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/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [isClosed_iff_frequently] | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ IsClosed t | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β (x : π), (βαΆ (y : π) in π x, y β t) β x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ IsClosed t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro x e | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β (x : π), (βαΆ (y : π) in π x, y β t) β x β t | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
β’ x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β (x : π), (βαΆ (y : π) in π x, y β t) β x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | by_contra xt | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
β’ x β t | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
β’ x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have pb : potential d x = abs b := by
apply tendsto_nhds_unique_of_frequently_eq potential_continuous.continuousAt
continuousAt_const
refine e.mp (eventually_of_forall ?_); intro z β¨_, hβ©; rw [β h.self_of_nhds, abs_bottcher] | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ False | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [β pb, potential_lt_one] at b1 | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | have e' : βαΆ y in π[{x}αΆ] x, y β t := by
simp only [frequently_nhdsWithin_iff, mem_compl_singleton_iff]
refine e.mp (eventually_of_forall fun z zt β¦ β¨zt, ?_β©)
contrapose xt; simp only [not_not] at xt β’; rwa [β xt] | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose xt | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ False | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
xt : Β¬False
β’ Β¬x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | clear xt | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
xt : Β¬False
β’ Β¬x β t | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ Β¬x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
xt : Β¬False
β’ Β¬x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [not_not] | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ Β¬x β t | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ Β¬x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | use b1 | case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ x β t | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ βαΆ (e : π) in π x, bottcher d e = b | Please generate a tactic in lean4 to solve the state.
STATE:
case left
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | cases' HolomorphicAt.eventually_eq_or_eventually_ne (bottcherHolomorphic d _ b1)
holomorphicAt_const with h h | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ βαΆ (e : π) in π x, bottcher d e = b | case right.inl
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π x, bottcher d w = ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
β’ βαΆ (e : π) in π x, bottcher d e = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | use h | case right.inl
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π x, bottcher d w = ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β b
β’ βαΆ (e : π) in π x, bottcher d e = b | Please generate a tactic in lean4 to solve the state.
STATE:
case right.inl
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π x, bottcher d w = ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β ?m.320207
β’ βαΆ (e : π) in π x, bottcher d e = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose h | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β b
β’ βαΆ (e : π) in π x, bottcher d e = b | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : Β¬βαΆ (e : π) in π x, bottcher d e = b
β’ Β¬βαΆ (w : π) in π[β ] x, bottcher d w β b | Please generate a tactic in lean4 to solve the state.
STATE:
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (w : π) in π[β ] x, bottcher d w β b
β’ βαΆ (e : π) in π x, bottcher d e = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [Filter.not_eventually, not_not] at h β’ | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : Β¬βαΆ (e : π) in π x, bottcher d e = b
β’ Β¬βαΆ (w : π) in π[β ] x, bottcher d w β b | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (x : π) in π x, Β¬bottcher d x = b
β’ βαΆ (x : π) in π[β ] x, bottcher d x = b | Please generate a tactic in lean4 to solve the state.
STATE:
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : Β¬βαΆ (e : π) in π x, bottcher d e = b
β’ Β¬βαΆ (w : π) in π[β ] x, bottcher d w β b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exact e'.mp (eventually_of_forall fun y yt β¦ yt.2.self_of_nhds) | case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (x : π) in π x, Β¬bottcher d x = b
β’ βαΆ (x : π) in π[β ] x, bottcher d x = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right.inr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
e' : βαΆ (y : π) in π[β ] x, y β t
h : βαΆ (x : π) in π x, Β¬bottcher d x = b
β’ βαΆ (x : π) in π[β ] x, bottcher d x = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply tendsto_nhds_unique_of_frequently_eq potential_continuous.continuousAt
continuousAt_const | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ potential d x = Complex.abs b | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ βαΆ (x : π) in π x, potential d x = Complex.abs b | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ potential d x = Complex.abs b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | refine e.mp (eventually_of_forall ?_) | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ βαΆ (x : π) in π x, potential d x = Complex.abs b | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ β x β t, potential d x = Complex.abs b | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ βαΆ (x : π) in π x, potential d x = Complex.abs b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro z β¨_, hβ© | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ β x β t, potential d x = Complex.abs b | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
z : π
leftβ : z β multibrotExt d
h : βαΆ (e : π) in π z, bottcher d e = b
β’ potential d z = Complex.abs b | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
β’ β x β t, potential d x = Complex.abs b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [β h.self_of_nhds, abs_bottcher] | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
z : π
leftβ : z β multibrotExt d
h : βαΆ (e : π) in π z, bottcher d e = b
β’ potential d z = Complex.abs b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
e : βαΆ (y : π) in π x, y β t
xt : x β t
z : π
leftβ : z β multibrotExt d
h : βαΆ (e : π) in π z, bottcher d e = b
β’ potential d z = Complex.abs b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [frequently_nhdsWithin_iff, mem_compl_singleton_iff] | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ βαΆ (y : π) in π[β ] x, y β t | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ βαΆ (x_1 : π) in π x, x_1 β t β§ x_1 β x | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ βαΆ (y : π) in π[β ] x, y β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | refine e.mp (eventually_of_forall fun z zt β¦ β¨zt, ?_β©) | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ βαΆ (x_1 : π) in π x, x_1 β t β§ x_1 β x | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
β’ z β x | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
β’ βαΆ (x_1 : π) in π x, x_1 β t β§ x_1 β x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | contrapose xt | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
β’ z β x | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : Β¬z β x
β’ Β¬x β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
xt : x β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
β’ z β x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [not_not] at xt β’ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : Β¬z β x
β’ Β¬x β t | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : z = x
β’ x β t | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : Β¬z β x
β’ Β¬x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rwa [β xt] | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : z = x
β’ x β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
x : π
b1 : x β multibrotExt d
e : βαΆ (y : π) in π x, y β t
pb : potential d x = Complex.abs b
z : π
zt : z β t
xt : z = x
β’ x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | rw [isOpen_iff_eventually] | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ IsOpen t | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β x β t, βαΆ (y : π) in π x, y β t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ IsOpen t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | intro e β¨m, hβ© | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β x β t, βαΆ (y : π) in π x, y β t | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (y : π) in π e, y β t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
β’ β x β t, βαΆ (y : π) in π x, y β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply (isOpen_multibrotExt.eventually_mem m).mp | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (y : π) in π e, y β t | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, x β multibrotExt d β x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (y : π) in π e, y β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | apply (eventually_eventually_nhds.mpr h).mp | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, x β multibrotExt d β x β t | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, (βαΆ (x : π) in π x, bottcher d x = b) β x β multibrotExt d β x β t | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, x β multibrotExt d β x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | exact eventually_of_forall fun f h m β¦ β¨m, hβ© | case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, (βαΆ (x : π) in π x, bottcher d x = b) β x β multibrotExt d β x β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
mβ : c β multibrotExt d
b : β := bottcher d c
hβ : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
e : π
m : e β multibrotExt d
h : βαΆ (e : π) in π e, bottcher d e = b
β’ βαΆ (x : π) in π e, (βαΆ (x : π) in π x, bottcher d x = b) β x β multibrotExt d β x β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcherNontrivial | [634, 1] | [670, 75] | simp only [tu, mem_univ] | cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
tu : t = univ
β’ 0 β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
b : β := bottcher d c
h : βαΆ (x : π) in π c, bottcher d x = b
b1 : Complex.abs b < 1
t : Set π := {c | c β multibrotExt d β§ βαΆ (e : π) in π c, bottcher d e = b}
tu : t = univ
β’ 0 β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | set s := superF d | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ bottcher d '' multibrotExt d = ball 0 1 | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d = ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ bottcher d '' multibrotExt d = ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply subset_antisymm | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d = ball 0 1 | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d β ball 0 1
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d = ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro w | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d β ball 0 1 | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ w β bottcher d '' multibrotExt d β w β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ bottcher d '' multibrotExt d β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [mem_image] | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ w β bottcher d '' multibrotExt d β w β ball 0 1 | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ (β x β multibrotExt d, bottcher d x = w) β w β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ w β bottcher d '' multibrotExt d β w β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro β¨c, m, eβ© | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ (β x β multibrotExt d, bottcher d x = w) β w β ball 0 1 | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ w β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
β’ (β x β multibrotExt d, bottcher d x = w) β w β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [β e] | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ w β ball 0 1 | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ bottcher d c β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ w β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | clear e w | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ bottcher d c β ball 0 1 | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
c : π
m : c β multibrotExt d
β’ bottcher d c β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
w : β
c : π
m : c β multibrotExt d
e : bottcher d c = w
β’ bottcher d c β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | induction c using OnePoint.rec | case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
c : π
m : c β multibrotExt d
β’ bottcher d c β ball 0 1 | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ bottcher d β β ball 0 1
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : βxβ β multibrotExt d
β’ bottcher d βxβ β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
c : π
m : c β multibrotExt d
β’ bottcher d c β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [bottcher, fill_inf] | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ bottcher d β β ball 0 1 | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ 0 β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ bottcher d β β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact mem_ball_self one_pos | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ 0 β ball 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
m : β β multibrotExt d
β’ 0 β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [multibrotExt_coe] at m | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : βxβ β multibrotExt d
β’ bottcher d βxβ β ball 0 1 | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ bottcher d βxβ β ball 0 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : βxβ β multibrotExt d
β’ bottcher d βxβ β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [bottcher, fill_coe, bottcher', mem_ball, Complex.dist_eq, sub_zero] | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ bottcher d βxβ β ball 0 1 | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ Complex.abs (β―.bottcher xβ βxβ) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ bottcher d βxβ β ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact s.bottcher_lt_one (multibrotPost m) | case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ Complex.abs (β―.bottcher xβ βxβ) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hβ
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
xβ : β
m : xβ β multibrot d
β’ Complex.abs (β―.bottcher xβ βxβ) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine _root_.trans ?_ interior_subset | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β bottcher d '' multibrotExt d | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β interior (bottcher d '' multibrotExt d) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine IsPreconnected.relative_clopen (convex_ball _ _).isPreconnected ?_ ?_ ?_ | case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β interior (bottcher d '' multibrotExt d) | case a.refine_1
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (ball 0 1 β© bottcher d '' multibrotExt d).Nonempty
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β interior (bottcher d '' multibrotExt d)
case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© closure (bottcher d '' multibrotExt d) β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β interior (bottcher d '' multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | use 0, mem_ball_self one_pos, β | case a.refine_1
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (ball 0 1 β© bottcher d '' multibrotExt d).Nonempty | case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β β multibrotExt d β§ bottcher d β = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_1
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ (ball 0 1 β© bottcher d '' multibrotExt d).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [multibrotExt_inf, bottcher, fill_inf, true_and_iff] | case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β β multibrotExt d β§ bottcher d β = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β β multibrotExt d β§ bottcher d β = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [IsOpen.interior_eq] | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β interior (bottcher d '' multibrotExt d) | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β bottcher d '' multibrotExt d
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β interior (bottcher d '' multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact inter_subset_right _ _ | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β bottcher d '' multibrotExt d
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d) | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© bottcher d '' multibrotExt d β bottcher d '' multibrotExt d
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [isOpen_iff_eventually] | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d) | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β x β bottcher d '' multibrotExt d, βαΆ (y : β) in π x, y β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ IsOpen (bottcher d '' multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro z β¨c, m, eβ© | case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β x β bottcher d '' multibrotExt d, βαΆ (y : β) in π x, y β bottcher d '' multibrotExt d | case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (y : β) in π z, y β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ β x β bottcher d '' multibrotExt d, βαΆ (y : β) in π x, y β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [β e, (bottcherNontrivial m).nhds_eq_map_nhds, Filter.eventually_map] | case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (y : β) in π z, y β bottcher d '' multibrotExt d | case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (a : π) in π c, bottcher d a β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (y : β) in π z, y β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact
(isOpen_multibrotExt.eventually_mem m).mp (eventually_of_forall fun e m β¦ by use e, m) | case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (a : π) in π c, bottcher d a β bottcher d '' multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_2
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
m : c β multibrotExt d
e : bottcher d c = z
β’ βαΆ (a : π) in π c, bottcher d a β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | use e, m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
mβ : c β multibrotExt d
eβ : bottcher d c = z
e : π
m : e β multibrotExt d
β’ bottcher d e β bottcher d '' multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
z : β
c : π
mβ : c β multibrotExt d
eβ : bottcher d c = z
e : π
m : e β multibrotExt d
β’ bottcher d e β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro x β¨x1, mβ© | case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© closure (bottcher d '' multibrotExt d) β bottcher d '' multibrotExt d | case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
x1 : x β ball 0 1
m : x β closure (bottcher d '' multibrotExt d)
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
β’ ball 0 1 β© closure (bottcher d '' multibrotExt d) β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | simp only [mem_ball, Complex.dist_eq, sub_zero] at x1 | case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
x1 : x β ball 0 1
m : x β closure (bottcher d '' multibrotExt d)
β’ x β bottcher d '' multibrotExt d | case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
x1 : x β ball 0 1
m : x β closure (bottcher d '' multibrotExt d)
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rcases exists_between x1 with β¨b, xb, b1β© | case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | set t := {e | potential d e β€ b} | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have ct : IsCompact t := (isClosed_le potential_continuous continuous_const).isCompact | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have ts : t β multibrotExt d := by
intro c m; rw [β potential_lt_one]; exact lt_of_le_of_lt m b1 | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have mt : x β closure (bottcher d '' t) := by
rw [mem_closure_iff_frequently] at m β’; apply m.mp
have lt : βαΆ y : β in π x, abs y < b :=
Complex.continuous_abs.continuousAt.eventually_lt continuousAt_const xb
refine lt.mp (eventually_of_forall fun y lt m β¦ ?_)
rcases m with β¨c, _, cyβ©; rw [β cy]; rw [β cy, abs_bottcher] at lt
exact β¨c, lt.le, rflβ© | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply image_subset _ ts | case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' multibrotExt d | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [IsClosed.closure_eq] at mt | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' t | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β bottcher d '' t
β’ x β bottcher d '' t
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ x β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact mt | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β bottcher d '' t
β’ x β bottcher d '' t
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t) | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β bottcher d '' t
β’ x β bottcher d '' t
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply IsCompact.isClosed | case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t) | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsCompact (bottcher d '' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsClosed (bottcher d '' t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply IsCompact.image_of_continuousOn ct | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsCompact (bottcher d '' t) | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ IsCompact (bottcher d '' t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine ContinuousOn.mono ?_ ts | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) t | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) (multibrotExt d) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact (bottcherHolomorphic d).continuousOn | case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) (multibrotExt d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.refine_3.intro.intro.a.hs
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
mt : x β closure (bottcher d '' t)
β’ ContinuousOn (bottcher d) (multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | intro c m | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
β’ t β multibrotExt d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
β’ t β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [β potential_lt_one] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ c β multibrotExt d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ potential d c < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ c β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact lt_of_le_of_lt m b1 | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ potential d c < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
c : π
m : c β t
β’ potential d c < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [mem_closure_iff_frequently] at m β’ | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ x β closure (bottcher d '' t) | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : x β closure (bottcher d '' multibrotExt d)
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ x β closure (bottcher d '' t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | apply m.mp | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' t | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | have lt : βαΆ y : β in π x, abs y < b :=
Complex.continuous_abs.continuousAt.eventually_lt continuousAt_const xb | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
lt : βαΆ (y : β) in π x, Complex.abs y < b
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | refine lt.mp (eventually_of_forall fun y lt m β¦ ?_) | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
lt : βαΆ (y : β) in π x, Complex.abs y < b
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
m : y β bottcher d '' multibrotExt d
β’ y β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
lt : βαΆ (y : β) in π x, Complex.abs y < b
β’ βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d β x β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rcases m with β¨c, _, cyβ© | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
m : y β bottcher d '' multibrotExt d
β’ y β bottcher d '' t | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ y β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
mβ : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
m : y β bottcher d '' multibrotExt d
β’ y β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [β cy] | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ y β bottcher d '' t | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ y β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | rw [β cy, abs_bottcher] at lt | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
c : π
lt : potential d c < b
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
lt : Complex.abs y < b
c : π
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_surj | [673, 1] | [708, 81] | exact β¨c, lt.le, rflβ© | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
c : π
lt : potential d c < b
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
s : Super (f d) d β := superF d
x : β
m : βαΆ (x : β) in π x, x β bottcher d '' multibrotExt d
x1 : Complex.abs x < 1
b : β
xb : Complex.abs x < b
b1 : b < 1
t : Set π := {e | potential d e β€ b}
ct : IsCompact t
ts : t β multibrotExt d
ltβ : βαΆ (y : β) in π x, Complex.abs y < b
y : β
c : π
lt : potential d c < b
leftβ : c β multibrotExt d
cy : bottcher d c = y
β’ bottcher d c β bottcher d '' t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | set s := superF d | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
β’ Tendsto (fun z => β―.bottcher c βz * z) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
β’ Tendsto (fun z => β―.bottcher c βz * z) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have e : βαΆ z : β in atInf, s.bottcher c z * z = s.bottcherNear c z * z := by
suffices e : βαΆ z : β in atInf, s.bottcher c z = s.bottcherNear c z by
exact e.mp (eventually_of_forall fun z e β¦ by rw [e])
refine coe_tendsto_inf.eventually (p := fun z β¦ s.bottcher c z = s.bottcherNear c z) ?_
apply s.bottcher_eq_bottcherNear | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rw [Filter.tendsto_congr' e] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun z => s.bottcher c βz * z) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | clear e | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
e : βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have m := bottcherNear_monic (s.superNearC.s (mem_univ c)) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : HasDerivAt (bottcherNear (s.fl c) d) 1 0
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [hasDerivAt_iff_tendsto, sub_zero, bottcherNear_zero, smul_eq_mul, mul_one,
Metric.tendsto_nhds_nhds, Real.dist_eq, Complex.norm_eq_abs, Complex.dist_eq, abs_mul,
abs_of_nonneg (Complex.abs.nonneg _), abs_inv] at m | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : HasDerivAt (bottcherNear (s.fl c) d) 1 0
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : HasDerivAt (bottcherNear (s.fl c) d) 1 0
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [Metric.tendsto_nhds, atInf_basis.eventually_iff, true_and_iff, mem_setOf,
Complex.dist_eq, Complex.norm_eq_abs] | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ β Ξ΅ > 0, β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ Tendsto (fun x => s.bottcherNear c βx * x) atInf (π 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | intro e ep | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ β Ξ΅ > 0, β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < Ξ΅ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
β’ β Ξ΅ > 0, β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rcases m e ep with β¨r, rp, hβ© | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | use 1 / r | case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β β¦x : ββ¦, 1 / r < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β i, β β¦x : ββ¦, i < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | intro z zr | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β β¦x : ββ¦, 1 / r < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
β’ β β¦x : ββ¦, 1 / r < Complex.abs x β Complex.abs (s.bottcherNear c βx * x - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have az0 : abs z β 0 := (lt_trans (one_div_pos.mpr rp) zr).ne' | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have z0 : z β 0 := Complex.abs.ne_zero_iff.mp az0 | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | have zir : abs (zβ»ΒΉ) < r := by
simp only [one_div, map_invβ] at zr β’; exact inv_lt_of_inv_lt rp zr | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | specialize @h zβ»ΒΉ zir | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : (Complex.abs zβ»ΒΉ)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ - zβ»ΒΉ) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
h : β {x : β}, Complex.abs x < r β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [map_invβ, inv_inv, β Complex.abs.map_mul, sub_mul, inv_mul_cancel z0,
mul_comm z _] at h | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : (Complex.abs zβ»ΒΉ)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ - zβ»ΒΉ) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : (Complex.abs zβ»ΒΉ)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ - zβ»ΒΉ) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [Super.bottcherNear, extChartAt_inf, PartialEquiv.trans_apply,
coePartialEquiv_symm_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, RiemannSphere.inv_inf,
toComplex_zero, sub_zero, inv_coe z0, toComplex_coe] | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (s.bottcherNear c βz * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | exact h | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
m : β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, Complex.abs x < Ξ΄ β (Complex.abs x)β»ΒΉ * Complex.abs (bottcherNear (s.fl c) d x - x) < Ξ΅
e : β
ep : e > 0
r : β
rp : r > 0
z : β
zr : 1 / r < Complex.abs z
az0 : Complex.abs z β 0
z0 : z β 0
zir : Complex.abs zβ»ΒΉ < r
h : Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
β’ Complex.abs (bottcherNear (s.fl c) d zβ»ΒΉ * z - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | suffices e : βαΆ z : β in atInf, s.bottcher c z = s.bottcherNear c z by
exact e.mp (eventually_of_forall fun z e β¦ by rw [e]) | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (z : β) in atInf, s.bottcher c βz = s.bottcherNear c βz | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (z : β) in atInf, s.bottcher c βz * z = s.bottcherNear c βz * z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | refine coe_tendsto_inf.eventually (p := fun z β¦ s.bottcher c z = s.bottcherNear c z) ?_ | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (z : β) in atInf, s.bottcher c βz = s.bottcherNear c βz | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (y : π) in π β, (fun z => s.bottcher c z = s.bottcherNear c z) y | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (z : β) in atInf, s.bottcher c βz = s.bottcherNear c βz
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | apply s.bottcher_eq_bottcherNear | cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (y : π) in π β, (fun z => s.bottcher c z = s.bottcherNear c z) y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : β
s : Super (f d) d β := superF d
β’ βαΆ (y : π) in π β, (fun z => s.bottcher c z = s.bottcherNear c z) y
TACTIC:
|
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