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 ⟨k, kb, nk⟩ | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h✝ : g a = g b
n : ℕ
h : ∃ k ≤ b, g n = g k
⊢ ∃ k ≤ b, g (n + 1) = g k | case succ.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
⊢ ∃ k ≤ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h✝ : g a = g b
n : ℕ
h : ∃ k ≤ b, g n = g k
⊢ ∃ k ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
⊢ ∃ k ≤ b, g (n + 1) = g k | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ ∃ k ≤ b, g (n + 1) = g k
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
⊢ ∃ k ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ ∃ k ≤ b, g (n + 1) = g k
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ g (n + 1) = g (a + 1)
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ ∃ k ≤ b, g (n + 1) = g k
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ 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 [← hg, ← hg, Function.iterate_succ_apply', Function.iterate_succ_apply', hg, hg, nk, e, h] | case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ g (n + 1) = g (a + 1)
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : k = b
⊢ g (n + 1) = g (a + 1)
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ 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 k + 1, Nat.succ_le_iff.mpr (Ne.lt_of_le e kb) | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ b, g (n + 1) = g k | case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ g (n + 1) = g (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ ∃ k ≤ 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 [← hg, ← hg, Function.iterate_succ_apply', Function.iterate_succ_apply', hg, hg, nk] | case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ g (n + 1) = g (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
a b : ℕ
ab : a < b
g : ℕ → ℂ
hg : ∀ (n : ℕ), (f' d c)^[n] c = g n
h : g a = g b
n k : ℕ
kb : k ≤ b
nk : g n = g k
e : ¬k = b
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
⊢ c ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
⊢ c ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
⊢ c ∈ multibrot d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
i1 : (f d c)^[n + 1] ↑c = ↑c
⊢ c ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
⊢ c ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
i1 : (f d c)^[n + 1] ↑c = ↑c
⊢ c ∈ multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
i1 : (f d c)^[n + 1] ↑c = ↑c
⊢ c ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
⊢ (f d c)^[0] ↑c = ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
⊢ (f d c)^[0] ↑c = ↑c
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
⊢ (f d c)^[n + 1] ↑c = ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : (f d c)^[n] ↑c = 0
i0 : (f d c)^[0] ↑c = ↑c
⊢ (f d c)^[n + 1] ↑c = ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | multibrot_zero | [450, 1] | [451, 70] | apply multibrot_of_zero | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 ∈ multibrot d | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (f d 0)^[?n] ↑0 = 0
case n
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (f d 0)^[?n] ↑0 = 0
case n
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ℕ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (f d 0)^[?n] ↑0 = 0
case n
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ℕ
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
⊢ c ∉ multibrot d | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
⊢ c ∉ multibrot d
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
⊢ c ∉ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
⊢ c ∉ multibrot d
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ multibrot d | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : 2 < Complex.abs c
⊢ c ∉ multibrot d
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ 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 [multibrot_coe, not_not] | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ multibrot d | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ c ∉ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : ¬2 < Complex.abs c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
s : ℝ
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
s : ℝ
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : 2 < Complex.abs ((f' d c)^[n] c)
c2 : Complex.abs c ≤ 2
s : ℝ
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ 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 ≤ s := by linarith | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ 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 ≤ s - 1 := by linarith | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
⊢ 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 ≤ s := by linarith | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => (f' d c)^[n] c) atTop atInf | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 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 [← Filter.tendsto_add_atTop_iff_nat n] | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] c)) atTop atTop | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] c)) atTop atTop
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n_1 => Complex.abs ((f' d c)^[n_1 + n] c)) atTop atTop | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => s * (s - 1) ^ n) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n_1 => Complex.abs ((f' d c)^[n_1 + n] c)) atTop atTop
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => s * (s - 1) ^ n) atTop atTop | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => (s - 1) ^ n) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => s * (s - 1) ^ n) atTop atTop
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => (s - 1) ^ n) atTop atTop | case neg.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 1 < s - 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ Tendsto (fun n => (s - 1) ^ n) atTop atTop
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 1 < s - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ 1 ≤ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
⊢ 1 ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
⊢ 1 ≤ s - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
⊢ 0 ≤ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
⊢ 0 ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
⊢ s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ ∀ (k : ℕ), s * (s - 1) ^ k ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
⊢ s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c) | case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ s * (s - 1) ^ 0 ≤ Complex.abs ((f' d c)^[0 + n] c)
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs ((f' d c)^[k + 1 + n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
⊢ s * (s - 1) ^ k ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ s * (s - 1) ^ 0 ≤ Complex.abs ((f' d c)^[0 + n] c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
⊢ s * (s - 1) ^ 0 ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs ((f' d c)^[k + 1 + n] c) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs ((f' d c)^[k + 1 + n] c)
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c ((f' d c)^[k + n] c)) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
z : ℂ
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c ((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] | rw [hz] at p | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
z : ℂ
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
p : s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
z : ℂ
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have ss1 : 1 ≤ s * (s - 1) ^ k := by bound | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | have k2 : k ≤ k * 2 := by linarith | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z)
TACTIC:
|
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
_ ≥ abs (z ^ d) - abs c := by bound
_ = abs z ^ d - abs c := by rw [Complex.abs.map_pow]
_ ≥ (s * (s - 1) ^ k) ^ d - 2 := by bound
_ ≥ (s * (s - 1) ^ k) ^ 2 - 2 := by bound
_ = s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 := by rw [mul_pow, pow_mul, mul_one]
_ ≥ s ^ 2 * (s - 1) ^ k - s * (s - 1) ^ k := by bound
_ = s * ((s - 1) ^ k * (s - 1)) := by ring
_ = s * (s - 1) ^ (k + 1) := by rw [pow_succ] | case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s * (s - 1) ^ (k + 1) ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
⊢ 1 ≤ s * (s - 1) ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
⊢ 1 ≤ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
⊢ k ≤ k * 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
⊢ k ≤ k * 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs z ^ d - Complex.abs c ≥ (s * (s - 1) ^ k) ^ d - 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ Complex.abs z ^ d - Complex.abs c ≥ (s * (s - 1) ^ k) ^ d - 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ (s * (s - 1) ^ k) ^ d - 2 ≥ (s * (s - 1) ^ k) ^ 2 - 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ (s * (s - 1) ^ k) ^ d - 2 ≥ (s * (s - 1) ^ k) ^ 2 - 2
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ (s * (s - 1) ^ k) ^ 2 - 2 = s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ (s * (s - 1) ^ k) ^ 2 - 2 = s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | not_multibrot_of_two_lt | [453, 1] | [480, 52] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 ≥ s ^ 2 * (s - 1) ^ k - s * (s - 1) ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s ^ 2 * (s - 1) ^ (k * 2) - 2 * 1 ≥ s ^ 2 * (s - 1) ^ k - 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] | ring | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s ^ 2 * (s - 1) ^ k - s * (s - 1) ^ k = s * ((s - 1) ^ k * (s - 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s ^ 2 * (s - 1) ^ k - s * (s - 1) ^ k = s * ((s - 1) ^ k * (s - 1))
TACTIC:
|
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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s * ((s - 1) ^ k * (s - 1)) = s * (s - 1) ^ (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
k : ℕ
z : ℂ
p : s * (s - 1) ^ k ≤ Complex.abs z
hz : (f' d c)^[k + n] c = z
ss1 : 1 ≤ s * (s - 1) ^ k
k2 : k ≤ k * 2
⊢ s * ((s - 1) ^ k * (s - 1)) = s * (s - 1) ^ (k + 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 0 < s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
c2 : Complex.abs c ≤ 2
s : ℝ
h : 2 < s
hs : Complex.abs ((f' d c)^[n] c) = s
s1 : 1 ≤ s
s1' : 1 ≤ s - 1
s0 : 0 ≤ s
b : ∀ (k : ℕ), s * (s - 1) ^ k ≤ Complex.abs ((f' d c)^[k + n] c)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ multibrot d = ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2 | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ multibrot d = ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ c ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ c ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ c ∈ ⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2 | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2) → c ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d ↔ ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2 | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
n : ℕ
⊢ Complex.abs ((f' d c)^[n] c) ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ c ∈ multibrot d → ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
n : ℕ
⊢ Complex.abs ((f' d c)^[n] c) ≤ 2 | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : ¬Complex.abs ((f' d c)^[n] c) ≤ 2
⊢ c ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
m : c ∈ multibrot d
n : ℕ
⊢ Complex.abs ((f' d c)^[n] c) ≤ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : ¬Complex.abs ((f' d c)^[n] c) ≤ 2
⊢ c ∉ multibrot d | case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : 2 < Complex.abs ((f' d c)^[n] c)
⊢ c ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : ¬Complex.abs ((f' d c)^[n] c) ≤ 2
⊢ c ∉ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : 2 < Complex.abs ((f' d c)^[n] c)
⊢ c ∉ multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
n : ℕ
m : 2 < Complex.abs ((f' d c)^[n] c)
⊢ c ∉ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2) → c ∈ multibrot d | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2
⊢ c ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ (∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2) → c ∈ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2
⊢ c ∈ multibrot d | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : c ∉ multibrot d
⊢ ¬∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2
⊢ c ∈ 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 ⊢ | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : c ∉ multibrot d
⊢ ¬∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 2 | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
⊢ ∃ x, 2 < Complex.abs ((f' d c)^[x] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : c ∉ multibrot d
⊢ ¬∀ (i : ℕ), Complex.abs ((f' d c)^[i] c) ≤ 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 ⟨n, h⟩ | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
⊢ ∃ x, 2 < Complex.abs ((f' d c)^[x] c) | case h.mpr.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ ∃ x, 2 < Complex.abs ((f' d c)^[x] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
⊢ ∃ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ ∃ x, 2 < Complex.abs ((f' d c)^[x] c) | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ 2 < Complex.abs ((f' d c)^[n] c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ ∃ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ 2 < Complex.abs ((f' d c)^[n] c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
h✝ : ∀ (x : ℝ), ∀ᶠ (a : ℕ) in atTop, x ≤ Complex.abs ((f' d c)^[a] c)
n : ℕ
h : 3 ≤ Complex.abs ((f' d c)^[n] c)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (multibrot d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsClosed (multibrot d) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsClosed (multibrot d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsClosed (⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsClosed (⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2) | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (i : ℕ), IsClosed ((fun c => (f' d c)^[i] c) ⁻¹' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsClosed (⋂ n, (fun c => (f' d c)^[n] c) ⁻¹' 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (i : ℕ), IsClosed ((fun c => (f' d c)^[i] c) ⁻¹' closedBall 0 2) | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ IsClosed ((fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (i : ℕ), IsClosed ((fun c => (f' d c)^[i] c) ⁻¹' 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ IsClosed ((fun c => (f' d c)^[n] c) ⁻¹' closedBall 0 2) | case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ Continuous fun c => (f' d c)^[n] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ IsClosed ((fun c => (f' d c)^[n] c) ⁻¹' 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ Continuous fun c => (f' d c)^[n] c | case h.zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => (f' d c)^[0] c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => (f' d c)^[0] c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | case h.zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => (f' d c)^[0] c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.zero
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ Continuous fun c => c
case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => (f' d c)^[n + 1] c | case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ Continuous fun c => f' d c ((f' d c)^[n] c) | case h.succ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ ∀ (x : ℂ), 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ ∀ (x : ℂ), ContinuousAt (fun c => f' d c ((f' d c)^[n] c)) x | case h.succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
c : ℂ
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
⊢ ∀ (x : ℂ), 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.comp₂ continuousAt_id h.continuousAt | case h.succ
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
c : ℂ
⊢ 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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
n : ℕ
h : Continuous fun c => (f' d c)^[n] c
c : ℂ
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsOpen (multibrotExt d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (OnePoint.some ⁻¹' multibrotExt d)ᶜ ∧ IsOpen (OnePoint.some ⁻¹' multibrotExt d)
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (OnePoint.some ⁻¹' multibrotExt d)ᶜ ∧ IsOpen (OnePoint.some ⁻¹' multibrotExt d)
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (multibrot d) ∧ IsOpen (multibrot d)ᶜ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (OnePoint.some ⁻¹' multibrotExt d)ᶜ ∧ IsOpen (OnePoint.some ⁻¹' multibrotExt d)
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (multibrot d) ∧ IsOpen (multibrot d)ᶜ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsCompact (multibrot d) ∧ IsOpen (multibrot d)ᶜ
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ 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 : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∞ ∈ 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 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | generalize hg : fl (f d) ∞ c = g | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have ct : c⁻¹ ∈ {z : ℂ | abs z < (max 16 (abs c / 2))⁻¹} := by
simp only [mem_setOf, map_inv₀]
apply inv_lt_inv_of_lt; bound; refine max_lt lo (half_lt_self (lt_trans (by norm_num) lo)) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have mem : c ∉ multibrot d := multibrot_two_lt (lt_trans (by norm_num) lo) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have nz : ∀ n, (f d c)^[n] c ≠ 0 := by
intro n; contrapose mem; simp only [not_not] at mem ⊢; exact multibrot_of_zero mem | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have iter : ∀ n, ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹) := by
intro n; induction' n with n h
have cp : c ≠ 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] ↑c = ((g^[n] c⁻¹ : ℂ) : 𝕊)⁻¹ := by rw [← h, inv_inv]
simp only [Function.iterate_succ_apply', e]
generalize hz : g^[n] c⁻¹ = z
simp only [← hg, fl, extChartAt_inf, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply,
invEquiv_apply, RiemannSphere.inv_inf, coePartialEquiv_symm_apply, toComplex_zero, sub_zero,
Function.comp, add_zero, PartialEquiv.coe_trans_symm, PartialEquiv.symm_symm,
coePartialEquiv_apply, Equiv.toPartialEquiv_symm_apply, invEquiv_symm]
rw [coe_toComplex]
simp only [Ne, inv_eq_inf, ← hz, ← h, inv_inv, ← Function.iterate_succ_apply' (f d c)]
apply nz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | have b := mem | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : c ∉ multibrot d
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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 ⟨n, eq, _⟩ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | clear attracts b | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
b : (c, ↑c) ∈ ⋯.basin
attracts : ∀ᶠ (x : ℕ) in atTop, s.bottcher c ((f d c)^[x] ↑c) = s.bottcherNear c ((f d c)^[x] ↑c)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
eq : s.bottcher c ((f d c)^[n] ↑c) = s.bottcherNear c ((f d c)^[n] ↑c)
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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] ↑c) = bottcher' d c ^ d ^ n := s.bottcher_eqn_iter n | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n := by
rw [← hg]; exact bottcherNear_eqn_iter (superNearF d c) ct n | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : s.bottcher c ((f d c)^[n] ↑c) = bottcherNear g d (g^[n] c⁻¹)
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
iter : ∀ (n : ℕ), ((f d c)^[n] ↑c)⁻¹ = ↑(g^[n] c⁻¹)
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
e0 : s.bottcher c ((f d c)^[n] ↑c) = bottcher' d c ^ d ^ n
e1 : bottcherNear g d (g^[n] c⁻¹) = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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 c⁻¹) := by
apply (pow_left_inj (Complex.abs.nonneg _) (Complex.abs.nonneg _)
(pow_ne_zero n (d_ne_zero d))).mp
simp only [← Complex.abs.map_pow, eq] | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_bound | [513, 1] | [554, 59] | rw [ae, ← hg] | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹ | case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcherNear (fl (f d) ∞ c) d c⁻¹) ≤ 3 * Complex.abs c⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcher' d c) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcherNear (fl (f d) ∞ c) d c⁻¹) ≤ 3 * Complex.abs c⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
lo : 16 < Complex.abs c
s : Super (f d) d ∞ := superF d
g : ℂ → ℂ
hg : fl (f d) ∞ c = g
ct : c⁻¹ ∈ {z | Complex.abs z < (max 16 (Complex.abs c / 2))⁻¹}
mem : c ∉ multibrot d
nz : ∀ (n : ℕ), (f d c)^[n] ↑c ≠ 0
n : ℕ
right✝ : (c, (f d c)^[n] ↑c) ∈ s.near
eq : bottcher' d c ^ d ^ n = bottcherNear g d c⁻¹ ^ d ^ n
ae : Complex.abs (bottcher' d c) = Complex.abs (bottcherNear g d c⁻¹)
⊢ Complex.abs (bottcherNear (fl (f d) ∞ c) d c⁻¹) ≤ 3 * Complex.abs c⁻¹
TACTIC:
|
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