url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | rcases(mem_image _ _ _).mp m with β¨β¨c', zβ©, β¨zp, zc, zaβ©, ccβ© | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : (c', z).1 = c
zp : s.potential (c', z).1 (c', z).2 β€ q
zc : Critical (f (c', z).1) (c', z).2
za : (c', z).2 β a
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only at cc za zc zp | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : (c', z).1 = c
zp : s.potential (c', z).1 (c', z).2 β€ q
zc : Critical (f (c', z).1) (c', z).2
za : (c', z).2 β a
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
zp : s.potential c' z β€ q
zc : Critical (f c') z
za : z β a
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : (c', z).1 = c
zp : s.potential (c', z).1 (c', z).2 β€ q
zc : Critical (f (c', z).1) (c', z).2
za : (c', z).2 β a
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [cc] at za zc zp | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
zp : s.potential c' z β€ q
zc : Critical (f c') z
za : z β a
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
zp : s.potential c' z β€ q
zc : Critical (f c') z
za : z β a
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | clear cc c' | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
c' : β
z : S
cc : c' = c
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | simp only [Ne, β s.potential_eq_zero_of_onePreimage c] at za | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.p c β€ q | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
za : z β a
zc : Critical (f c) z
zp : s.potential c z β€ q
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine _root_.trans (csInf_le s.bddBelow_ps ?_) zp | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.p c β€ q | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β s.ps c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.p c β€ q
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | right | case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β s.ps c | case intro.mk.intro.intro.intro.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β 0 β§ β z_1, s.potential c z_1 = s.potential c z β§ Critical (f c) z_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β s.ps c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | use za, z, rfl, zc | case intro.mk.intro.intro.intro.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β 0 β§ β z_1, s.potential c z_1 = s.potential c z β§ Critical (f c) z_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro.h
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
p : β
h : βαΆ (x : β) in π c, s.p x β€ p
q : β
pq : p < q
q1 : q < 1
t : Set (β Γ S) := {x | s.potential x.1 x.2 β€ q β§ Critical (f x.1) x.2 β§ x.2 β a}
ct : IsClosed t
u : Set β := Prod.fst '' t
cu : IsClosed u
m : c β u
z : S
zc : Critical (f c) z
zp : s.potential c z β€ q
za : Β¬s.potential c z = 0
β’ s.potential c z β 0 β§ β z_1, s.potential c z_1 = s.potential c z β§ Critical (f c) z_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | contrapose p | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
p0 : s.potential c z β 0
β’ Β¬Precritical (f c) z | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : ¬¬Precritical (f c) z
β’ Β¬Postcritical s c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
p0 : s.potential c z β 0
β’ Β¬Precritical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [Postcritical, not_not, not_forall, not_lt] at p β’ | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : ¬¬Precritical (f c) z
β’ Β¬Postcritical s c z | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : Precritical (f c) z
β’ s.p c β€ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : ¬¬Precritical (f c) z
β’ Β¬Postcritical s c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | rcases p with β¨n, pβ© | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : Precritical (f c) z
β’ s.p c β€ s.potential c z | case intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p0 : s.potential c z β 0
p : Precritical (f c) z
β’ s.p c β€ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | trans s.potential c ((f c)^[n] z) | case intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c z | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c ((f c)^[n] z)
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β€ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | refine csInf_le s.bddBelow_ps (Or.inr β¨?_, (f c)^[n] z, rfl, pβ©) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c ((f c)^[n] z) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.p c β€ s.potential c ((f c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [s.potential_eqn_iter] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β 0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | exact pow_ne_zero _ p0 | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [s.potential_eqn_iter] | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β€ s.potential c z | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β€ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c ((f c)^[n] z) β€ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | exact pow_le_of_le_one s.potential_nonneg s.potential_le_one (pow_ne_zero _ s.d0) | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β€ s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
s : Super f d a
p0 : s.potential c z β 0
n : β
p : Critical (f c) ((f c)^[n] z)
β’ s.potential c z ^ d ^ n β€ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | apply p.not_precritical | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ Β¬Precritical (f c) z | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ s.potential c z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ Β¬Precritical (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | simp only [Ne, s.potential_eq_zero_of_onePreimage] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ s.potential c z β 0 | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ Β¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ s.potential c z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | exact za | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ Β¬z = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
za : z β a
instβ : OnePreimage s
β’ Β¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | set f := fun x : β Γ S β¦ s.p x.1 - s.potential x.1 x.2 | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ IsOpen s.post | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
β’ IsOpen s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
β’ IsOpen s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | have fc : LowerSemicontinuous f :=
(s.lowerSemicontinuous_p.comp continuous_fst).add
(Continuous.potential s).neg.lowerSemicontinuous | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
β’ IsOpen s.post | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
β’ IsOpen s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
β’ IsOpen s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | have e : s.post = f β»ΒΉ' Ioi 0 :=
Set.ext fun _ β¦ by
simp only [Super.post, mem_setOf, Postcritical, mem_preimage, mem_Ioi, sub_pos, f] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
β’ IsOpen s.post | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
β’ IsOpen s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | rw [e] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen s.post | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen (f β»ΒΉ' Ioi 0) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | exact fc.isOpen_preimage _ | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen (f β»ΒΉ' Ioi 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f β»ΒΉ' Ioi 0
β’ IsOpen (f β»ΒΉ' Ioi 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | simp only [Super.post, mem_setOf, Postcritical, mem_preimage, mem_Ioi, sub_pos, f] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
xβ : β Γ S
β’ xβ β s.post β xβ β f β»ΒΉ' Ioi 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
f : β Γ S β β := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
xβ : β Γ S
β’ xβ β s.post β xβ β f β»ΒΉ' Ioi 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.eventually | [161, 1] | [163, 90] | refine (s.isOpen_post.eventually_mem ?_).mp (eventually_of_forall fun _ m β¦ m) | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
instβ : OnePreimage s
β’ βαΆ (p : β Γ S) in π (c, z), Postcritical s p.1 p.2 | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
instβ : OnePreimage s
β’ (c, z) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
instβ : OnePreimage s
β’ βαΆ (p : β Γ S) in π (c, z), Postcritical s p.1 p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.eventually | [161, 1] | [163, 90] | exact p | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
instβ : OnePreimage s
β’ (c, z) β s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
s : Super f d a
p : Postcritical s c z
instβ : OnePreimage s
β’ (c, z) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.post_a | [174, 1] | [175, 82] | simp only [Super.post, Postcritical, s.potential_a, mem_setOf] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
β’ (c, a) β s.post | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
β’ 0 < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
β’ (c, a) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.post_a | [174, 1] | [175, 82] | exact s.p_pos c | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
β’ 0 < s.p c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
cβ : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
c : β
β’ 0 < s.p c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | rcases p with β¨c, zβ© | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, f p.1 p.2) β s.post | case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ ((c, z).1, f (c, z).1 (c, z).2) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, f p.1 p.2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | simp only [Super.post, mem_setOf, Postcritical, s.potential_eqn] | case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ ((c, z).1, f (c, z).1 (c, z).2) β s.post | case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ s.potential c z ^ d < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ ((c, z).1, f (c, z).1 (c, z).2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | exact lt_of_le_of_lt (pow_le_of_le_one s.potential_nonneg s.potential_le_one s.d0) m | case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ s.potential c z ^ d < s.p c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
cβ : β
a zβ z0 z1 : S
d n : β
sβ s : Super f d a
c : β
z : S
m : (c, z) β s.post
β’ s.potential c z ^ d < s.p c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | induction' n with n h | S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
β’ (p.1, (f p.1)^[n] p.2) β s.post | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, (f p.1)^[0] p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
β’ (p.1, (f p.1)^[n] p.2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | simp only [Function.iterate_zero_apply] | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, (f p.1)^[0] p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, (f p.1)^[0] p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | exact m | case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
β’ (p.1, p.2) β s.post
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | simp only [Function.iterate_succ_apply'] | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, f p.1 ((f p.1)^[n] p.2)) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, (f p.1)^[n + 1] p.2) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | exact s.stays_post h | case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, f p.1 ((f p.1)^[n] p.2)) β s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
instββ΄ : TopologicalSpace S
instβΒ³ : CompactSpace S
instβΒ² : T3Space S
instβΒΉ : ChartedSpace β S
instβ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
p : β Γ S
m : p β s.post
n : β
h : (p.1, (f p.1)^[n] p.2) β s.post
β’ (p.1, f p.1 ((f p.1)^[n] p.2)) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | rcases tendsto_atTop_nhds.mp (s.basin_attracts m) {z | (c, z) β s.post} (s.post_a c)
(s.isOpen_post.snd_preimage c) with β¨n, hβ© | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
β’ β n, (c, (f c)^[n] z) β s.post | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : β (n_1 : β), n β€ n_1 β (f c)^[n_1] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
β’ β n, (c, (f c)^[n] z) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | specialize h n (le_refl n) | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : β (n_1 : β), n β€ n_1 β (f c)^[n_1] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (f c)^[n] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : β (n_1 : β), n β€ n_1 β (f c)^[n_1] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | simp only [mem_setOf] at h | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (f c)^[n] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (c, (f c)^[n] z) β s.post
β’ β n, (c, (f c)^[n] z) β s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (f c)^[n] z β {z | (c, z) β s.post}
β’ β n, (c, (f c)^[n] z) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | use n, h | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (c, (f c)^[n] z) β s.post
β’ β n, (c, (f c)^[n] z) β s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d nβ : β
sβ s : Super f d a
instβ : OnePreimage s
m : (c, z) β s.basin
n : β
h : (c, (f c)^[n] z) β s.post
β’ β n, (c, (f c)^[n] z) β s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rcases((Filter.eventually_ge_atTop n).and (s.eventually_noncritical β¨_, rβ©)).exists with
β¨m, nm, mcβ© | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | have r' := s.iter_stays_near' r nm | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | replace h := h.nonconst | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : NontrivialHolomorphicAt (s.bottcherNearIter m c) z
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : NontrivialHolomorphicAt (s.bottcherNearIter m c) z
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | refine β¨(s.bottcherNearIter_holomorphic r).along_snd, ?_β© | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ NontrivialHolomorphicAt (s.bottcherNearIter n c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | contrapose h | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : Β¬βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z
β’ Β¬βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
β’ βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Filter.not_frequently, not_not] at h β’ | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : Β¬βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z
β’ Β¬βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter m c x = s.bottcherNearIter m c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : Β¬βαΆ (w : S) in π z, s.bottcherNearIter n c w β s.bottcherNearIter n c z
β’ Β¬βαΆ (w : S) in π z, s.bottcherNearIter m c w β s.bottcherNearIter m c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [β Nat.sub_add_cancel nm] | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter m c x = s.bottcherNearIter m c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter (m - n + n) c x = s.bottcherNearIter (m - n + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter m c x = s.bottcherNearIter m c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | generalize hk : m - n = k | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter (m - n + n) c x = s.bottcherNearIter (m - n + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
hk : m - n = k
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
β’ βαΆ (x : S) in π z, s.bottcherNearIter (m - n + n) c x = s.bottcherNearIter (m - n + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | clear hk nm mc r' p m | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
hk : m - n = k
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
hk : m - n = k
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | have er : βαΆ w in π z, (c, (f c)^[n] w) β s.near :=
(continuousAt_const.prod (s.continuousAt_iter continuousAt_const
continuousAt_id)).eventually_mem (s.isOpen_near.mem_nhds r) | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | refine (h.and er).mp (eventually_of_forall ?_) | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ β (x : S),
s.bottcherNearIter n c x = s.bottcherNearIter n c z β§ (c, (f c)^[n] x) β s.near β
s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ βαΆ (x : S) in π z, s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | intro x β¨e, mβ© | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ β (x : S),
s.bottcherNearIter n c x = s.bottcherNearIter n c z β§ (c, (f c)^[n] x) β s.near β
s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNearIter n c x = s.bottcherNearIter n c z
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
β’ β (x : S),
s.bottcherNearIter n c x = s.bottcherNearIter n c z β§ (c, (f c)^[n] x) β s.near β
s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Super.bottcherNearIter] at e | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNearIter n c x = s.bottcherNearIter n c z
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNear c ((f c)^[n] x) = s.bottcherNear c ((f c)^[n] z)
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNearIter n c x = s.bottcherNearIter n c z
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Super.bottcherNearIter, Function.iterate_add_apply, s.bottcherNear_eqn_iter m,
s.bottcherNear_eqn_iter r, e] | case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNear c ((f c)^[n] x) = s.bottcherNear c ((f c)^[n] z)
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
instβ : OnePreimage s
h : βαΆ (x : S) in π z, s.bottcherNearIter n c x = s.bottcherNearIter n c z
k : β
er : βαΆ (w : S) in π z, (c, (f c)^[n] w) β s.near
x : S
e : s.bottcherNear c ((f c)^[n] x) = s.bottcherNear c ((f c)^[n] z)
m : (c, (f c)^[n] x) β s.near
β’ s.bottcherNearIter (k + n) c x = s.bottcherNearIter (k + n) c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | by_cases p0 : s.potential c z = 0 | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : Β¬s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [s.potential_eq_zero_of_onePreimage] at p0 | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [p0] | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | exact s.bottcherNearIter_nontrivial_a | case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : z = a
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | exact nontrivialHolomorphicAt_of_mfderiv_ne_zero (s.bottcherNearIter_holomorphic r').along_snd
(s.bottcherNearIter_mfderiv_ne_zero mc (p.not_precritical p0)) | case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : Β¬s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
r : (c, (f c)^[n] z) β s.near
p : Postcritical s c z
instβ : OnePreimage s
m : β
nm : n β€ m
mc : mfderiv I I (s.bottcherNear c) ((f c)^[m] z) β 0
r' : (c, (f c)^[m] z) β s.near
p0 : Β¬s.potential c z = 0
β’ NontrivialHolomorphicAt (s.bottcherNearIter m c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | contrapose m | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : βαΆ (w : S) in π z, s.potential c z β€ s.potential c w
β’ z = a | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ Β¬βαΆ (w : S) in π z, s.potential c z β€ s.potential c w | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : βαΆ (w : S) in π z, s.potential c z β€ s.potential c w
β’ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [Filter.not_eventually, not_le] | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ Β¬βαΆ (w : S) in π z, s.potential c z β€ s.potential c w | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ Β¬βαΆ (w : S) in π z, s.potential c z β€ s.potential c w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rcases s.nice_nz p.basin z (le_refl _) with β¨near, ncβ© | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (f c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((f c)^[k] z) β 0
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | set f : S β β := s.bottcherNearIter (s.nz c z) c | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (f c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((f c)^[k] z) β 0
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
f : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super f d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (f c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((f c)^[k] z) β 0
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | have o : π (f z) = Filter.map f (π z) :=
(nontrivialHolomorphicAt_of_mfderiv_ne_zero (s.bottcherNearIter_holomorphic near).along_snd
(s.bottcherNearIter_mfderiv_ne_zero (nc _ (le_refl _))
(p.not_precritical ((s.potential_ne_zero _).mpr m)))).nhds_eq_map_nhds | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | have e : βαΆ x : β in π (f z), abs x < abs (f z) := by
apply frequently_smaller; contrapose m; simp only [not_not] at m β’
replace m := (s.bottcherNear_eq_zero near).mp m
rw [s.preimage_eq] at m; exact m | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (x : β) in π (f z), Complex.abs x < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [o, Filter.frequently_map] at e | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (x : β) in π (f z), Complex.abs x < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (x : β) in π (f z), Complex.abs x < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply e.mp | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply (((s.isOpen_preimage _).snd_preimage c).eventually_mem near).mp | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z,
x β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}} β
Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z, Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | refine eventually_of_forall fun w m lt β¦ ?_ | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z,
x β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}} β
Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : w β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}}
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
β’ βαΆ (x : S) in π z,
x β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}} β
Complex.abs (f x) < Complex.abs (f z) β s.potential c x < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [mem_setOf, mem_setOf] at m | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : w β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}}
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : ((c, w).1, (fβ (c, w).1)^[s.nz c z] (c, w).2) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : w β {b | (c, b) β {p | (p.1, (fβ p.1)^[s.nz c z] p.2) β s.near}}
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only at m | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : ((c, w).1, (fβ (c, w).1)^[s.nz c z] (c, w).2) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : ((c, w).1, (fβ (c, w).1)^[s.nz c z] (c, w).2) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [s.potential_eq m, s.potential_eq near, Super.potential'] | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] w)) ^ (βd ^ s.nz c z)β»ΒΉ <
Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] z)) ^ (βd ^ s.nz c z)β»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ s.potential c w < s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | exact Real.rpow_lt_rpow (Complex.abs.nonneg _) lt
(inv_pos.mpr (pow_pos (Nat.cast_pos.mpr s.dp) _)) | case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] w)) ^ (βd ^ s.nz c z)β»ΒΉ <
Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] z)) ^ (βd ^ s.nz c z)β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
mβ : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
e : βαΆ (a : S) in π z, Complex.abs (f a) < Complex.abs (f z)
w : S
m : (c, (fβ c)^[s.nz c z] w) β s.near
lt : Complex.abs (f w) < Complex.abs (f z)
β’ Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] w)) ^ (βd ^ s.nz c z)β»ΒΉ <
Complex.abs (s.bottcherNear c ((fβ c)^[s.nz c z] z)) ^ (βd ^ s.nz c z)β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply frequently_smaller | S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ βαΆ (x : β) in π (f z), Complex.abs x < Complex.abs (f z) | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ βαΆ (x : β) in π (f z), Complex.abs x < Complex.abs (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | contrapose m | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ f z β 0 | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : Β¬f z β 0
⒠¬¬z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
m : Β¬z = a
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
β’ f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [not_not] at m β’ | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : Β¬f z β 0
⒠¬¬z = a | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : f z = 0
β’ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : Β¬f z β 0
⒠¬¬z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | replace m := (s.bottcherNear_eq_zero near).mp m | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : f z = 0
β’ z = a | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : (fβ c)^[s.nz c z] z = a
β’ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : f z = 0
β’ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [s.preimage_eq] at m | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : (fβ c)^[s.nz c z] z = a
β’ z = a | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : z = a
β’ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : (fβ c)^[s.nz c z] z = a
β’ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | exact m | case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : z = a
β’ z = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
S : Type
instββ΅ : TopologicalSpace S
instββ΄ : CompactSpace S
instβΒ³ : T3Space S
instβΒ² : ChartedSpace β S
instβΒΉ : AnalyticManifold I S
fβ : β β S β S
c : β
a z z0 z1 : S
d n : β
sβ s : Super fβ d a
instβ : OnePreimage s
p : Postcritical s c z
near : (c, (fβ c)^[s.nz c z] z) β s.near
nc : β (k : β), s.nz c z β€ k β mfderiv I I (s.bottcherNear c) ((fβ c)^[k] z) β 0
f : S β β := s.bottcherNearIter (s.nz c z) c
o : π (f z) = Filter.map f (π z)
m : z = a
β’ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_contra bad | c : β
d : β
instβ : Fact (2 β€ d)
β’ InjOn (bottcher d) (multibrotExt d) | c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ InjOn (bottcher d) (multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [InjOn, not_forall, β ne_eq] at bad | c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False | c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases bad with β¨x, xm, y, ym, bxy, xyβ© | c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize hb : potential d x = b | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have b1 : b < 1 := by rwa [β hb, potential_lt_one] | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set u := {c | potential d c β€ b} | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t0 := u ΓΛ’ u | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t1 := {q : π Γ π | bottcher d q.1 = bottcher d q.2 β§ q β t0} | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t2 := {q : π Γ π | q.1 β q.2 β§ q β t1} | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear x xm y ym bxy xy hb | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ue : u β multibrotExt d := by intro c m; rw [β potential_lt_one]; exact lt_of_le_of_lt m b1 | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t01 : t1 β t0 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12 : t2 β t1 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have uc : IsClosed u := isClosed_le potential_continuous continuous_const | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t0c : IsClosed t0 := uc.prod uc | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12' : closure t2 β t1 := by rw [β t1c.closure_eq]; exact closure_mono t12 | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2c' : IsCompact (closure t2) := isClosed_closure.isCompact | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2ne' : (closure t2).Nonempty := t2ne.closure | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pc : Continuous fun q : π Γ π β¦ potential d q.1 := potential_continuous.comp continuous_fst | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
β’ False
TACTIC:
|
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