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
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | simp only [g, z0, if_false] | case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
z : β
z0 : Β¬z = 0
β’ f z = z ^ d * g f d z | case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
z : β
z0 : Β¬z = 0
β’ f z = z ^ d * (f z / z ^ d) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
z : β
z0 : Β¬z = 0
β’ f z = z ^ d * g f d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | field_simp [z0] | case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
z : β
z0 : Β¬z = 0
β’ f z = z ^ d * (f z / z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
z : β
z0 : Β¬z = 0
β’ f z = z ^ d * (f z / z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rcases fa.exists_ball_analyticOn with β¨r, rp, faβ© | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
fa : AnalyticAt β f c
β’ AnalyticAt β (g f d) c | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
β’ AnalyticAt β (g f d) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
fa : AnalyticAt β f c
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have o : IsOpen (ball c r) := isOpen_ball | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
β’ AnalyticAt β (g f d) c | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
β’ AnalyticAt β (g f d) c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | generalize ht : ball c r = t | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
β’ AnalyticAt β (g f d) c | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
t : Set β
ht : ball c r = t
β’ AnalyticAt β (g f d) c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [ht] at fa o | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
t : Set β
ht : ball c r = t
β’ AnalyticAt β (g f d) c | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticAt β (g f d) c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
fa : AnalyticOn β f (ball c r)
o : IsOpen (ball c r)
t : Set β
ht : ball c r = t
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | suffices h : AnalyticOn β (g f d) t by rw [β ht] at h; exact h _ (mem_ball_self rp) | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticAt β (g f d) c | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have ga : DifferentiableOn β (g f d) (t \ {0}) := by
have e : β z : β, z β t \ {0} β g f d z = f z / z ^ d := by
intro z zs; simp only [Set.mem_diff, Set.mem_singleton_iff] at zs
simp only [g, zs.2, if_false]
rw [differentiableOn_congr e]
apply DifferentiableOn.div (fa.mono (Set.diff_subset _ _)).differentiableOn
exact (Differentiable.pow differentiable_id _).differentiableOn
intro z zs; exact pow_ne_zero _ (Set.mem_diff_singleton.mp zs).2 | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticOn β (g f d) t | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ AnalyticOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ AnalyticOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [analyticOn_iff_differentiableOn o] | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ AnalyticOn β (g f d) t | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ DifferentiableOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ AnalyticOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | by_cases t0 : (0 : β) β t | case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ DifferentiableOn β (g f d) t | case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ DifferentiableOn β (g f d) t
case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : Β¬0 β t
β’ DifferentiableOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
β’ DifferentiableOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Set.not_not_mem] at t0 | case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : Β¬0 β t
β’ DifferentiableOn β (g f d) t | case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ DifferentiableOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : Β¬0 β t
β’ DifferentiableOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact (Complex.differentiableOn_compl_singleton_and_continuousAt_iff (o.mem_nhds t0)).mp β¨ga, gcβ© | case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
gc : ContinuousAt (g f d) 0
β’ DifferentiableOn β (g f d) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
gc : ContinuousAt (g f d) 0
β’ DifferentiableOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [β ht] at h | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
h : AnalyticOn β (g f d) t
β’ AnalyticAt β (g f d) c | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
h : AnalyticOn β (g f d) (ball c r)
β’ AnalyticAt β (g f d) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
h : AnalyticOn β (g f d) t
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact h _ (mem_ball_self rp) | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
h : AnalyticOn β (g f d) (ball c r)
β’ AnalyticAt β (g f d) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
h : AnalyticOn β (g f d) (ball c r)
β’ AnalyticAt β (g f d) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have e : β z : β, z β t \ {0} β g f d z = f z / z ^ d := by
intro z zs; simp only [Set.mem_diff, Set.mem_singleton_iff] at zs
simp only [g, zs.2, if_false] | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ DifferentiableOn β (g f d) (t \ {0}) | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (g f d) (t \ {0}) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ DifferentiableOn β (g f d) (t \ {0})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [differentiableOn_congr e] | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (g f d) (t \ {0}) | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => f x / x ^ d) (t \ {0}) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (g f d) (t \ {0})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | apply DifferentiableOn.div (fa.mono (Set.diff_subset _ _)).differentiableOn | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => f x / x ^ d) (t \ {0}) | case hd
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => x ^ d) (t \ {0})
case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => f x / x ^ d) (t \ {0})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact (Differentiable.pow differentiable_id _).differentiableOn | case hd
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => x ^ d) (t \ {0})
case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0 | case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hd
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ DifferentiableOn β (fun x => x ^ d) (t \ {0})
case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0 | case hx
f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
z : β
zs : z β t \ {0}
β’ z ^ d β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
β’ β x β t \ {0}, x ^ d β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact pow_ne_zero _ (Set.mem_diff_singleton.mp zs).2 | case hx
f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
z : β
zs : z β t \ {0}
β’ z ^ d β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
e : β z β t \ {0}, g f d z = f z / z ^ d
z : β
zs : z β t \ {0}
β’ z ^ d β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ β z β t \ {0}, g f d z = f z / z ^ d | f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t \ {0}
β’ g f d z = f z / z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
β’ β z β t \ {0}, g f d z = f z / z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Set.mem_diff, Set.mem_singleton_iff] at zs | f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t \ {0}
β’ g f d z = f z / z ^ d | f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t β§ Β¬z = 0
β’ g f d z = f z / z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t \ {0}
β’ g f d z = f z / z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [g, zs.2, if_false] | f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t β§ Β¬z = 0
β’ g f d z = f z / z ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
z : β
zs : z β t β§ Β¬z = 0
β’ g f d z = f z / z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Set.diff_singleton_eq_self t0] at ga | case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ DifferentiableOn β (g f d) t | case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) t
t0 : 0 β t
β’ DifferentiableOn β (g f d) t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ DifferentiableOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact ga | case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) t
t0 : 0 β t
β’ DifferentiableOn β (g f d) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) t
t0 : 0 β t
β’ DifferentiableOn β (g f d) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Metric.continuousAt_iff] | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ ContinuousAt (g f d) 0 | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ ContinuousAt (g f d) 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro e ep | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < Ξ΅ | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
e : β
ep : e > 0
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
β’ β Ξ΅ > 0, β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < Ξ΅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rcases Metric.eventually_nhds_iff.mp
(Asymptotics.isBigOWith_iff.mp (s.approx.forall_isBigOWith (by linarith : e / 2 > 0))) with
β¨t, tp, hβ© | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
e : β
ep : e > 0
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e | case intro.intro
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
e : β
ep : e > 0
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | use t, tp | case intro.intro
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e | case right
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β {x : β}, dist x 0 < t β dist (g f d x) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β Ξ΄ > 0, β {x : β}, dist x 0 < Ξ΄ β dist (g f d x) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | case right
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β {x : β}, dist x 0 < t β dist (g f d x) (g f d 0) < e | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
z : β
zs : dist z 0 < t
β’ dist (g f d z) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
f : β β β
d : β
z : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
β’ β {x : β}, dist x 0 < t β dist (g f d x) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | specialize h zs | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
z : β
zs : dist z 0 < t
β’ dist (g f d z) (g f d 0) < e | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : βf z - z ^ dβ β€ e / 2 * βz ^ dβ
β’ dist (g f d z) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
h : β β¦y : ββ¦, dist y 0 < t β βf y - y ^ dβ β€ e / 2 * βy ^ dβ
z : β
zs : dist z 0 < t
β’ dist (g f d z) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Complex.norm_eq_abs] at h | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : βf z - z ^ dβ β€ e / 2 * βz ^ dβ
β’ dist (g f d z) (g f d 0) < e | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ dist (g f d z) (g f d 0) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : βf z - z ^ dβ β€ e / 2 * βz ^ dβ
β’ dist (g f d z) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [g, Complex.dist_eq] | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ dist (g f d z) (g f d 0) < e | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ dist (g f d z) (g f d 0) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | by_cases z0 : z = 0 | case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e | case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e
case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case right
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [z0, if_false, eq_self_iff_true, if_true] | case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e | case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1) < e | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | calc
abs (f z / z ^ d - 1) = abs (f z * (z ^ d)β»ΒΉ - 1) := by rw [div_eq_mul_inv]
_ = abs ((f z - z ^ d) * (z ^ d)β»ΒΉ) := by
rw [mul_sub_right_distrib, mul_inv_cancel (pow_ne_zero d z0)]
_ = abs (f z - z ^ d) * (abs (z ^ d))β»ΒΉ := by rw [Complex.abs.map_mul, map_invβ]
_ β€ e / 2 * abs (z ^ d) * (abs (z ^ d))β»ΒΉ := by bound
_ = e / 2 * (abs (z ^ d) * (abs (z ^ d))β»ΒΉ) := by ring
_ β€ e / 2 * 1 := by bound
_ = e / 2 := by ring
_ < e := half_lt_self ep | case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1) < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | linarith | f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
e : β
ep : e > 0
β’ e / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
tβ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
t : Set β
o : IsOpen t
fa : AnalyticOn β f t
ht : ball c r = t
ga : DifferentiableOn β (g f d) (t \ {0})
t0 : 0 β t
e : β
ep : e > 0
β’ e / 2 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [z0, sub_self, AbsoluteValue.map_zero] | case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e | case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ 0 < e | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ Complex.abs ((if z = 0 then 1 else f z / z ^ d) - if True then 1 else f 0 / 0 ^ d) < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact ep | case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ 0 < e | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : z = 0
β’ 0 < e
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [div_eq_mul_inv] | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1) = Complex.abs (f z * (z ^ d)β»ΒΉ - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1) = Complex.abs (f z * (z ^ d)β»ΒΉ - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [mul_sub_right_distrib, mul_inv_cancel (pow_ne_zero d z0)] | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z * (z ^ d)β»ΒΉ - 1) = Complex.abs ((f z - z ^ d) * (z ^ d)β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z * (z ^ d)β»ΒΉ - 1) = Complex.abs ((f z - z ^ d) * (z ^ d)β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Complex.abs.map_mul, map_invβ] | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs ((f z - z ^ d) * (z ^ d)β»ΒΉ) = Complex.abs (f z - z ^ d) * (Complex.abs (z ^ d))β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs ((f z - z ^ d) * (z ^ d)β»ΒΉ) = Complex.abs (f z - z ^ d) * (Complex.abs (z ^ d))β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | bound | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z - z ^ d) * (Complex.abs (z ^ d))β»ΒΉ β€ e / 2 * Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ Complex.abs (f z - z ^ d) * (Complex.abs (z ^ d))β»ΒΉ β€ e / 2 * Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | ring | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ = e / 2 * (Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ = e / 2 * (Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | bound | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * (Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ) β€ e / 2 * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * (Complex.abs (z ^ d) * (Complex.abs (z ^ d))β»ΒΉ) β€ e / 2 * 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | ring | f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * 1 = e / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
tβΒΉ : Set β
s : SuperAt f d
c : β
faβ : AnalyticAt β f c
r : β
rp : 0 < r
tβ : Set β
o : IsOpen tβ
fa : AnalyticOn β f tβ
ht : ball c r = tβ
ga : DifferentiableOn β (g f d) (tβ \ {0})
t0 : 0 β tβ
e : β
ep : e > 0
t : β
tp : t > 0
z : β
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) β€ e / 2 * Complex.abs (z ^ d)
z0 : Β¬z = 0
β’ e / 2 * 1 = e / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z z0 zs | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
β’ β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : z β ball 0 r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
β’ β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs, lt_min_iff] at zs | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : z β ball 0 r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : Complex.abs z < r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : z β ball 0 r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | specialize gs zs | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : Complex.abs z < r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (g f d z - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gs : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
z0 : z β 0
zs : Complex.abs z < r
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [g, z0, if_false, eq_self_iff_true, if_true] at gs | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (g f d z - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (f z / z ^ d - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (g f d z - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | exact gs.le | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (f z / z ^ d - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
z : β
z0 : z β 0
zs : Complex.abs z < r
gs : Complex.abs (f z / z ^ d - 1) < 1 / 4
β’ Complex.abs (f z / z ^ d - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z zs | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ β {z : β}, z β ball 0 r β Complex.abs z β€ 1 / 2 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ Complex.abs z β€ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ β {z : β}, z β ball 0 r β Complex.abs z β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs] at zs | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ Complex.abs z β€ 1 / 2 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : Complex.abs z < r
β’ Complex.abs z β€ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ Complex.abs z β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | exact le_trans zs.le r2 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : Complex.abs z < r
β’ Complex.abs z β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : Complex.abs z < r
β’ Complex.abs z β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z zs | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ MapsTo f (ball 0 r) (ball 0 r) | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ f z β ball 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ MapsTo f (ball 0 r) (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs] at zs gs β’ | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ f z β ball 0 r | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ Complex.abs (f z) < r | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
gs : β {z : β}, z β 0 β z β ball 0 r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z : β
zs : z β ball 0 r
β’ f z β ball 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | by_cases z0 : z = 0 | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ Complex.abs (f z) < r | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : z = 0
β’ Complex.abs (f z) < r
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z) < r | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
β’ Complex.abs (f z) < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | calc abs (f z)
_ = abs (f z / z ^ d * z ^ d) := by rw [div_mul_cancelβ _ (pow_ne_zero d z0)]
_ = abs (f z / z ^ d - 1 + 1) * abs z ^ d := by
simp only [AbsoluteValue.map_mul, Complex.abs_pow, sub_add_cancel]
_ β€ (abs (f z / z ^ d - 1) + abs (1 : β)) * r ^ d := by bound
_ β€ (1 / 4 + abs (1 : β)) * r ^ d := by bound [gs z0 zs]
_ β€ 5 / 4 * r ^ (d - 1) * r := by
rw [mul_assoc, β pow_succ, Nat.sub_add_cancel (le_trans one_le_two s.d2)]; norm_num
_ β€ 5 / 4 * (1 / 2 : β) ^ (d - 1) * r := by bound
_ β€ 5 / 4 * (1 / 2 : β) ^ (2 - 1) * r := by bound
_ = 5 / 8 * r := by norm_num
_ < r := by linarith | case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z) < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z) < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [z0, s.f0, rp, AbsoluteValue.map_zero] | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : z = 0
β’ Complex.abs (f z) < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : z = 0
β’ Complex.abs (f z) < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | rw [div_mul_cancelβ _ (pow_ne_zero d z0)] | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z) = Complex.abs (f z / z ^ d * z ^ d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z) = Complex.abs (f z / z ^ d * z ^ d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [AbsoluteValue.map_mul, Complex.abs_pow, sub_add_cancel] | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d * z ^ d) = Complex.abs (f z / z ^ d - 1 + 1) * Complex.abs z ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d * z ^ d) = Complex.abs (f z / z ^ d - 1 + 1) * Complex.abs z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1 + 1) * Complex.abs z ^ d β€ (Complex.abs (f z / z ^ d - 1) + Complex.abs 1) * r ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ Complex.abs (f z / z ^ d - 1 + 1) * Complex.abs z ^ d β€ (Complex.abs (f z / z ^ d - 1) + Complex.abs 1) * r ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound [gs z0 zs] | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (Complex.abs (f z / z ^ d - 1) + Complex.abs 1) * r ^ d β€ (1 / 4 + Complex.abs 1) * r ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (Complex.abs (f z / z ^ d - 1) + Complex.abs 1) * r ^ d β€ (1 / 4 + Complex.abs 1) * r ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | rw [mul_assoc, β pow_succ, Nat.sub_add_cancel (le_trans one_le_two s.d2)] | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (1 / 4 + Complex.abs 1) * r ^ d β€ 5 / 4 * r ^ (d - 1) * r | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (1 / 4 + Complex.abs 1) * r ^ d β€ 5 / 4 * r ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (1 / 4 + Complex.abs 1) * r ^ d β€ 5 / 4 * r ^ (d - 1) * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | norm_num | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (1 / 4 + Complex.abs 1) * r ^ d β€ 5 / 4 * r ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ (1 / 4 + Complex.abs 1) * r ^ d β€ 5 / 4 * r ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * r ^ (d - 1) * r β€ 5 / 4 * (1 / 2) ^ (d - 1) * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * r ^ (d - 1) * r β€ 5 / 4 * (1 / 2) ^ (d - 1) * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * (1 / 2) ^ (d - 1) * r β€ 5 / 4 * (1 / 2) ^ (2 - 1) * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * (1 / 2) ^ (d - 1) * r β€ 5 / 4 * (1 / 2) ^ (2 - 1) * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | norm_num | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * (1 / 2) ^ (2 - 1) * r = 5 / 8 * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 4 * (1 / 2) ^ (2 - 1) * r = 5 / 8 * r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | linarith | f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 8 * r < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperAt f d
r : β
rp : 0 < r
r2 : r β€ 1 / 2
fa : AnalyticOn β f (ball 0 r)
gsβ : β {z : β}, Complex.abs z < r β Complex.abs (g f d z - 1) < 1 / 4
z : β
zs : Complex.abs z < r
gs : β {z : β}, z β 0 β Complex.abs z < r β Complex.abs (f z / z ^ d - 1) β€ 1 / 4
z0 : Β¬z = 0
β’ 5 / 8 * r < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | rcases s.fa0.exists_ball_analyticOn with β¨r0, r0p, faβ© | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
β’ β t, SuperNear f d t | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
β’ β t, SuperNear f d t | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
β’ β t, SuperNear f d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | rcases Metric.continuousAt_iff.mp (s.ga_of_fa (fa 0 (mem_ball_self r0p))).continuousAt (1 / 4)
(by norm_num) with
β¨r1, r1p, gsβ© | case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
β’ β t, SuperNear f d t | case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
β’ β t, SuperNear f d t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
β’ β t, SuperNear f d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | set r := min r0 (min r1 (1 / 2)) | case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
β’ β t, SuperNear f d t | case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ β t, SuperNear f d t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
β’ β t, SuperNear f d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | use ball 0 r | case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ β t, SuperNear f d t | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ SuperNear f d (ball 0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ β t, SuperNear f d t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have rp : 0 < r := by bound | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ SuperNear f d (ball 0 r) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
β’ SuperNear f d (ball 0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ SuperNear f d (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have r2 : r β€ 1 / 2 := le_trans (min_le_right _ _) (min_le_right _ _) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
β’ SuperNear f d (ball 0 r) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
β’ SuperNear f d (ball 0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
β’ SuperNear f d (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have rr1 : r β€ r1 := le_trans (min_le_right r0 _) (min_le_left r1 _) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
β’ SuperNear f d (ball 0 r) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
β’ SuperNear f d (ball 0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
β’ SuperNear f d (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | simp only [g0, dist_zero_right, Complex.norm_eq_abs, Complex.dist_eq, sub_zero] at gs | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
β’ SuperNear f d (ball 0 r) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
gs : β {x : β}, Complex.abs x < r1 β Complex.abs (g f d x - 1) < 1 / 4
β’ SuperNear f d (ball 0 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
β’ SuperNear f d (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | exact s.super_on_ball rp r2 (fa.mono (Metric.ball_subset_ball (min_le_left r0 _))) (fun {z} zr β¦
gs (lt_of_lt_of_le zr rr1)) | case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
gs : β {x : β}, Complex.abs x < r1 β Complex.abs (g f d x - 1) < 1 / 4
β’ SuperNear f d (ball 0 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
r : β := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r β€ 1 / 2
rr1 : r β€ r1
gs : β {x : β}, Complex.abs x < r1 β Complex.abs (g f d x - 1) < 1 / 4
β’ SuperNear f d (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | norm_num | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
β’ 1 / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
β’ 1 / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | bound | f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ 0 < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperAt f d
r0 : β
r0p : 0 < r0
fa : AnalyticOn β f (ball 0 r0)
r1 : β
r1p : r1 > 0
gs : β {x : β}, dist x 0 < r1 β dist (g f d x) (g f d 0) < 1 / 4
r : β := min r0 (min r1 (1 / 2))
β’ 0 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | by_cases z0 : z = 0 | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
β’ Complex.abs (g f d z - 1) β€ 1 / 4 | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : Β¬z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
β’ Complex.abs (g f d z - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | simp only [z0, g0, sub_self, AbsoluteValue.map_zero, one_div, inv_nonneg, zero_le_bit0,
zero_le_one] | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4 | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ 0 β€ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | norm_num | case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ 0 β€ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z = 0
β’ 0 β€ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | simp only [g, z0, if_false, s.gs' z0 zt] | case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : Β¬z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : Β¬z = 0
β’ Complex.abs (g f d z - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | have h := s.gs zt | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
β’ g f d z β 0 | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Complex.abs (g f d z - 1) β€ 1 / 4
β’ g f d z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
β’ g f d z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | contrapose h | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Complex.abs (g f d z - 1) β€ 1 / 4
β’ g f d z β 0 | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Β¬g f d z β 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Complex.abs (g f d z - 1) β€ 1 / 4
β’ g f d z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | simp only [not_not] at h | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Β¬g f d z β 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : Β¬g f d z β 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | simp only [h] | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4 | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (0 - 1) β€ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (g f d z - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | norm_num | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (0 - 1) β€ 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
h : g f d z = 0
β’ Β¬Complex.abs (0 - 1) β€ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.f_ne_zero | [209, 1] | [210, 87] | simp only [s.fg, mul_ne_zero (pow_ne_zero _ z0) (s.g_ne_zero zt), Ne, not_false_iff] | f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z β 0
β’ f z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
zβ : β
t : Set β
s : SuperNear f d t
z : β
zt : z β t
z0 : z β 0
β’ f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_eqn | [245, 1] | [248, 68] | intro n | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ β (n : β), term f d n (f z) = term f d (n + 1) z ^ d | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
n : β
β’ term f d n (f z) = term f d (n + 1) z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ β (n : β), term f d n (f z) = term f d (n + 1) z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_eqn | [245, 1] | [248, 68] | simp only [term, β Function.iterate_succ_apply, pow_mul_nat, div_mul, pow_succ' _ (n + 1),
mul_div_cancel_leftβ _ s.dz, Nat.succ_eq_add_one, Nat.cast_mul] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
n : β
β’ term f d n (f z) = term f d (n + 1) z ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
n : β
β’ term f d n (f z) = term f d (n + 1) z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | rw [term] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * term f d 0 z) ^ d | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d (f^[0] z) ^ (1 / β(d ^ (0 + 1)))) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * term f d 0 z) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | simp only [Function.iterate_zero, id, pow_one, one_div] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d (f^[0] z) ^ (1 / β(d ^ (0 + 1)))) ^ d | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d z ^ (β(d ^ (0 + 1)))β»ΒΉ) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d (f^[0] z) ^ (1 / β(d ^ (0 + 1)))) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | rw [mul_pow, pow_mul_nat, zero_add, pow_one, inv_mul_cancel _] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d z ^ (β(d ^ (0 + 1)))β»ΒΉ) ^ d | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = z ^ d * g f d z ^ 1
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ βd β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = (z * g f d z ^ (β(d ^ (0 + 1)))β»ΒΉ) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | rw [s.fg] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = z ^ d * g f d z ^ 1 | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ z ^ d * g f d z = z ^ d * g f d z ^ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ f z = z ^ d * g f d z ^ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | simp only [Complex.cpow_one] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ z ^ d * g f d z = z ^ d * g f d z ^ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ z ^ d * g f d z = z ^ d * g f d z ^ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | simp only [Ne, Nat.cast_eq_zero] | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ βd β 0 | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ Β¬d = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ βd β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | exact (gt_of_ge_of_gt s.d2 (by norm_num)).ne' | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ Β¬d = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ Β¬d = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_base | [255, 1] | [260, 50] | norm_num | f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
d : β
z : β
t : Set β
s : SuperNear f d t
β’ 2 > 0
TACTIC:
|
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