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 | tendsto_iter_atInf | [61, 1] | [65, 100] | exact Filter.Tendsto.atTop_mul_const (by linarith) (tendsto_pow_atTop_atTop_of_one_lt one_lt_two) | 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
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop | 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
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | linarith | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 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)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | rw [← Complex.abs.ne_zero_iff] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ z ^ d + c ≠ 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ z ^ d + c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | apply ne_of_gt | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | have z1 : 1 ≤ abs z := le_trans (by norm_num) z3 | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | calc abs (z ^ d + c)
_ ≥ abs (z ^ d) - abs c := by bound
_ = abs z ^ d - abs c := by rw [Complex.abs.map_pow]
_ ≥ abs z ^ 2 - abs z := by bound
_ = abs z * (abs z - 1) := by ring
_ ≥ 3 * (3 - 1) := by bound
_ > 0 := by norm_num | case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 0 < Complex.abs (z ^ d + c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
⊢ 1 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d + c) ≥ Complex.abs (z ^ d) - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | rw [Complex.abs.map_pow] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs (z ^ d) - Complex.abs c = Complex.abs z ^ d - Complex.abs c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ 2 - Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ d - Complex.abs c ≥ Complex.abs z ^ 2 - Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | ring | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ 2 - Complex.abs z = Complex.abs z * (Complex.abs z - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z ^ 2 - Complex.abs z = Complex.abs z * (Complex.abs z - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | bound | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z * (Complex.abs z - 1) ≥ 3 * (3 - 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ Complex.abs z * (Complex.abs z - 1) ≥ 3 * (3 - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_ne_zero | [67, 1] | [77, 25] | norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 3 * (3 - 1) > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
cz : Complex.abs c ≤ Complex.abs z
z3 : 3 ≤ Complex.abs z
z1 : 1 ≤ Complex.abs z
⊢ 3 * (3 - 1) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | rw [Real.le_log_iff_exp_le (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 1.0986 ≤ (Complex.abs z).log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 1.0986 ≤ (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | refine le_trans ?_ z3 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | exact (exp_div_lt).le | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_log_abs_z | [90, 1] | [95, 24] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have z0 : 0 < abs z := lt_of_lt_of_le (by norm_num) z3 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have z0' : z ≠ 0 := by exact nnnorm_pos.mp z0 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have i1 : 1 / abs z ≤ 1 := by rw [one_div_le z0]; exact le_trans (by norm_num) z3; norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have s1 : 1 - 1 / abs z < 1 := by rw [tsub_lt_iff_tsub_lt]; norm_num; exact z0'; exact i1; rfl | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | have l1 := le_log_abs_z z3 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact div_nonneg (neg_nonneg.mpr (Real.log_nonpos (sub_nonneg.mpr i1) s1.le)) (by positivity) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact nnnorm_pos.mp z0 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
⊢ z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | rw [one_div_le z0] | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / Complex.abs z ≤ 1 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / 1 ≤ Complex.abs z
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / Complex.abs z ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact le_trans (by norm_num) z3 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / 1 ≤ Complex.abs z
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / 1 ≤ Complex.abs z
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 0 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / 1 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
⊢ 1 / 1 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | rw [tsub_lt_iff_tsub_lt] | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 - 1 / Complex.abs z < 1 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 - 1 < 1 / Complex.abs z
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 - 1 / Complex.abs z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | norm_num | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 - 1 < 1 / Complex.abs z
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ ¬z = 0
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 - 1 < 1 / Complex.abs z
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact z0' | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ ¬z = 0
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ ¬z = 0
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | exact i1 | case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 / Complex.abs z ≤ 1
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | rfl | case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
⊢ 1 ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_inner_nonneg | [97, 1] | [105, 96] | positivity | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ ↑d * (Complex.abs z).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
d : ℕ
z : ℂ
z3 : 3 ≤ Complex.abs z
z0 : 0 < Complex.abs z
z0' : z ≠ 0
i1 : 1 / Complex.abs z ≤ 1
s1 : 1 - 1 / Complex.abs z < 1
l1 : 1.0986 ≤ (Complex.abs z).log
⊢ 0 ≤ ↑d * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | rw [f_error, le_neg, neg_zero] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ f_error d z | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | 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
z3 : 3 ≤ Complex.abs z
⊢ 0 ≤ f_error d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | have d0 : 0 < d := d_pos d | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | 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
z3 : 3 ≤ Complex.abs z
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | have l1 : 1 ≤ log (abs z) := le_trans (by norm_num) (le_log_abs_z z3) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | 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
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | apply Real.log_nonpos | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0 | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
case h'x
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ 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)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)).log ≤ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
⊢ 1 ≤ 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)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
⊢ 1 ≤ 1.0986
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | simp only [one_div, neg_div, sub_nonneg, neg_le] | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -1 ≤ (1 - (Complex.abs z)⁻¹).log / (↑d * (Complex.abs z).log) | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | rw [le_div_iff (by positivity), neg_one_mul, neg_le] | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -1 ≤ (1 - (Complex.abs z)⁻¹).log / (↑d * (Complex.abs z).log) | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ ↑d * (Complex.abs z).log | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -1 ≤ (1 - (Complex.abs z)⁻¹).log / (↑d * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | trans 2 | case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ ↑d * (Complex.abs z).log | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ 2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 ≤ ↑d * (Complex.abs z).log | Please generate a tactic in lean4 to solve the state.
STATE:
case hx
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ ↑d * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 < ↑d * (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)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 < ↑d * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | refine le_trans (neg_log_one_sub_le_linear (c := 2) (by positivity) (by norm_num) ?_) ?_ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ 2 | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((2 - 1) * 2)⁻¹ + 1)⁻¹
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 * (Complex.abs z)⁻¹ ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ (Complex.abs z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 < 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 < 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | exact le_trans (inv_le_inv_of_le (by positivity) z3) (by norm_num) | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((2 - 1) * 2)⁻¹ + 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((2 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 3⁻¹ ≤ min 1 (((2 - 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
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 3⁻¹ ≤ min 1 (((2 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | simp only [zero_lt_two, mul_le_iff_le_one_right] | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 * (Complex.abs z)⁻¹ ≤ 2 | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 * (Complex.abs z)⁻¹ ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | apply inv_le_one | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ 1 | case refine_2.ha
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 ≤ Complex.abs z | 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)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | linarith | case refine_2.ha
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.ha
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | exact le_trans (by norm_num) (mul_le_mul (two_le_cast_d d) l1 zero_le_one (by positivity)) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 ≤ ↑d * (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)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 ≤ ↑d * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 ≤ 2 * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 2 ≤ 2 * 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ ↑d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 0 ≤ ↑d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_nonneg | [107, 1] | [120, 41] | linarith [f_error_inner_nonneg d z3] | case h'x
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h'x
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
d0 : 0 < d
l1 : 1 ≤ (Complex.abs z).log
⊢ 1 - -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have z3 := le_trans b3 bz | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | replace lb := (Real.le_log_iff_exp_le (by positivity)).mpr (le_trans lb bz) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have l0' : 0 < log (abs z) := trans l0 lb | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have inner_le : -log (1 - 1 / abs z) ≤ s / abs z := by
rw [one_div, div_eq_mul_inv]
apply neg_log_one_sub_le_linear (by positivity) s1
exact le_trans (inv_le_inv_of_le (by positivity) bz) bs | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have dm : 2 * log (abs z) ≤ d * log (abs z) := by bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have div_le : -log (1 - 1 / abs z) / (d * log (abs z)) ≤ t := by
have sz : s / abs z ≤ s / b := div_le_div_of_nonneg_left (by positivity) (by positivity) bz
exact le_trans (div_le_div (by positivity) (le_trans inner_le sz) (by positivity)
(le_trans (mul_le_mul_of_nonneg_left lb (by norm_num)) dm)) st | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ f_error d z ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | refine le_trans (neg_log_one_sub_le_linear (f_error_inner_nonneg d z3) c1 ?_) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log) | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (-(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)) ≤ g / (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ f_error d z ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
⊢ 0 < 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
lb : l.exp ≤ b
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
⊢ 0 < Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | rw [one_div, div_eq_mul_inv] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ s * (Complex.abs z)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | apply neg_log_one_sub_le_linear (by positivity) s1 | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ s * (Complex.abs z)⁻¹ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ -(1 - (Complex.abs z)⁻¹).log ≤ s * (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | exact le_trans (inv_le_inv_of_le (by positivity) bz) bs | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((s - 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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ (Complex.abs z)⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ 0 ≤ (Complex.abs z)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ 0 ≤ (Complex.abs z)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ 0 < b | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
⊢ 0 < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | bound | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
⊢ 2 * (Complex.abs z).log ≤ ↑d * (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
⊢ 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | have sz : s / abs z ≤ s / b := div_le_div_of_nonneg_left (by positivity) (by positivity) bz | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | exact le_trans (div_le_div (by positivity) (le_trans inner_le sz) (by positivity)
(le_trans (mul_le_mul_of_nonneg_left lb (by norm_num)) dm)) st | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ 0 ≤ s | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ 0 ≤ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ 0 < b | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
⊢ 0 < b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 ≤ s / b | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 ≤ s / b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 < 2 * l | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 < 2 * l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
sz : s / Complex.abs z ≤ s / b
⊢ 0 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | exact le_trans div_le tc | case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | refine le_trans (mul_le_mul_of_nonneg_left (div_le_div (by positivity) inner_le
(by positivity) dm) (by positivity)) ?_ | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (-(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)) ≤ g / (Complex.abs z * (Complex.abs z).log) | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (s / Complex.abs z / (2 * (Complex.abs z).log)) ≤ g / (Complex.abs z * (Complex.abs z).log) | 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (-(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log)) ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | simp only [div_eq_mul_inv, ←mul_assoc, mul_inv, mul_comm _ (2⁻¹ : ℝ)] | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (s / Complex.abs z / (2 * (Complex.abs z).log)) ≤ g / (Complex.abs z * (Complex.abs z).log) | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 2⁻¹ * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ c * (s / Complex.abs z / (2 * (Complex.abs z).log)) ≤ g / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | norm_num | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 2⁻¹ * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 2⁻¹ * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | simp only [mul_assoc _ (abs z)⁻¹ _] | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹) ≤ g * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹ ≤ g * (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | exact mul_le_mul_of_nonneg_right csg (by positivity) | case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹) ≤ g * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 1 / 2 * c * s * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹) ≤ g * ((Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ s / 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ s / Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 < 2 * (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)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 < 2 * (Complex.abs z).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ 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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_generic | [122, 1] | [152, 57] | positivity | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ (Complex.abs z)⁻¹ * (Complex.abs z).log⁻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 l s t c g : ℝ
z : ℂ
bz : b ≤ Complex.abs z
b3 : autoParam (3 ≤ b) _auto✝
st : autoParam (s / b / (2 * l) ≤ t) _auto✝
tc : autoParam (t ≤ min 1 (((c - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
bs : autoParam (b⁻¹ ≤ min 1 (((s - 1) * 2)⁻¹ + 1)⁻¹) _auto✝
csg : autoParam (1 / 2 * c * s ≤ g) _auto✝
l0 : autoParam (0 < l) _auto✝
c1 : autoParam (1 < c) _auto✝
s1 : autoParam (1 < s) _auto✝
z3 : 3 ≤ Complex.abs z
lb : l ≤ (Complex.abs z).log
l0' : 0 < (Complex.abs z).log
inner_le : -(1 - 1 / Complex.abs z).log ≤ s / Complex.abs z
dm : 2 * (Complex.abs z).log ≤ ↑d * (Complex.abs z).log
div_le : -(1 - 1 / Complex.abs z).log / (↑d * (Complex.abs z).log) ≤ t
⊢ 0 ≤ (Complex.abs z)⁻¹ * (Complex.abs z).log⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z3 | [154, 1] | [158, 34] | refine f_error_le_generic d 3 1.0986 (s := 1.25) (t := 0.1897) (c := 1.1171) _ z3 ?_ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ f_error d z ≤ 0.699 / (Complex.abs z * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | 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
z3 : 3 ≤ Complex.abs z
⊢ f_error d z ≤ 0.699 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z3 | [154, 1] | [158, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | 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
z3 : 3 ≤ Complex.abs z
⊢ exp 1.0986 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z3 | [154, 1] | [158, 34] | exact (exp_div_lt).le | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z3 : 3 ≤ Complex.abs z
⊢ (5493 / 5000).exp ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | refine f_error_le_generic d (b := 4) (l := 1.3862) (s := 1.167) (t := 0.106) (c := 1.06)
(g := _) z4 (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
z4 : 4 ≤ Complex.abs z
⊢ f_error d z ≤ 0.619 / (Complex.abs z * (Complex.abs z).log) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ exp 1.3862 ≤ 4 | 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
z4 : 4 ≤ Complex.abs z
⊢ f_error d z ≤ 0.619 / (Complex.abs z * (Complex.abs z).log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ exp 1.3862 ≤ 4 | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ (6931 / 5000).exp ≤ 4 | 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
z4 : 4 ≤ Complex.abs z
⊢ exp 1.3862 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | exact (exp_div_lt).le | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ (6931 / 5000).exp ≤ 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
z4 : 4 ≤ Complex.abs z
⊢ (6931 / 5000).exp ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ 3 ≤ 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
z4 : 4 ≤ Complex.abs z
⊢ 3 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ 1.167 / 4 / (2 * 1.3862) ≤ 0.106 | 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
z4 : 4 ≤ Complex.abs z
⊢ 1.167 / 4 / (2 * 1.3862) ≤ 0.106
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ 0.106 ≤ min 1 (((1.06 - 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
z4 : 4 ≤ Complex.abs z
⊢ 0.106 ≤ min 1 (((1.06 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ 4⁻¹ ≤ min 1 (((1.167 - 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
z4 : 4 ≤ Complex.abs z
⊢ 4⁻¹ ≤ min 1 (((1.167 - 1) * 2)⁻¹ + 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | f_error_le_of_z4 | [160, 1] | [166, 34] | norm_num | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
z4 : 4 ≤ Complex.abs z
⊢ 1 / 2 * 1.06 * 1.167 ≤ 0.619 | 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
z4 : 4 ≤ Complex.abs z
⊢ 1 / 2 * 1.06 * 1.167 ≤ 0.619
TACTIC:
|
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