url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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| tactic
stringlengths 1
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| state_after
stringlengths 6
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x y : ℝ
x0 : 0 ≤ x
xy : x ≤ y
⊢ dene x ≥ dene y
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℝ
z1 : 1 < z
k : ℝ
l0 : 0 < z.log
⊢ 0 < z
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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