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/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) (le_trans b3 bz) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have z3 : 3 ≤ abs z := le_trans (by norm_num) (le_trans b3 bz) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have l0 : 1 < log (abs z) := lt_of_lt_of_le (by norm_num) (le_log_abs_z z3) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | generalize hbb : (b^(d-1)-1)^d * b^(d-1) - 1 = bb at b11 bb3 bs2 b0'' si | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have fz : (abs z)^d - abs c ≤ abs (f' d c z) := by
calc abs (z^d + c)
_ ≥ abs (z^d) - abs c := by bound
_ = (abs z)^d - abs c := by rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have fz' : (b^(d-1)-1) * abs z ≤ abs (f' d c z) := by
calc abs (f' d c z)
_ ≥ (abs z)^d - abs c := fz
_ ≥ (abs z)^d - abs z := by bound
_ = (abs z)^(d-1) * abs z - abs z := by rw [←pow_succ, Nat.sub_add_cancel (d_ge_one d)]
_ = ((abs z)^(d-1) - 1) * abs z := by rw [sub_one_mul]
_ ≥ (b^(d-1)-1) * abs z := by bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have zfz : abs z ≤ abs (f' d c z) := le_self_iter d z3 cz 1 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have zffz : abs z ≤ abs (f' d c (f' d c z)) := le_self_iter d z3 cz 2 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have bfz : b ≤ abs (f' d c z) := le_trans bz zfz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have ffz : bb * abs z ≤ abs (f' d c (f' d c z)) := by
calc abs ((f' d c z)^d + c)
_ ≥ abs ((f' d c z)^d) - abs c := by bound
_ = (abs (f' d c z))^d - abs c := by rw [Complex.abs.map_pow]
_ ≥ ((b^(d-1)-1) * abs z)^d - abs z := by bound
_ = (b^(d-1)-1)^d * (abs z)^(d-1) * abs z - abs z := by
rw [mul_assoc, ←pow_succ, mul_pow, Nat.sub_add_cancel (d_ge_one d)]
_ ≥ (b^(d-1)-1)^d * b^(d-1) * abs z - abs z := by bound
_ = bb * abs z := by rw [←hbb, sub_one_mul] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | have e0 : f_error d z ≤ s0 / (abs z * log (abs z)) := bs0 bz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [iter_error_peel z3 cz, iter_error_peel (le_trans b3 bfz) (le_trans cz zfz), ←add_assoc] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ f_error d z + f_error d (f' d c z) + iter_error d c (f' d c (f' d c z)) ≤ i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ iter_error d c z ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine le_trans (add_le_add (add_le_add e0 e1) e2) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ f_error d z + f_error d (f' d c z) + iter_error d c (f' d c (f' d c z)) ≤ i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ s0 / (Complex.abs z * (Complex.abs z).log) + s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) +
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ f_error d z + f_error d (f' d c z) + iter_error d c (f' d c (f' d c z)) ≤ i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | simp only [←add_div, le_refl] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ s0 / (Complex.abs z * (Complex.abs z).log) + s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) +
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ (s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb)) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ s0 / (Complex.abs z * (Complex.abs z).log) + s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) +
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | exact div_le_div_of_nonneg_right si (by positivity) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ (s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb)) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ (s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb)) / (Complex.abs z * (Complex.abs z).log) ≤
i / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
⊢ 3 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
⊢ 3 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 < 1.0986 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
bs2 :
∀ {w : ℂ},
((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b ≤ Complex.abs w →
f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b11 : 11 ≤ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1
bb3 : 3 ≤ ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b
b0' : 0 < b ^ (d - 1) - 1
b0'' : 0 < 1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹
si :
s0 + s1 / (b ^ (d - 1) - 1) +
s2 / ((1 - (((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1) * b - 1)⁻¹) * ((b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1)) ≤
i
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 < 1.0986
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | calc abs (z^d + c)
_ ≥ abs (z^d) - abs c := by bound
_ = (abs z)^d - abs c := by rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | calc abs (f' d c z)
_ ≥ (abs z)^d - abs c := fz
_ ≥ (abs z)^d - abs z := by bound
_ = (abs z)^(d-1) * abs z - abs z := by rw [←pow_succ, Nat.sub_add_cancel (d_ge_one d)]
_ = ((abs z)^(d-1) - 1) * abs z := by rw [sub_one_mul]
_ ≥ (b^(d-1)-1) * abs z := by bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ d - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ d - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [←pow_succ, Nat.sub_add_cancel (d_ge_one d)] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ d - Complex.abs z = Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ d - Complex.abs z = Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [sub_one_mul] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z = (Complex.abs z ^ (d - 1) - 1) * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z = (Complex.abs z ^ (d - 1) - 1) * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z ^ (d - 1) - 1) * Complex.abs z ≥ (b ^ (d - 1) - 1) * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z ^ (d - 1) - 1) * Complex.abs z ≥ (b ^ (d - 1) - 1) * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | calc abs ((f' d c z)^d + c)
_ ≥ abs ((f' d c z)^d) - abs c := by bound
_ = (abs (f' d c z))^d - abs c := by rw [Complex.abs.map_pow]
_ ≥ ((b^(d-1)-1) * abs z)^d - abs z := by bound
_ = (b^(d-1)-1)^d * (abs z)^(d-1) * abs z - abs z := by
rw [mul_assoc, ←pow_succ, mul_pow, Nat.sub_add_cancel (d_ge_one d)]
_ ≥ (b^(d-1)-1)^d * b^(d-1) * abs z - abs z := by bound
_ = bb * abs z := by rw [←hbb, sub_one_mul] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z ^ d + c) ≥ Complex.abs (f' d c z ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z ^ d + c) ≥ Complex.abs (f' d c z ^ d) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z ^ d) - Complex.abs c = Complex.abs (f' d c z) ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z ^ d) - Complex.abs c = Complex.abs (f' d c z) ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z) ^ d - Complex.abs c ≥ ((b ^ (d - 1) - 1) * Complex.abs z) ^ d - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ Complex.abs (f' d c z) ^ d - Complex.abs c ≥ ((b ^ (d - 1) - 1) * Complex.abs z) ^ d - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [mul_assoc, ←pow_succ, mul_pow, Nat.sub_add_cancel (d_ge_one d)] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ ((b ^ (d - 1) - 1) * Complex.abs z) ^ d - Complex.abs z =
(b ^ (d - 1) - 1) ^ d * Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ ((b ^ (d - 1) - 1) * Complex.abs z) ^ d - Complex.abs z =
(b ^ (d - 1) - 1) ^ d * Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) ^ d * Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z ≥
(b ^ (d - 1) - 1) ^ d * b ^ (d - 1) * Complex.abs z - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) ^ d * Complex.abs z ^ (d - 1) * Complex.abs z - Complex.abs z ≥
(b ^ (d - 1) - 1) ^ d * b ^ (d - 1) * Complex.abs z - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [←hbb, sub_one_mul] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) * Complex.abs z - Complex.abs z = bb * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
⊢ (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) * Complex.abs z - Complex.abs z = bb * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine le_trans (bs1 ?_) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * b ≤ Complex.abs (f' d c z)
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 / (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log) ≤
s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | exact le_trans (mul_le_mul_of_nonneg_left bz b0'.le) fz' | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * b ≤ Complex.abs (f' d c z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * b ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | simp only [div_eq_mul_inv, mul_inv, ←mul_assoc _ (abs z)⁻¹, mul_assoc s1 _ (abs z)⁻¹] | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 / (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log) ≤
s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * ((Complex.abs (f' d c z))⁻¹ * (Complex.abs (f' d c z)).log⁻¹) ≤
s1 * ((b ^ (d - 1) - 1)⁻¹ * (Complex.abs z)⁻¹) * (Complex.abs z).log⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 / (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log) ≤
s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | simp only [←mul_inv, mul_assoc s1] | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * ((Complex.abs (f' d c z))⁻¹ * (Complex.abs (f' d c z)).log⁻¹) ≤
s1 * ((b ^ (d - 1) - 1)⁻¹ * (Complex.abs z)⁻¹) * (Complex.abs z).log⁻¹ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log)⁻¹ ≤
s1 * ((b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * ((Complex.abs (f' d c z))⁻¹ * (Complex.abs (f' d c z)).log⁻¹) ≤
s1 * ((b ^ (d - 1) - 1)⁻¹ * (Complex.abs z)⁻¹) * (Complex.abs z).log⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine mul_le_mul_of_nonneg_left (inv_le_inv_of_le (by positivity) ?_) s1p | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log)⁻¹ ≤
s1 * ((b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log)⁻¹ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log ≤ Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ s1 * (Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log)⁻¹ ≤
s1 * ((b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | exact mul_le_mul fz' (Real.log_le_log (by positivity) zfz) (by positivity)
(le_trans b0.le bfz) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log ≤ Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ (b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log ≤ Complex.abs (f' d c z) * (Complex.abs (f' d c z)).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < (b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < (b ^ (d - 1) - 1) * Complex.abs z * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine le_trans (iter_error_weak d bb3 s2p bs2 ?_ (le_trans cz zffz)) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ bb * b ≤ Complex.abs (f' d c (f' d c z))
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 / (1 - (bb * b - 1)⁻¹) / (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log) ≤
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | exact le_trans (mul_le_mul_of_nonneg_left bz (by positivity)) ffz | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ bb * b ≤ Complex.abs (f' d c (f' d c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ bb * b ≤ Complex.abs (f' d c (f' d c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ bb | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ bb
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | simp only [div_eq_mul_inv, mul_assoc s2] | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 / (1 - (bb * b - 1)⁻¹) / (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log) ≤
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 * ((1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹) ≤
s2 * (((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 / (1 - (bb * b - 1)⁻¹) / (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log) ≤
s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine mul_le_mul_of_nonneg_left ?_ s2p | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 * ((1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹) ≤
s2 * (((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ s2 * ((1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹) ≤
s2 * (((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | simp only [←mul_inv, ←mul_assoc] | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ ((1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹)⁻¹ * (Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb)⁻¹ * (Complex.abs z * (Complex.abs z).log)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine inv_le_inv_of_le (by positivity) ?_ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ ((1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log)⁻¹ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log ≤
(1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ ((1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log)⁻¹ ≤
((1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | refine mul_le_mul ?_ (Real.log_le_log z0 zffz) (by positivity) (by positivity) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log ≤
(1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log ≤
(1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) * (Complex.abs (f' d c (f' d c z))).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | rw [mul_assoc] | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * (bb * Complex.abs z) ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | exact mul_le_mul_of_nonneg_left ffz (by positivity) | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * (bb * Complex.abs z) ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ (1 - (bb * b - 1)⁻¹) * (bb * Complex.abs z) ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < (1 - (bb * b - 1)⁻¹) * bb * Complex.abs z * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ (1 - (bb * b - 1)⁻¹) * Complex.abs (f' d c (f' d c z))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ 1 - (bb * b - 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ 1 - (bb * b - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le | [276, 1] | [342, 54] | positivity | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ Complex.abs z * (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
i b s0 s1 s2 : ℝ
c : ℂ
b3 : 3 ≤ b
s1p : 0 ≤ s1
s2p : 0 ≤ s2
bs0 : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s0 / (Complex.abs w * (Complex.abs w).log)
bs1 : ∀ {w : ℂ}, (b ^ (d - 1) - 1) * b ≤ Complex.abs w → f_error d w ≤ s1 / (Complex.abs w * (Complex.abs w).log)
b0' : 0 < b ^ (d - 1) - 1
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b0 : 0 < b
z0 : 0 < Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 1 < (Complex.abs z).log
bb : ℝ
hbb : (b ^ (d - 1) - 1) ^ d * b ^ (d - 1) - 1 = bb
b11 : 11 ≤ bb
bb3 : 3 ≤ bb * b
bs2 : ∀ {w : ℂ}, bb * b ≤ Complex.abs w → f_error d w ≤ s2 / (Complex.abs w * (Complex.abs w).log)
b0'' : 0 < 1 - (bb * b - 1)⁻¹
si : s0 + s1 / (b ^ (d - 1) - 1) + s2 / ((1 - (bb * b - 1)⁻¹) * bb) ≤ i
fz : Complex.abs z ^ d - Complex.abs c ≤ Complex.abs (f' d c z)
fz' : (b ^ (d - 1) - 1) * Complex.abs z ≤ Complex.abs (f' d c z)
zfz : Complex.abs z ≤ Complex.abs (f' d c z)
zffz : Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
bfz : b ≤ Complex.abs (f' d c z)
ffz : bb * Complex.abs z ≤ Complex.abs (f' d c (f' d c z))
e0 : f_error d z ≤ s0 / (Complex.abs z * (Complex.abs z).log)
e1 : f_error d (f' d c z) ≤ s1 / (b ^ (d - 1) - 1) / (Complex.abs z * (Complex.abs z).log)
e2 : iter_error d c (f' d c (f' d c z)) ≤ s2 / ((1 - (bb * b - 1)⁻¹) * bb) / (Complex.abs z * (Complex.abs z).log)
⊢ 0 ≤ Complex.abs z * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b3 : (3:ℝ) ≤ 3^(d-1) := by
calc (3:ℝ)^(d-1)
_ ≥ 3^(2-1) := by bound
_ = 3 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b3 : 3 ≤ 3 ^ (d - 1)
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | generalize hb3 : (3:ℝ)^(d-1) = t3 at b3 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b3 : 3 ≤ 3 ^ (d - 1)
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
b3 : 3 ≤ 3 ^ (d - 1)
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b2 : (2:ℝ) ≤ t3 - 1 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
b2 : 2 ≤ t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | generalize hb2 : t3 - 1 = t2 at b2 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
b2 : 2 ≤ t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
b2 : 2 ≤ t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have t2p : 0 ≤ t2 := by positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b6 : (6:ℝ) ≤ t2 * 3 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b11 : (11:ℝ) ≤ t2^d * t3 - 1 := by
calc t2^d * t3 - 1
_ ≥ 2^d * 3 - 1 := by bound
_ ≥ 2^2 * 3 - 1 := by bound
_ = 11 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t2 ^ d * t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | generalize hb11 : t2^d * t3 - 1 = t11 at b11 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t2 ^ d * t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t2 ^ d * t3 - 1
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b33 : (33:ℝ) ≤ t11 * 3 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
b33 : 33 ≤ t11 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | generalize hb33 : t11 * 3 = t33 at b33 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
b33 : 33 ≤ t11 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
b33 : 33 ≤ t11 * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11 := by
have h : 1 ≤ t33 - 1 := by linarith
calc (1 - (t33 - 1)⁻¹) * t11
_ ≥ (1 - (33 - 1)⁻¹) * 11 := by bound
_ ≥ 10.65 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | simp only [←hb2, ←hb3, ←hb11, ←hb33] at b2 b6 b11 b33 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | refine iter_error_le _ (by norm_num) (by norm_num) (by norm_num)
(bs0 := f_error_le_of_z3 d)
(bs1 := fun {_} bz ↦ f_error_le_of_z6 d (le_trans b6 bz))
(bs2 := fun {_} bz ↦ f_error_le_of_z33 d (le_trans b33 bz))
b11 (le_trans (by norm_num) b33) (by positivity) ?_ ?_ z3 cz | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log) | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 < 1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0.699 + 0.565 / (3 ^ (d - 1) - 1) +
0.512 /
((1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹) * ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1)) ≤
1.03 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ iter_error d c z ≤ 1.03 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | calc (3:ℝ)^(d-1)
_ ≥ 3^(2-1) := by bound
_ = 3 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ≤ 3 ^ (d - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ≤ 3 ^ (d - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ^ (d - 1) ≥ 3 ^ (2 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ^ (d - 1) ≥ 3 ^ (2 - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ^ (2 - 1) = 3 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ^ (2 - 1) = 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
⊢ 2 ≤ t3 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
⊢ 2 ≤ t3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
⊢ 0 ≤ t2 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
⊢ 0 ≤ t2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
⊢ 6 ≤ t2 * 3 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
⊢ 6 ≤ t2 * 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | calc t2^d * t3 - 1
_ ≥ 2^d * 3 - 1 := by bound
_ ≥ 2^2 * 3 - 1 := by bound
_ = 11 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 11 ≤ t2 ^ d * t3 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 11 ≤ t2 ^ d * t3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ t2 ^ d * t3 - 1 ≥ 2 ^ d * 3 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ t2 ^ d * t3 - 1 ≥ 2 ^ d * 3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 2 ^ d * 3 - 1 ≥ 2 ^ 2 * 3 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 2 ^ d * 3 - 1 ≥ 2 ^ 2 * 3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 2 ^ 2 * 3 - 1 = 11 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
⊢ 2 ^ 2 * 3 - 1 = 11
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
⊢ 33 ≤ t11 * 3 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
⊢ 33 ≤ t11 * 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | have h : 1 ≤ t33 - 1 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | calc (1 - (t33 - 1)⁻¹) * t11
_ ≥ (1 - (33 - 1)⁻¹) * 11 := by bound
_ ≥ 10.65 := by norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ 1 ≤ t33 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
⊢ 1 ≤ t33 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ (1 - (t33 - 1)⁻¹) * t11 ≥ (1 - (33 - 1)⁻¹) * 11 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ (1 - (t33 - 1)⁻¹) * t11 ≥ (1 - (33 - 1)⁻¹) * 11
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ (1 - (33 - 1)⁻¹) * 11 ≥ 10.65 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
b2 : 2 ≤ t2
t2p : 0 ≤ t2
b6 : 6 ≤ t2 * 3
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
b11 : 11 ≤ t11
t33 : ℝ
hb33 : t11 * 3 = t33
b33 : 33 ≤ t33
h : 1 ≤ t33 - 1
⊢ (1 - (33 - 1)⁻¹) * 11 ≥ 10.65
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 3 ≤ 3 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 3 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 ≤ 0.565 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 ≤ 0.565
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 ≤ 0.512 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 ≤ 0.512
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 3 ≤ 33 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 3 ≤ 33
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 < 3 ^ (d - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 < 3 ^ (d - 1) - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | exact sub_pos.mpr (inv_lt_one (by linarith)) | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 < 1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹ | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0 < 1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 1 < ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 1 < ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | simp only [hb2, hb3, hb11, hb33] at b2 b3 b6 b11 b33 ⊢ | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0.699 + 0.565 / (3 ^ (d - 1) - 1) +
0.512 /
((1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹) * ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1)) ≤
1.03 | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0.699 + 0.565 / t2 + 0.512 / ((1 - (t33 - 1)⁻¹) * t11) ≤ 1.03 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ 3 ^ (d - 1) - 1
b6 : 6 ≤ (3 ^ (d - 1) - 1) * 3
b11 : 11 ≤ (3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1
b33 : 33 ≤ ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3
⊢ 0.699 + 0.565 / (3 ^ (d - 1) - 1) +
0.512 /
((1 - (((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1) * 3 - 1)⁻¹) * ((3 ^ (d - 1) - 1) ^ d * 3 ^ (d - 1) - 1)) ≤
1.03
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | exact le_trans (add_le_add (add_le_add_left
(div_le_div_of_nonneg_left (by norm_num) (by norm_num) b2) _)
(div_le_div_of_nonneg_left (by norm_num) (by norm_num) b10)) (by norm_num) | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0.699 + 0.565 / t2 + 0.512 / ((1 - (t33 - 1)⁻¹) * t11) ≤ 1.03 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0.699 + 0.565 / t2 + 0.512 / ((1 - (t33 - 1)⁻¹) * t11) ≤ 1.03
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 ≤ 0.565 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 ≤ 0.565
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 < 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 ≤ 0.512 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 ≤ 0.512
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_le_of_z3 | [344, 1] | [379, 81] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 < 10.65 | 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
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
t3 : ℝ
hb3 : 3 ^ (d - 1) = t3
b3 : 3 ≤ t3
t2 : ℝ
hb2 : t3 - 1 = t2
t2p : 0 ≤ t2
t11 : ℝ
hb11 : t2 ^ d * t3 - 1 = t11
t33 : ℝ
hb33 : t11 * 3 = t33
b10 : 10.65 ≤ (1 - (t33 - 1)⁻¹) * t11
b2 : 2 ≤ t2
b6 : 6 ≤ t2 * 3
b11 : 11 ≤ t11
b33 : 33 ≤ t33
⊢ 0 < 10.65
TACTIC:
|
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