url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rcases h with β¨k, kb, nkβ© | case succ
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
hβ : g a = g b
n : β
h : β k β€ b, g n = g k
β’ β k β€ b, g (n + 1) = g k | case succ.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
β’ β k β€ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
hβ : g a = g b
n : β
h : β k β€ b, g n = g k
β’ β k β€ b, g (n + 1) = g k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | by_cases e : k = b | case succ.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
β’ β k β€ b, g (n + 1) = g k | case pos
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ β k β€ b, g (n + 1) = g k
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
β’ β k β€ b, g (n + 1) = g k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use a + 1, Nat.succ_le_iff.mpr ab | case pos
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ β k β€ b, g (n + 1) = g k
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k ... | case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ g (n + 1) = g (a + 1)
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ β k β€ b, g (n + 1) = g k
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [β hg, β hg, Function.iterate_succ_apply', Function.iterate_succ_apply', hg, hg, nk, e, h] | case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ g (n + 1) = g (a + 1)
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k :... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ β k β€ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : k = b
β’ g (n + 1) = g (a + 1)
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | use k + 1, Nat.succ_le_iff.mpr (Ne.lt_of_le e kb) | case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ β k β€ b, g (n + 1) = g k | case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ g (n + 1) = g (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ β k β€ b, g (n + 1) = g k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_repeat | [424, 1] | [441, 61] | rw [β hg, β hg, Function.iterate_succ_apply', Function.iterate_succ_apply', hg, hg, nk] | case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ g (n + 1) = g (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c : β
d : β
instβ : Fact (2 β€ d)
a b : β
ab : a < b
g : β β β
hg : β (n : β), (f' d c)^[n] c = g n
h : g a = g b
n k : β
kb : k β€ b
nk : g n = g k
e : Β¬k = b
β’ g (n + 1) = g (k + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_zero | [444, 1] | [447, 75] | have i0 : (f d c)^[0] c = c := by rw [Function.iterate_zero_apply] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_zero | [444, 1] | [447, 75] | have i1 : (f d c)^[n + 1] c = c := by simp only [Function.iterate_succ_apply', h, f_0] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
β’ c β multibrot d | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
i1 : (f d c)^[n + 1] βc = βc
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_zero | [444, 1] | [447, 75] | exact multibrot_of_repeat (Nat.zero_lt_succ _) (_root_.trans i0 i1.symm) | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
i1 : (f d c)^[n + 1] βc = βc
β’ c β multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
i1 : (f d c)^[n + 1] βc = βc
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_zero | [444, 1] | [447, 75] | rw [Function.iterate_zero_apply] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
β’ (f d c)^[0] βc = βc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
β’ (f d c)^[0] βc = βc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_of_zero | [444, 1] | [447, 75] | simp only [Function.iterate_succ_apply', h, f_0] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
β’ (f d c)^[n + 1] βc = βc | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : (f d c)^[n] βc = 0
i0 : (f d c)^[0] βc = βc
β’ (f d c)^[n + 1] βc = βc
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_zero | [450, 1] | [451, 70] | apply multibrot_of_zero | c : β
d : β
instβ : Fact (2 β€ d)
β’ 0 β multibrot d | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ (f d 0)^[?n] β0 = 0
case n
c : β
d : β
instβ : Fact (2 β€ d)
β’ β | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ 0 β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_zero | [450, 1] | [451, 70] | rw [Function.iterate_zero_apply, coe_zero] | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ (f d 0)^[?n] β0 = 0
case n
c : β
d : β
instβ : Fact (2 β€ d)
β’ β | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ (f d 0)^[?n] β0 = 0
case n
c : β
d : β
instβ : Fact (2 β€ d)
β’ β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | by_cases c2 : 2 < abs c | c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
β’ c β multibrot d | case pos
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
β’ c β multibrot d
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | exact multibrot_two_lt c2 | case pos
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
β’ c β multibrot d
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
β’ c β multibrot d
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | simp only [multibrot_coe, not_not] | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | simp only [not_lt] at c2 | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Β¬2 < Complex.abs c
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | generalize hs : abs ((f' d c)^[n] c) = s | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
s : β
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [hs] at h | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
s : β
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c β€ 2
s : β
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have s1 : 1 β€ s := by linarith | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have s1' : 1 β€ s - 1 := by linarith | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have s0 : 0 β€ s := by linarith | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | simp only [tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun x => Complex.abs ((f' d c)^[x] c)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => (f' d c)^[... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [β Filter.tendsto_add_atTop_iff_nat n] | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun x => Complex.abs ((f' d c)^[x] c)) atTop atTop | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n_1 => Complex.abs ((f' d c)^[n_1 + n] c)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun x => Complex.ab... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | apply Filter.tendsto_atTop_mono b | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n_1 => Complex.abs ((f' d c)^[n_1 + n] c)) atTop atTop | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => s * (s - 1) ^ n) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n_1 => Complex.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | refine Filter.Tendsto.mul_atTop (by linarith) tendsto_const_nhds ?_ | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => s * (s - 1) ^ n) atTop atTop | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => (s - 1) ^ n) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => s * (s - 1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | apply tendsto_pow_atTop_atTop_of_one_lt | case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => (s - 1) ^ n) atTop atTop | case neg.h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ 1 < s - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ Tendsto (fun n => (s - 1) ^ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | case neg.h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ 1 < s - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ 1 < s - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
β’ 1 β€ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
β’ 1 β€ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
β’ 1 β€ s - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
β’ 1 β€ s - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
β’ 0 β€ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
β’ 0 β€ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | intro k | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c) | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
β’ s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | induction' k with k p | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
β’ s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c) | case zero
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ s * (s - 1) ^ 0 β€ Complex.abs ((f' d c)^[0 + n] c)
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.a... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
β’ s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | simp only [pow_zero, mul_one, zero_add, Nat.zero_eq, hs, le_refl] | case zero
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ s * (s - 1) ^ 0 β€ Complex.abs ((f' d c)^[0 + n] c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
β’ s * (s - 1) ^ 0 β€ Complex.abs ((f' d c)^[0 + n] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | simp only [Nat.succ_add, Function.iterate_succ_apply'] | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs ((f' d c)^[k + 1 + n] c) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c ((f' d c)^[k + n] c)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ s * (s - 1) ^ (k + 1) β€ Complex.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | generalize hz : (f' d c)^[k + n] c = z | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c ((f' d c)^[k + n] c)) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
z : β
hz : (f' d c)^[k + n] c = z
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ s * (s - 1) ^ (k + 1) β€ Complex.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [hz] at p | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
z : β
hz : (f' d c)^[k + n] c = z
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
p : s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
z : β
hz : (f' d c)^[k + n] c = z
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have ss1 : 1 β€ s * (s - 1) ^ k := by bound | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
β’ s * (s - 1) ^ (k ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have k2 : k β€ k * 2 := by linarith | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (f' d c z) | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | calc abs (f' d c z)
_ = abs (z ^ d + c) := rfl
_ β₯ abs (z ^ d) - abs c := by bound
_ = abs z ^ d - abs c := by rw [Complex.abs.map_pow]
_ β₯ (s * (s - 1) ^ k) ^ d - 2 := by bound
_ β₯ (s * (s - 1) ^ k) ^ 2 - 2 := by bound
_ = s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 := by rw [mul_pow, pow_mul, mul_one]
_ β₯ s ^ 2 *... | case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ s * (s - 1) ^ (k + 1) β€ Complex.abs (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
β’ 1 β€ s * (s - 1) ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
β’ 1 β€ s * (s - 1) ^ k
TACTIC:... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
β’ k β€ k * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
β’ k... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ Complex.abs (z ^ d + c) β₯ Complex.abs (z ^ d) -... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [Complex.abs.map_pow] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ Complex.abs (z ^ d) - Complex.abs c = Complex.a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ Complex.abs z ^ d - Complex.abs c β₯ (s * (s - 1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ (s * (s - 1) ^ k) ^ d - 2 β₯ (s * (s - 1) ^ k) ^... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [mul_pow, pow_mul, mul_one] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ (s * (s - 1) ^ k) ^ 2 - 2 = s ^ 2 * (s - 1) ^ (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 β₯ s ^ 2 * (s ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | ring | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ s ^ 2 * (s - 1) ^ k - s * (s - 1) ^ k = s * ((s... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | rw [pow_succ] | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 : k β€ k * 2
β’ s * ((s - 1) ^ k * (s - 1)) = s * (s - 1) ^ (k ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
k : β
z : β
p : s * (s - 1) ^ k β€ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 β€ s * (s - 1) ^ k
k2 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | linarith | c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ 0 < s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
n : β
c2 : Complex.abs c β€ 2
s : β
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 β€ s
s1' : 1 β€ s - 1
s0 : 0 β€ s
b : β (k : β), s * (s - 1) ^ k β€ Complex.abs ((f' d c)^[k + n] c)
β’ 0 < s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | apply Set.ext | c : β
d : β
instβ : Fact (2 β€ d)
β’ multibrot d = β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2 | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (x : β), x β multibrot d β x β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ multibrot d = β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | intro c | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (x : β), x β multibrot d β x β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2 | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β c β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (x : β), x β multibrot d β x β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | simp only [mem_iInter, mem_preimage, mem_closedBall, Complex.dist_eq, sub_zero] | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β c β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2 | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β c β β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | constructor | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2 | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ (β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2) β c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | intro m n | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2 | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
n : β
β’ Complex.abs ((f' d c)^[n] c) β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ c β multibrot d β β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | contrapose m | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
n : β
β’ Complex.abs ((f' d c)^[n] c) β€ 2 | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : Β¬Complex.abs ((f' d c)^[n] c) β€ 2
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
m : c β multibrot d
n : β
β’ Complex.abs ((f' d c)^[n] c) β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | simp only [not_le] at m | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : Β¬Complex.abs ((f' d c)^[n] c) β€ 2
β’ c β multibrot d | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : 2 < Complex.abs ((f' d c)^[n] c)
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : Β¬Complex.abs ((f' d c)^[n] c) β€ 2
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | exact not_multibrot_of_two_lt m | case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : 2 < Complex.abs ((f' d c)^[n] c)
β’ c β multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
n : β
m : 2 < Complex.abs ((f' d c)^[n] c)
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | intro h | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ (β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2) β c β multibrot d | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
β’ c β multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
β’ (β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2) β c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | contrapose h | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
β’ c β multibrot d | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : c β multibrot d
β’ Β¬β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
β’ c β multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | simp only [multibrot_coe, tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs,
not_not, not_forall, not_le, Filter.tendsto_atTop, not_exists] at h β’ | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : c β multibrot d
β’ Β¬β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2 | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : c β multibrot d
β’ Β¬β (i : β), Complex.abs ((f' d c)^[i] c) β€ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | rcases(h 3).exists with β¨n, hβ© | case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c) | case h.mpr.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
h : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | use n | case h.mpr.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c) | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ 2 < Complex.abs ((f' d c)^[n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ β x, 2 < Complex.abs ((f' d c)^[x] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_eq_le_two | [482, 1] | [490, 52] | linarith | case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ 2 < Complex.abs ((f' d c)^[n] c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
hβ : β (x : β), βαΆ (a : β) in atTop, x β€ Complex.abs ((f' d c)^[a] c)
n : β
h : 3 β€ Complex.abs ((f' d c)^[n] c)
β’ 2 < Complex.abs ((f' d c)^[n] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | refine IsCompact.of_isClosed_subset (isCompact_closedBall _ _) ?_ multibrot_subset_closedBall | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (multibrot d) | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (multibrot d) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (multibrot d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | rw [multibrot_eq_le_two] | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (multibrot d) | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (multibrot d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | apply isClosed_iInter | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2) | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (i : β), IsClosed ((fun c => (f' d c)^[i] c) β»ΒΉ' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsClosed (β n, (fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | intro n | case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (i : β), IsClosed ((fun c => (f' d c)^[i] c) β»ΒΉ' closedBall 0 2) | case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ IsClosed ((fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
β’ β (i : β), IsClosed ((fun c => (f' d c)^[i] c) β»ΒΉ' closedBall 0 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | refine IsClosed.preimage ?_ Metric.isClosed_ball | case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ IsClosed ((fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2) | case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ Continuous fun c => (f' d c)^[n] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ IsClosed ((fun c => (f' d c)^[n] c) β»ΒΉ' closedBall 0 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | induction' n with n h | case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ Continuous fun c => (f' d c)^[n] c | case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => (f' d c)^[0] c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
d : β
instβ : Fact (2 β€ d)
n : β
β’ Continuous fun c => (f' d c)^[n] c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | simp only [Function.iterate_zero_apply] | case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => (f' d c)^[0] c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => (f' d c)^[0] c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | exact continuous_id | case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.zero
c : β
d : β
instβ : Fact (2 β€ d)
β’ Continuous fun c => c
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | simp only [Function.iterate_succ_apply'] | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => f' d c ((f' d c)^[n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => (f' d c)^[n + 1] c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | rw [continuous_iff_continuousAt] | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => f' d c ((f' d c)^[n] c) | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ β (x : β), ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ Continuous fun c => f' d c ((f' d c)^[n] c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | intro c | case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ β (x : β), ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) x | case h.succ
cβ : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
c : β
β’ ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.succ
c : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
β’ β (x : β), ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isCompact_multibrot | [493, 1] | [499, 87] | exact (analytic_f' _ (mem_univ _)).continuousAt.compβ continuousAt_id h.continuousAt | case h.succ
cβ : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
c : β
β’ ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.succ
cβ : β
d : β
instβ : Fact (2 β€ d)
n : β
h : Continuous fun c => (f' d c)^[n] c
c : β
β’ ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isOpen_multibrotExt | [502, 1] | [506, 25] | rw [OnePoint.isOpen_iff_of_mem'] | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsOpen (multibrotExt d) | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (OnePoint.some β»ΒΉ' multibrotExt d)αΆ β§ IsOpen (OnePoint.some β»ΒΉ' multibrotExt d)
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsOpen (multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isOpen_multibrotExt | [502, 1] | [506, 25] | simp only [coe_preimage_multibrotExt, compl_compl] | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (OnePoint.some β»ΒΉ' multibrotExt d)αΆ β§ IsOpen (OnePoint.some β»ΒΉ' multibrotExt d)
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (multibrot d) β§ IsOpen (multibrot d)αΆ
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (OnePoint.some β»ΒΉ' multibrotExt d)αΆ β§ IsOpen (OnePoint.some β»ΒΉ' multibrotExt d)
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isOpen_multibrotExt | [502, 1] | [506, 25] | use isCompact_multibrot, isCompact_multibrot.isClosed.isOpen_compl | c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (multibrot d) β§ IsOpen (multibrot d)αΆ
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ IsCompact (multibrot d) β§ IsOpen (multibrot d)αΆ
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | isOpen_multibrotExt | [502, 1] | [506, 25] | exact multibrotExt_inf | c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ β β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | set s := superF d | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | generalize hg : fl (f d) β c = g | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have ct : cβ»ΒΉ β {z : β | abs z < (max 16 (abs c / 2))β»ΒΉ} := by
simp only [mem_setOf, map_invβ]
apply inv_lt_inv_of_lt; bound; refine max_lt lo (half_lt_self (lt_trans (by norm_num) lo)) | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have mem : c β multibrot d := multibrot_two_lt (lt_trans (by norm_num) lo) | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have nz : β n, (f d c)^[n] c β 0 := by
intro n; contrapose mem; simp only [not_not] at mem β’; exact multibrot_of_zero mem | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have iter : β n, ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ) := by
intro n; induction' n with n h
have cp : c β 0 := Complex.abs.ne_zero_iff.mp (lt_trans (by norm_num) lo).ne'
simp only [Function.iterate_zero_apply, inv_coe cp, toComplex_coe]
have e : (f d c)^[n] βc = ((g^[n] cβ»ΒΉ : β) : π)β»ΒΉ := by rw [β h, inv_inv]
si... | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
β’ Complex.abs (bottcher' d c) β€ 3 * Complex.abs cβ»ΒΉ | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
β’ Complex.abs (bott... | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
β’ Complex.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have b := mem | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
β’ Complex.abs (bott... | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : c β multibrot d... | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [multibrot_basin', not_not] at b | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : c β multibrot d... | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : (c, βc) β β―.bas... | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have attracts := (s.basin_attracts b).eventually (s.bottcher_eq_bottcherNear c) | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : (c, βc) β β―.bas... | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : (c, βc) β β―.bas... | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rcases (attracts.and (s.basin_stays b)).exists with β¨n, eq, _β© | cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b : (c, βc) β β―.bas... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b ... | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | clear attracts b | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
b ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | simp only [Super.bottcherNear, extChartAt_inf, PartialEquiv.trans_apply,
coePartialEquiv_symm_apply, Equiv.toPartialEquiv_apply, invEquiv_apply, RiemannSphere.inv_inf,
toComplex_zero, sub_zero, Super.fl, hg, iter, toComplex_coe] at eq | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have e0 : s.bottcher c ((f d c)^[n] βc) = bottcher' d c ^ d ^ n := s.bottcher_eqn_iter n | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have e1 : bottcherNear g d (g^[n] cβ»ΒΉ) = bottcherNear g d cβ»ΒΉ ^ d ^ n := by
rw [β hg]; exact bottcherNear_eqn_iter (superNearF d c) ct n | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [e0, e1] at eq | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | clear e0 e1 iter | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
iter : β (n : β), ((f d c)^[n] βc)β»ΒΉ = β(g^[n] cβ»ΒΉ)
n ... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have ae : abs (bottcher' d c) = abs (bottcherNear g d cβ»ΒΉ) := by
apply (pow_left_inj (Complex.abs.nonneg _) (Complex.abs.nonneg _)
(pow_ne_zero n (d_ne_zero d))).mp
simp only [β Complex.abs.map_pow, eq] | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [ae, β hg] | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | exact bottcherNear_le (superNearF d c) ct | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] βc β 0
n : β
rightβ : (c, (f d c)^[n] βc) β s.near
eq : bottc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
c : β
lo : 16 < Complex.abs c
s : Super (f d) d β := superF d
g : β β β
hg : fl (f d) β c = g
ct : cβ»ΒΉ β {z | Complex.abs z < (max 16 (Complex.abs c / 2))β»ΒΉ}
mem : c β multibrot d
nz : β (n : β), (f d c)^[n] ... |
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