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
|
---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
hasDerivAt_ene
|
[136, 1]
|
[140, 68]
|
simp only [mul_neg, ← Real.exp_add, neg_add_eq_sub] at h
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x
⊢ HasDerivAt ene (-dene x) x
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x
⊢ HasDerivAt ene (-dene x) x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
hasDerivAt_ene
|
[136, 1]
|
[140, 68]
|
exact h
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x
⊢ HasDerivAt ene (-dene x) x
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
refine Real.exp_le_exp.mpr ?_
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ dene x ≥ dene y
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ y - y.exp ≤ x - x.exp
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
exact a x0 (le_trans x0 xy) xy
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
a : AntitoneOn (fun x => x - x.exp) (Ici 0)
⊢ y - y.exp ≤ x - x.exp
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
have hd : ∀ x, HasDerivAt (fun x ↦ x - exp x) (1 - exp x) x :=
fun x ↦ (hasDerivAt_id x).sub (Real.hasDerivAt_exp x)
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
have d : Differentiable ℝ (fun x ↦ x - exp x) := fun x ↦ (hd x).differentiableAt
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
|
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
apply antitoneOn_of_deriv_nonpos (convex_Ici _)
|
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
|
case hf
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)
case hf'
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
exact d.continuous.continuousOn
|
case hf
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
exact d.differentiableOn
|
case hf'
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
intro x m
|
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0
|
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : x ∈ interior (Ici 0)
⊢ deriv (fun x => x - x.exp) x ≤ 0
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m
|
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : x ∈ interior (Ici 0)
⊢ deriv (fun x => x - x.exp) x ≤ 0
|
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : 0 < x
⊢ deriv (fun x => x - x.exp) x ≤ 0
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_anti
|
[148, 1]
|
[160, 33]
|
simp only [(hd x).deriv, sub_nonpos, Real.one_le_exp m.le]
|
case hf'_nonpos
c z : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
x✝ y : ℝ
x0 : 0 ≤ x✝
xy : x✝ ≤ y
hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x
d : Differentiable ℝ fun x => x - x.exp
x : ℝ
m : 0 < x
⊢ deriv (fun x => x - x.exp) x ≤ 0
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_eq
|
[162, 1]
|
[169, 60]
|
have l0 : 0 < log z := Real.log_pos z1
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_eq
|
[162, 1]
|
[169, 60]
|
simp only [dene, sub_eq_add_neg, Real.exp_add, Real.exp_log l0, mul_comm _ (exp (-k)), ←neg_mul]
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_eq
|
[162, 1]
|
[169, 60]
|
apply congr_arg₂ _ rfl
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_eq
|
[162, 1]
|
[169, 60]
|
rw [mul_comm, Real.exp_mul, Real.exp_log (by positivity)]
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
dene_eq
|
[162, 1]
|
[169, 60]
|
positivity
|
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ 0 < z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have z1 : 1 < abs z := by linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have z4 : 4 ≤ abs z := by linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have l1 : 1.386 < log (abs z) := lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) z4)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [potential_error, dene_eq z1]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have ie := iter_error_le_of_z4 d z4 cz
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log)
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have ie0 := iter_error_nonneg d (by linarith) cz
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log)
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log)
ie0 : 0 ≤ iter_error d c z
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
generalize hk : (0.8095 : ℝ) = k at ie
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log)
ie0 : 0 ≤ iter_error d c z
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have k0 : 0 < k := by rw [← hk]; norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
k0 : 0 < k
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
generalize hx : abs z = x at ie z1 l0 l1 z4
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
k0 : 0 < k
⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
ie : iter_error d c z ≤ k / (x * x.log)
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / x ^ (1 + (-k / (x * x.log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
generalize iter_error d c z = i at ie ie0
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
ie : iter_error d c z ≤ k / (x * x.log)
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤
k / x ^ (1 + (-k / (x * x.log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
have x0 : 0 < x := by rw [←hx]; linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
refine le_trans (mul_le_mul_of_nonneg_left ie (by positivity)) ?_
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [←mul_assoc, div_eq_inv_mul, mul_inv, mul_comm _ (log x)]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [←mul_assoc, mul_comm _ (log x)⁻¹, inv_mul_cancel l0.ne', one_mul]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [mul_assoc]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
refine le_trans (mul_le_of_le_one_left (by positivity) (Real.exp_le_one_iff.mpr (by linarith))) ?_
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [←mul_assoc, ←Real.rpow_neg_one x, ←Real.rpow_add (lt_trans zero_lt_one z1),
←Real.rpow_neg x0.le]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
refine mul_le_mul_of_nonneg_right (Real.rpow_le_rpow_of_exponent_le z1.le ?_) k0.le
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
simp only [Real.rpow_neg_one, mul_neg, neg_add_rev, le_add_neg_iff_add_le, neg_add_cancel_right,
neg_le_neg_iff, Real.exp_le_exp, ←mul_inv, ←div_eq_inv_mul k, mul_comm _ x, ie]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
⊢ 4 ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
⊢ 0 < 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ 0 < 1.386
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log)
⊢ 3 ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
rw [← hk]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < k
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < 0.8095
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
z1 : 1 < Complex.abs z
z4 : 4 ≤ Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
ie0 : 0 ≤ iter_error d c z
k : ℝ
hk : 0.8095 = k
ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log)
⊢ 0 < 0.8095
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
rw [←hx]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
⊢ 0 < x
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
⊢ 0 < Complex.abs z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
⊢ 0 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ 0 ≤ (-i).exp * x.log * x ^ (-(-i).exp)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ 0 ≤ x ^ (-(-i).exp) * (x⁻¹ * k)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le
|
[179, 1]
|
[204, 84]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
k : ℝ
hk : 0.8095 = k
k0 : 0 < k
x : ℝ
hx : Complex.abs z = x
z1 : 1 < x
l0 : 0 < x.log
l1 : 1.386 < x.log
z4 : 4 ≤ x
i : ℝ
ie : i ≤ k / (x * x.log)
ie0 : 0 ≤ i
x0 : 0 < x
⊢ -i ≤ 0
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
have z0 : 0 < abs z := by linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
have l1 : 1.386 < log (abs z) :=
lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) (by linarith))
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
refine le_trans (potential_error_le d b4 bz cz) (div_le_div_of_nonneg_left (by norm_num)
(by positivity) ?_)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
refine Real.rpow_le_rpow_of_exponent_le (by linarith) ?_
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
simp only [add_comm (1:ℝ), ←sub_le_iff_le_add]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
refine le_trans ij (Real.exp_le_exp.mpr ?_)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
simp only [neg_div, neg_le_neg_iff, div_mul_eq_div_div]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
rw [div_le_iff l0, mul_comm j, ←div_le_iff j0]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
trans 0.8095 / b / j
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / b / j ≤ (Complex.abs z).log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
⊢ 0 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
⊢ 0 < 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
⊢ 4 ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
⊢ 0 < 1.386
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0 ≤ 0.8095
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0 < Complex.abs z ^ i
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
linarith
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 1 ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
gcongr
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
rw [Real.le_log_iff_exp_le z0]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ 0.8095 / b / j ≤ (Complex.abs z).log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ (0.8095 / b / j).exp ≤ Complex.abs z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le'
|
[206, 1]
|
[227, 25]
|
exact le_trans bj bz
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
i j b : ℝ
c z : ℂ
b4 : 4 ≤ b
bz : b ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
j0 : 0 < j
ij : i - 1 ≤ (-j).exp
bj : (0.8095 / b / j).exp ≤ b
z0 : 0 < Complex.abs z
l1 : 1.386 < (Complex.abs z).log
l0 : 0 < (Complex.abs z).log
⊢ (0.8095 / b / j).exp ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
apply potential_error_le' d _ (j := 0.146) (b := 4) (by norm_num) z4 cz (by norm_num)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.864
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1.864 - 1 ≤ (-0.146).exp
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (0.8095 / 4 / 0.146).exp ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 4 ≤ 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 0.146
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
norm_num
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1.864 - 1 ≤ (-0.146).exp
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 108 / 125 ≤ (-(73 / 500)).exp
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
exact (lt_exp_neg_div).le
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 108 / 125 ≤ (-(73 / 500)).exp
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
norm_num
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (0.8095 / 4 / 0.146).exp ≤ 4
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (1619 / 1168).exp ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z4
|
[229, 1]
|
[235, 36]
|
exact (exp_div_lt).le
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (1619 / 1168).exp ≤ 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
apply potential_error_le' d _ (j := 0.0753) (b := 6) (by norm_num) z6 cz (by norm_num)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.927
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1.927 - 1 ≤ (-753e-4).exp
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (0.8095 / 6 / 753e-4).exp ≤ 6
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 4 ≤ 6
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < 753e-4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
norm_num
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1.927 - 1 ≤ (-753e-4).exp
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 927 / 1000 ≤ (-(753 / 10000)).exp
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
exact (lt_exp_neg_div).le
|
case ij
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 927 / 1000 ≤ (-(753 / 10000)).exp
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
norm_num
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (0.8095 / 6 / 753e-4).exp ≤ 6
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (8095 / 4518).exp ≤ 6
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
potential_error_le_of_z6
|
[237, 1]
|
[243, 36]
|
exact (exp_div_lt).le
|
case bj
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z6 : 6 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (8095 / 4518).exp ≤ 6
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
have hl : 1.38 ≤ log (abs z) := by
rw [Real.le_log_iff_exp_le (by positivity)]
norm_num
exact le_trans (exp_div_lt).le z4
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ iter_error d c z ≤ (Complex.abs z).log.log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ iter_error d c z ≤ (Complex.abs z).log.log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
have hll : 0.32 ≤ log (log (abs z)) := by
rw [Real.le_log_iff_exp_le (by positivity)]
norm_num
exact le_trans (exp_div_lt).le hl
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ iter_error d c z ≤ (Complex.abs z).log.log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ iter_error d c z ≤ (Complex.abs z).log.log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
refine le_trans ?_ hll
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ iter_error d c z ≤ (Complex.abs z).log.log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ iter_error d c z ≤ 0.32
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
apply le_trans (iter_error_le_of_z4 d z4 cz)
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ iter_error d c z ≤ 0.32
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
rw [div_le_iff' (by positivity), ←div_le_iff (by norm_num)]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
refine le_trans (by norm_num) (mul_le_mul z4 hl (by positivity) (by positivity))
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
rw [Real.le_log_iff_exp_le (by positivity)]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 1.38 ≤ (Complex.abs z).log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ exp 1.38 ≤ Complex.abs z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ exp 1.38 ≤ Complex.abs z
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (69 / 50).exp ≤ Complex.abs z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
exact le_trans (exp_div_lt).le z4
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (69 / 50).exp ≤ Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 0 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
rw [Real.le_log_iff_exp_le (by positivity)]
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ 0.32 ≤ (Complex.abs z).log.log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ exp 0.32 ≤ (Complex.abs z).log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ exp 0.32 ≤ (Complex.abs z).log
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ (8 / 25).exp ≤ (Complex.abs z).log
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
exact le_trans (exp_div_lt).le hl
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ (8 / 25).exp ≤ (Complex.abs z).log
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
⊢ 0 < (Complex.abs z).log
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
positivity
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 < Complex.abs z * (Complex.abs z).log
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0 < 0.32
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
iter_error_le_log_log_abs
|
[245, 1]
|
[259, 83]
|
norm_num
|
c✝ z✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
hl : 1.38 ≤ (Complex.abs z).log
hll : 0.32 ≤ (Complex.abs z).log.log
⊢ 0.8095 / 0.32 ≤ 4 * 1.38
|
no goals
|
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