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 | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1.02 / 33 / (2 * 3.49) ≤ 45e-4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1.02 / 33 / (2 * 3.49) ≤ 45e-4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 45e-4 ≤ min 1 (((1.003 - 1) * 2)⁻¹ + 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)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 45e-4 ≤ min 1 (((1.003 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 33⁻¹ ≤ min 1 (((1.02 - 1) * 2)⁻¹ + 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)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 33⁻¹ ≤ min 1 (((1.02 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1 / 2 * 1.003 * 1.02 ≤ 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)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1 / 2 * 1.003 * 1.02 ≤ 0.512
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 0 < 3.49 | 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)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 0 < 3.49
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z33 | [184, 1] | [190, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1 < 1.003 | 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)
z : ℂ
z33 : 33 ≤ Complex.abs z
⊢ 1 < 1.003
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | refine f_error_le_generic d (b := 140) (l := 4.94) (s := 1.004) (t := 0.00073) (c := 1.0004)
(g := _) z140 (lb := ?_) (b3 := by norm_num) (st := by norm_num) (tc := by norm_num)
(bs := by norm_num) (csg := by norm_num) (l0 := by norm_num) (c1 := by norm_num) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ f_error d z ≤ 0.5023 / (Complex.abs z * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ exp 4.94 ≤ 140 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ f_error d z ≤ 0.5023 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ exp 4.94 ≤ 140 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ (247 / 50).exp ≤ 140 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ exp 4.94 ≤ 140
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | exact (exp_div_lt).le | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ (247 / 50).exp ≤ 140 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ (247 / 50).exp ≤ 140
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 3 ≤ 140 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 3 ≤ 140
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1.004 / 140 / (2 * 4.94) ≤ 73e-5 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1.004 / 140 / (2 * 4.94) ≤ 73e-5
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 73e-5 ≤ min 1 (((1.0004 - 1) * 2)⁻¹ + 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 73e-5 ≤ min 1 (((1.0004 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 140⁻¹ ≤ min 1 (((1.004 - 1) * 2)⁻¹ + 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 140⁻¹ ≤ min 1 (((1.004 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1 / 2 * 1.0004 * 1.004 ≤ 0.5023 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1 / 2 * 1.0004 * 1.004 ≤ 0.5023
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 0 < 4.94 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 0 < 4.94
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z140 | [192, 1] | [198, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1 < 1.0004 | 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)
z : ℂ
z140 : 140 ≤ Complex.abs z
⊢ 1 < 1.0004
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have mf : ∀ k (w : ℂ), b ≤ abs w → abs c ≤ abs w → (b-1)^k * abs w ≤ abs ((f' d c)^[k] w) :=
fun k w bw cw ↦ iter_large d b (by linarith) bw cw k | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
⊢ iter_error_sum d c z N ≤ s / (1 - (b - 1)⁻¹) / (C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have z3 : 3 ≤ abs z := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have l0 : 0 < log (abs z) := lt_of_lt_of_le (by norm_num) (le_log_abs_z z3) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have b1 : 1 < b - 1 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have fb : ∀ k, f_error d ((f' d c)^[k] z) ≤ s / ((b-1)^k * abs z * log (abs z)) := by
intro k
have mk := mf k z bz cz
have mk' : abs z ≤ abs ((f' d c)^[k] z) :=
le_trans (le_mul_of_one_le_left (Complex.abs.nonneg _) (one_le_pow_of_one_le b1.le _)) mk
refine le_trans (bs (le_trans bz mk')) ?_
refine div_le... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | simp only [div_eq_mul_inv, ←mul_assoc, mul_inv, mul_comm _ ((b-1)^_)⁻¹,
mul_comm _ (1-(b-1)⁻¹)⁻¹] at fb ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | simp only [mul_assoc] at fb ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | generalize ht : s * ((Complex.abs z)⁻¹ * (log (Complex.abs z))⁻¹) = t at fb | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have t0 : 0 ≤ t := by rw [←ht]; positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | apply le_trans (Finset.sum_le_sum (fun k _ ↦ fb k)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | simp only [mul_comm _ t, ←Finset.mul_sum, ←inv_pow] at fb ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | exact mul_le_mul_of_nonneg_left (partial_geometric_bound _ (by positivity) (inv_lt_one b1)) t0 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
k : ℕ
w : ℂ
bw : b ≤ Complex.abs w
cw : Complex.abs... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | intro k | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have mk := mf k z bz cz | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | have mk' : abs z ≤ abs ((f' d c)^[k] z) :=
le_trans (le_mul_of_one_le_left (Complex.abs.nonneg _) (one_le_pow_of_one_le b1.le _)) mk | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | refine le_trans (bs (le_trans bz mk')) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | refine div_le_div_of_nonneg_left s0 (by positivity) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | exact mul_le_mul mk (Real.log_le_log (by positivity) mk') l0.le (by positivity) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | rw [←ht] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_sum_weak | [213, 1] | [238, 97] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
mf :
∀ (k : ℕ) (w : ℂ),
b ≤ Complex.abs w → C... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | apply summable_of_sum_le | 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
⊢ Summable fun n => f_error d ((f' d c)^[n] z) | case hf
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
⊢ 0 ≤ fun n => f_error d ((f' d c)^[n] z)
case h
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
⊢ ... | 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
⊢ Summable fun n => f_error d ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | intro n | case hf
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
⊢ 0 ≤ fun n => f_error d ((f' d c)^[n] z) | case hf
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
n : ℕ
⊢ 0 n ≤ (fun n => f_error d ((f' d c)^[n] z)) n | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
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
⊢ 0 ≤ fun n => f_error d ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | exact f_error_nonneg (le_trans z3 (le_self_iter d z3 cz n)) | case hf
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
n : ℕ
⊢ 0 n ≤ (fun n => f_error d ((f' d c)^[n] z)) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
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
n : ℕ
⊢ 0 n ≤ (fun n => f_error d ((f' d c)^[n] z)) n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | intro N | case h
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
⊢ ∀ (u : Finset ℕ), (u.sum fun x => f_error d ((f' d c)^[x] z)) ≤ ?c | case h
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
N : Finset ℕ
⊢ (N.sum fun x => f_error d ((f' d c)^[x] z)) ≤ ?c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (u : Finset ℕ), (u.sum fun x => f_error d ((f' d c)^[x] z)) ≤ ?c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | exact iter_error_sum_weak d (le_refl _) (by norm_num) (f_error_le_of_z3 d) z3 cz | case h
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
N : Finset ℕ
⊢ (N.sum fun x => f_error d ((f' d c)^[x] z)) ≤ ?c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
N : Finset ℕ
⊢ (N.sum fun x => f_error d ((f' d c)^[x] z)) ≤ ?c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_summable | [240, 1] | [245, 94] | 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
N : Finset ℕ
⊢ 0 ≤ 0.699 | 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
N : Finset ℕ
⊢ 0 ≤ 0.699
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_peel | [247, 1] | [252, 43] | have h0 := sum_drop (iter_error_summable d z3 cz).hasSum | c✝ : ℂ
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 = f_error d z + iter_error d c (f' d c z) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 :
HasSum (fun n => f_error d ((f' d c)^[n + 1] z)) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d ((f' d c)^[0] z))
⊢ iter_error d c z = f_error d z + iter_error d c (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
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 = f_error d z + iter_error d c (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_peel | [247, 1] | [252, 43] | simp only [Function.iterate_succ_apply, zero_add, Function.iterate_zero, id_eq] at h0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 :
HasSum (fun n => f_error d ((f' d c)^[n + 1] z)) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d ((f' d c)^[0] z))
⊢ iter_error d c z = f_error d z + iter_error d c (f' d c z) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ iter_error d c z = f_error d z + iter_error d c (f' d c z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 :
HasSum (fun n => f_error d ((f' d c)^[n + 1] z)) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d ((f' d c)^[0] z))
⊢ iter_error d c z = f_error d z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_peel | [247, 1] | [252, 43] | simp only [iter_error, h0.tsum_eq] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ iter_error d c z = f_error d z + iter_error d c (f' d c z) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ ∑' (n : ℕ), f_error d ((f' d c)^[n] z) = f_error d z + (∑' (n : ℕ), f_error d ((f' d c)^[n] z) - f_error... | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ iter_error d c z = f_error d z + iter_error... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_peel | [247, 1] | [252, 43] | abel | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ ∑' (n : ℕ), f_error d ((f' d c)^[n] z) = f_error d z + (∑' (n : ℕ), f_error d ((f' d c)^[n] z) - f_error... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
h0 : HasSum (fun n => f_error d ((f' d c)^[n] (f' d c z))) (∑' (b : ℕ), f_error d ((f' d c)^[b] z) - f_error d z)
⊢ ∑' (n : ℕ), f_error d ((f' d c)^[n] z) = f_... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | have z3 : 3 ≤ abs z := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ s / (1 - (b - 1)⁻¹) / (Complex.abs z * (Com... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ iter_error d c z ≤ s / (1 - (b - 1)⁻¹) ... | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | have l0 : 0 < log (abs z) := lt_of_lt_of_le (by norm_num) (le_log_abs_z z3) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ iter_error d c z ≤ s / (1 - (b - 1)⁻¹) ... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ iter_error... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | have b1 : 1 < b - 1 := by linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ iter_error... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
b1 : 1 < b -... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | have b0 : 0 ≤ 1 - (b - 1)⁻¹ := sub_nonneg.mpr (inv_le_one b1.le) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
b1 : 1 < b -... | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
b1 : 1 < b -... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | refine tsum_le_of_sum_le' ?_ ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
b1 : 1 < b -... | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).lo... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ≤ Complex.abs z | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < 1.0986 | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 1 < b - 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | positivity | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).lo... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | intro N | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).lo... | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).lo... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak | [254, 1] | [265, 56] | exact iter_error_sum_weak d b3 s0 bs bz cz | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
l0 : 0 < (Complex.abs z).lo... | 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)
b s : ℝ
c : ℂ
b3 : 3 ≤ b
s0 : 0 ≤ s
bs : ∀ {w : ℂ}, b ≤ Complex.abs w → f_error d w ≤ s / (Complex.abs w * (Complex.abs w).log)
z : ℂ
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | refine le_trans (iter_error_weak d (b := 33) (by norm_num) (by norm_num)
(fun {_} bz ↦ f_error_le_of_z33 d bz) z33 cz) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 0.529 / (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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ 0.529 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | have l0 : 0 < log (abs z) :=
lt_of_lt_of_le (by norm_num) (le_log_abs_z (le_trans (by norm_num) z33)) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | exact div_le_div_of_nonneg_right (by norm_num) (by positivity) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (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✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 0.512 / (1 - (33 - 1)⁻¹) / (Complex.abs z * (Complex.abs z).log) ≤ 0.529 / (Complex.abs z * (Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 3 ≤ 33
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 ≤ 0.512
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 1.0986 | 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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 1.0986
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 0.512 / (1 - (33 - 1)⁻¹) ≤ 0.529 | 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
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (Complex.abs z).log
⊢ 0.512 / (1 - (33 - 1)⁻¹) ≤ 0.529
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_error_weak_of_z33 | [267, 1] | [274, 65] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (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✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z33 : 33 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
l0 : 0 < (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 | [276, 1] | [342, 54] | have b0 : 0 < b := lt_of_lt_of_le (by norm_num) b3 | 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},... | 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},... | 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 ≤ ... |
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},... | 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},... | 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 ≤ ... |
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},... | 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},... | 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 ≤ ... |
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},... | 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},... | 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 ≤ ... |
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},... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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) * a... | 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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 ... | 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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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},... | 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 ≤ ... |
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},... | 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 ≤ ... |
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},... | 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 ≤ ... |
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},... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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 -... | 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 ≤ ... |
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