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/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
have cw : abs (c * w ^ d) ≤ (abs z)⁻¹ := by
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
calc abs c * abs w ^ d
_ ≤ abs z * (abs z)⁻¹ ^ d := by bound
_ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound
_ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
have cw2 : abs (c * w ^ d) ≤ 1 / 2 := le_trans cw (le_trans i8 (by norm_num))
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
simp only [gl_f, gl]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (g (fl (f d) ∞ c) d w ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [Complex.inv_cpow, ← Complex.cpow_neg]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d)⁻¹ ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
swap
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ 0 < 16
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [one_div]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
apply inv_le_inv_of_le
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
|
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
exact le_trans (by norm_num) (le_trans c16.le cz)
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ 16
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
⊢ 8 ≤ 16
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ 1 / 8 ≤ 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ 1 / 8 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [map_inv₀] at wc
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * Complex.abs z⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
exact le_trans wc (mul_le_of_le_one_left (inv_nonneg.mpr (Complex.abs.nonneg _)) (by bound))
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ Complex.abs w ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ (5 / 8) ^ n ≤ 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
wc : Complex.abs w ≤ (5 / 8) ^ n * (Complex.abs z)⁻¹
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
⊢ (5 / 8) ^ n ≤ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
simp only [Complex.abs.map_mul, Complex.abs.map_pow]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
calc abs c * abs w ^ d
_ ≤ abs z * (abs z)⁻¹ ^ d := by bound
_ ≤ abs z * (abs z)⁻¹ ^ 2 := by bound
_ = (abs z)⁻¹ := by rw [pow_two]; field_simp [z0]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs c * Complex.abs w ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ d
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ d ≤ Complex.abs z * (Complex.abs z)⁻¹ ^ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [pow_two]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * (Complex.abs z)⁻¹ ^ 2 = (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
field_simp [z0]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
⊢ Complex.abs z * ((Complex.abs z)⁻¹ * (Complex.abs z)⁻¹) = (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ 1 / 8 ≤ 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
⊢ 1 / 8 ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
refine (lt_of_le_of_lt (le_abs_self _) (lt_of_le_of_lt ?_ (half_lt_self Real.pi_pos))).ne
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (1 + c * w ^ d).arg ≠ Real.pi
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [Complex.abs_arg_le_pi_div_two_iff, Complex.add_re, Complex.one_re]
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 0 ≤ 1 + (c * w ^ d).re
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ |(1 + c * w ^ d).arg| ≤ Real.pi / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
calc 1 + (c * w ^ d).re
_ ≥ 1 + -|(c * w ^ d).re| := by bound
_ = 1 - |(c * w ^ d).re| := by ring
_ ≥ 1 - abs (c * w ^ d) := by bound
_ ≥ 1 - 1 / 2 := by linarith
_ ≥ 0 := by norm_num
|
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 0 ≤ 1 + (c * w ^ d).re
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 0 ≤ 1 + (c * w ^ d).re
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 + (c * w ^ d).re ≥ 1 + -|(c * w ^ d).re|
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 + (c * w ^ d).re ≥ 1 + -|(c * w ^ d).re|
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
ring
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 + -|(c * w ^ d).re| = 1 - |(c * w ^ d).re|
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 + -|(c * w ^ d).re| = 1 - |(c * w ^ d).re|
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - |(c * w ^ d).re| ≥ 1 - Complex.abs (c * w ^ d)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - |(c * w ^ d).re| ≥ 1 - Complex.abs (c * w ^ d)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
linarith
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - Complex.abs (c * w ^ d) ≥ 1 - 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - Complex.abs (c * w ^ d) ≥ 1 - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - 1 / 2 ≥ 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ 1 - 1 / 2 ≥ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
have dn : abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ))) ≤ (1 / 2 : ℝ) ^ (n + 1) := by
simp only [Nat.cast_pow, one_div, map_neg_eq_map, map_inv₀, map_pow, Complex.abs_natCast,
inv_pow]
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
have d1 : abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ))) ≤ 1 := le_trans dn (by bound)
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
refine le_trans (pow_small ?_ d1) ?_
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs ((1 + c * w ^ d) ^ (-(1 / ↑(d ^ (n + 1)))) - 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
simp only [Nat.cast_pow, one_div, map_neg_eq_map, map_inv₀, map_pow, Complex.abs_natCast,
inv_pow]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
⊢ (1 / 2) ^ (n + 1) ≤ 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
⊢ (1 / 2) ^ (n + 1) ≤ 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [add_sub_cancel_left]
|
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2
|
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (c * w ^ d) ≤ 1 / 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (1 + c * w ^ d - 1) ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
exact cw2
|
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (c * w ^ d) ≤ 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ Complex.abs (c * w ^ d) ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
rw [add_sub_cancel_left]
|
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (1 + c * w ^ d - 1) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
calc 4 * abs (c * w ^ d) * abs (-(1 / ((d ^ (n + 1) : ℕ) : ℂ)))
_ ≤ 4 * (abs z)⁻¹ * (1/2 : ℝ) ^ (n + 1) := by bound
_ ≤ 2 * (1/2 : ℝ) ^ n * (abs z)⁻¹ := by
simp only [pow_succ, ←mul_assoc, mul_comm _ (1/2:ℝ)]; norm_num
simp only [mul_comm _ ((2:ℝ)^n)⁻¹, ←mul_assoc, le_refl]
|
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
bound
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * Complex.abs (c * w ^ d) * Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
simp only [pow_succ, ←mul_assoc, mul_comm _ (1/2:ℝ)]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ (n + 1) ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 1 / 2 * 4 * (Complex.abs z)⁻¹ * (1 / 2) ^ n ≤ 2 * (1 / 2) ^ n * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
term_approx
|
[89, 1]
|
[131, 66]
|
simp only [mul_comm _ ((2:ℝ)^n)⁻¹, ←mul_assoc, le_refl]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
s : Super (f d) d ∞ := superF d
z0 : Complex.abs z ≠ 0
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
i1 : (Complex.abs z)⁻¹ ≤ 1
w : ℂ
hw : (fl (f d) ∞ c)^[n] z⁻¹ = w
wc : Complex.abs w ≤ (Complex.abs z)⁻¹
cw : Complex.abs (c * w ^ d) ≤ (Complex.abs z)⁻¹
cw2 : Complex.abs (c * w ^ d) ≤ 1 / 2
dn : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ (1 / 2) ^ (n + 1)
d1 : Complex.abs (-(1 / ↑(d ^ (n + 1)))) ≤ 1
⊢ 2 * (Complex.abs z)⁻¹ * (2 ^ n)⁻¹ ≤ 2 * (2 ^ n)⁻¹ * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
set s := superF d
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs (⋯.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs (⋯.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
have i8 : (abs z)⁻¹ ≤ 1 / 8 := by
rw [one_div]; apply inv_le_inv_of_le; norm_num
exact le_trans (by norm_num) (le_trans c16.le cz)
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
simp only [bottcher_eq_bottcherNear_z c16 cz, bottcherNear, Complex.abs.map_mul, ← mul_sub_one,
pow_two, ← mul_assoc, map_inv₀, mul_comm (abs z)⁻¹]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤
16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (s.bottcher c ↑z - z⁻¹) ≤ 16 * (Complex.abs z)⁻¹ ^ 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
refine mul_le_mul_of_nonneg_right ?_ (inv_nonneg.mpr (Complex.abs.nonneg _))
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤
16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) * (Complex.abs z)⁻¹ ≤
16 * (Complex.abs z)⁻¹ * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
rcases term_prod_exists (superNearF d c) _ (inv_mem_t c16 cz) with ⟨p, h⟩
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
rw [h.tprod_eq]
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (∏' (n : ℕ), term (fl (f d) ∞ c) d n z⁻¹ - 1) ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
simp only [HasProd] at h
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : HasProd (fun n => term (fl (f d) ∞ c) d n z⁻¹) p
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
apply le_of_tendsto' (Filter.Tendsto.comp Complex.continuous_abs.continuousAt (h.sub_const 1))
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ Complex.abs (p - 1) ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
clear h
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
h : Tendsto (fun s => s.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) Filter.atTop (𝓝 p)
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
intro A
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
⊢ ∀ (c_1 : Finset ℕ),
(⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) c_1 ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
simp only [Function.comp]
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun x => (x.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) A ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
rw [(by norm_num : (16 : ℝ) = 4 * 4), mul_assoc]
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 16 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
refine dist_prod_one_le_abs_sum ?_ (by linarith)
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹)
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ Complex.abs ((A.prod fun x => term (fl (f d) ∞ c) d x z⁻¹) - 1) ≤ 4 * (4 * (Complex.abs z)⁻¹)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
refine le_trans (Finset.sum_le_sum fun n _ ↦ term_approx d (by linarith) cz n) ?_
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => Complex.abs (term (fl (f d) ∞ c) d n z⁻¹ - 1)) ≤ 4 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
simp only [mul_comm _ _⁻¹, ← mul_assoc, ← Finset.mul_sum]
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun i => 2 * (1 / 2) ^ i * (Complex.abs z)⁻¹) ≤ 4 * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
calc (abs z)⁻¹ * 2 * A.sum (fun n ↦ (1/2:ℝ)^n)
_ ≤ (abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ := by gcongr; apply partial_geometric_bound; repeat norm_num
_ = (abs z)⁻¹ * 4 := by ring
|
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun i => (1 / 2) ^ i) ≤ (Complex.abs z)⁻¹ * 4
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
rw [one_div]
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ (Complex.abs z)⁻¹ ≤ 1 / 8
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
apply inv_le_inv_of_le
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
|
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ (Complex.abs z)⁻¹ ≤ 8⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
norm_num
|
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case ha
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 0 < 8
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
exact le_trans (by norm_num) (le_trans c16.le cz)
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ 16
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
⊢ 8 ≤ 16
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
norm_num
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 16 = 4 * 4
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 16 = 4 * 4
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
linarith
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 4 * (Complex.abs z)⁻¹ ≤ 1 / 2
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 4 * (Complex.abs z)⁻¹ ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
linarith
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
n : ℕ
x✝ : n ∈ A
⊢ 16 < Complex.abs c
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
n : ℕ
x✝ : n ∈ A
⊢ 16 < Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
gcongr
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun n => (1 / 2) ^ n) ≤ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ ((Complex.abs z)⁻¹ * 2 * A.sum fun n => (1 / 2) ^ n) ≤ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
apply partial_geometric_bound
|
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹
|
case h.a0
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 0 ≤ 1 / 2
case h.a1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 1 / 2 < 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (A.sum fun n => (1 / 2) ^ n) ≤ (1 - 1 / 2)⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
repeat norm_num
|
case h.a0
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 0 ≤ 1 / 2
case h.a1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 1 / 2 < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.a0
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 0 ≤ 1 / 2
case h.a1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
norm_num
|
case h.a1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 1 / 2 < 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.a1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_approx_z
|
[134, 1]
|
[153, 33]
|
ring
|
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ = (Complex.abs z)⁻¹ * 4
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c16 : 16 < Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d ∞ := superF d
i8 : (Complex.abs z)⁻¹ ≤ 1 / 8
p : ℂ
A : Finset ℕ
⊢ (Complex.abs z)⁻¹ * 2 * (1 - 1 / 2)⁻¹ = (Complex.abs z)⁻¹ * 4
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
rw [HasDerivAt, HasDerivAtFilter, bottcher, hasFDerivAtFilter_iff_isLittleO, coe_zero, inv_zero',
fill_inf]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ HasDerivAt (fun z => bottcher d (↑z)⁻¹) 1 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ HasDerivAt (fun z => bottcher d (↑z)⁻¹) 1 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [sub_zero, ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
smul_eq_mul, mul_one]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x'
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - 0 - (ContinuousLinearMap.smulRight 1 1) (x' - 0)) =o[𝓝 0] fun x' => x' - 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
rw [Asymptotics.isLittleO_iff]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x'
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (fun x' => fill (bottcher' d) 0 (↑x')⁻¹ - x') =o[𝓝 0] fun x' => x'
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
intro k k0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ c * ‖x‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
rw [Metric.eventually_nhds_iff]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖fill (bottcher' d) 0 (↑x)⁻¹ - x‖ ≤ k * ‖x‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
refine ⟨min 16⁻¹ (k / 16), by bound, ?_⟩
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∃ ε > 0, ∀ ⦃y : ℂ⦄, dist y 0 < ε → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
intro z le
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : dist z 0 < min 16⁻¹ (k / 16)
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ ∀ ⦃y : ℂ⦄, dist y 0 < min 16⁻¹ (k / 16) → ‖fill (bottcher' d) 0 (↑y)⁻¹ - y‖ ≤ k * ‖y‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [Complex.dist_eq, sub_zero, lt_min_iff] at le
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : dist z 0 < min 16⁻¹ (k / 16)
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : dist z 0 < min 16⁻¹ (k / 16)
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [Complex.norm_eq_abs]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ ‖fill (bottcher' d) 0 (↑z)⁻¹ - z‖ ≤ k * ‖z‖
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
by_cases z0 : z = 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
|
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [inv_coe z0, fill_coe]
|
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
|
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
have b : abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ (16:ℝ) * (abs z⁻¹)⁻¹ ^ 2 := bottcher_approx d ?_
|
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ 16 < Complex.abs z⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
bound
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ min 16⁻¹ (k / 16) > 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
⊢ min 16⁻¹ (k / 16) > 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [z0, coe_zero, inv_zero', fill_inf, sub_zero, Complex.abs.map_zero,
MulZeroClass.mul_zero, le_refl]
|
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : z = 0
⊢ Complex.abs (fill (bottcher' d) 0 (↑z)⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [inv_inv] at b
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z⁻¹⁻¹) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
apply le_trans b
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ Complex.abs (bottcher' d z⁻¹ - z) ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
simp only [map_inv₀, inv_inv, pow_two, ← mul_assoc]
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2 ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
refine mul_le_mul_of_nonneg_right ?_ (Complex.abs.nonneg _)
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z ≤ k
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z * Complex.abs z ≤ k * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
calc 16 * abs z
_ ≤ 16 * (k / 16) := by linarith [le.2]
_ = k := by ring
|
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z ≤ k
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z ≤ k
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
linarith [le.2]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z ≤ 16 * (k / 16)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * Complex.abs z ≤ 16 * (k / 16)
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
ring
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * (k / 16) = k
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
b : Complex.abs (bottcher' d z⁻¹ - z) ≤ 16 * (Complex.abs z⁻¹)⁻¹ ^ 2
⊢ 16 * (k / 16) = k
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
rw [map_inv₀, lt_inv (by norm_num) (Complex.abs.pos_iff.mpr z0)]
|
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ 16 < Complex.abs z⁻¹
|
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs z < 16⁻¹
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ 16 < Complex.abs z⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
exact le.1
|
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs z < 16⁻¹
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg.refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ Complex.abs z < 16⁻¹
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_hasDerivAt_one
|
[161, 1]
|
[181, 81]
|
norm_num
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ 0 < 16
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
k : ℝ
k0 : 0 < k
z : ℂ
le : Complex.abs z < 16⁻¹ ∧ Complex.abs z < k / 16
z0 : ¬z = 0
⊢ 0 < 16
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_mfderiv_inf_ne_zero
|
[184, 1]
|
[195, 40]
|
simp only [mfderiv, (bottcherHolomorphic d _ multibrotExt_inf).mdifferentiableAt, if_pos,
writtenInExtChartAt, bottcher_inf, extChartAt_inf, extChartAt_eq_refl, Function.comp,
PartialEquiv.refl_coe, id, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, invEquiv_apply,
RiemannSphere.inv_inf, coePartialEquiv_symm_apply, toComplex_zero, PartialEquiv.coe_trans_symm,
PartialEquiv.symm_symm, coePartialEquiv_apply, Equiv.toPartialEquiv_symm_apply, invEquiv_symm,
ModelWithCorners.Boundaryless.range_eq_univ, fderivWithin_univ]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ mfderiv I I (bottcher d) ∞ ≠ 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ mfderiv I I (bottcher d) ∞ ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_mfderiv_inf_ne_zero
|
[184, 1]
|
[195, 40]
|
rw [bottcher_hasDerivAt_one.hasFDerivAt.fderiv]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ fderiv ℂ (fun x => bottcher d (↑x)⁻¹) 0 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Bottcher.lean
|
bottcher_mfderiv_inf_ne_zero
|
[184, 1]
|
[195, 40]
|
rw [Ne, ContinuousLinearMap.ext_iff, not_forall]
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
|
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∃ x, ¬(ContinuousLinearMap.smulRight 1 1) x = 0 x
|
Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ContinuousLinearMap.smulRight 1 1 ≠ 0
TACTIC:
|
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