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | exact mem_extChartAt_target 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
⊢ ↑(extChartAt I a) a ∈ (extChartAt I a).target
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [← 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ g 0 = 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a) = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ g 0 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | simp only [zero_add] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (0 + ↑(extChartAt I a) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (0 + ↑(extChartAt I a) a) = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.attracts | [367, 1] | [390, 78] | exact PartialEquiv.left_inv _ (mem_extChartAt_source _ _) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) 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
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
m : ((c, (f c)^[n] z).1, ↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a) ∈ s.near'
t : Tendsto (fun n_1 => (s.fl c)^[n_1] (↑(extChartAt I a) (c, (f c)^[n] z).2 - ↑(extChartAt I a) a)) atTop (𝓝 0)
g : ℂ → S
hg : (fun x => ↑(extChartAt I a).symm (x + ↑(extChartAt I a) a)) = g
gc : ContinuousAt g 0
⊢ ↑(extChartAt I a).symm (↑(extChartAt I a) a) = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isOpen_basin | [402, 1] | [403, 89] | simp only [Super.basin, setOf_exists] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen s.basin | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen (⋃ i, {x | (x.1, (f x.1)^[i] x.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
⊢ IsOpen s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.isOpen_basin | [402, 1] | [403, 89] | exact isOpen_iUnion fun n ↦ s.isOpen_preimage n | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ IsOpen (⋃ i, {x | (x.1, (f x.1)^[i] x.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
⊢ IsOpen (⋃ i, {x | (x.1, (f x.1)^[i] x.2) ∈ s.near})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | simp only [Super.basin, Set.mem_setOf] at 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.basin
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ 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
m : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ 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
m : (c, z) ∈ s.basin
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | rcases m with ⟨n, 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 : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near | case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ 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
m : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | rw [Filter.eventually_atTop] | case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near | case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ a_1, ∀ b ≥ a_1, (c, (f c)^[b] z) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ᶠ (n : ℕ) in atTop, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | use n | case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ a_1, ∀ b ≥ a_1, (c, (f c)^[b] z) ∈ s.near | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ b ≥ n, (c, (f c)^[b] z) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ a_1, ∀ b ≥ a_1, (c, (f c)^[b] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | intro k kn | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ b ≥ n, (c, (f c)^[b] z) ∈ s.near | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k] z) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
⊢ ∀ b ≥ n, (c, (f c)^[b] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | rw [← Nat.sub_add_cancel kn, Function.iterate_add_apply] | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k] z) ∈ s.near | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k - n] ((f c)^[n] z)) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_stays | [406, 1] | [412, 30] | exact s.iter_stays_near m _ | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k - n] ((f c)^[n] z)) ∈ s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
m : (c, (f c)^[n] z) ∈ s.near
k : ℕ
kn : k ≥ n
⊢ (c, (f c)^[k - n] ((f c)^[n] z)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_attracts | [415, 1] | [416, 43] | rcases m with ⟨n, 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.basin
⊢ Attracts (f c) z a | case 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
n : ℕ
m : ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near
⊢ Attracts (f c) 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.basin
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_attracts | [415, 1] | [416, 43] | exact s.attracts m | case 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
n : ℕ
m : ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near
⊢ Attracts (f c) z a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
n : ℕ
m : ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near
⊢ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_iff_attracts | [419, 1] | [423, 29] | 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
⊢ (c, z) ∈ s.basin ↔ Attracts (f c) 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
⊢ (c, z) ∈ s.basin → Attracts (f c) 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin | 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, z) ∈ s.basin ↔ Attracts (f c) z a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_iff_attracts | [419, 1] | [423, 29] | exact s.basin_attracts | 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
⊢ (c, z) ∈ s.basin → Attracts (f c) 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin | 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin | 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
⊢ (c, z) ∈ s.basin → Attracts (f c) 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_iff_attracts | [419, 1] | [423, 29] | intro h | 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin | 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
h : Attracts (f c) z a
⊢ (c, z) ∈ s.basin | 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
⊢ Attracts (f c) z a → (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNearM.lean | Super.basin_iff_attracts | [419, 1] | [423, 29] | rcases tendsto_atTop_nhds.mp h {z | (c, z) ∈ s.near} (s.mem_near c)
(s.isOpen_near.snd_preimage c) with ⟨n, h⟩ | 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
h : Attracts (f c) z a
⊢ (c, z) ∈ s.basin | 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 | 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
h : Attracts (f c) z a
⊢ (c, z) ∈ s.basin
TACTIC:
|
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