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stringlengths 6
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | ring_nf at h | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 - x + x * 2
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | rwa [mul_comm] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
h : (x * log 2).exp ≤ 1 + x
⊢ (log 2 * x).exp ≤ 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 < 1 + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 < 1 + x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 ≤ 1 - x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
⊢ 0 ≤ 1 - x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | le_log_one_add | [32, 1] | [39, 17] | abel | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ 1 - x + x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x0 : 0 ≤ x
x1 : x ≤ 1
x0' : 0 ≤ 1 - x
⊢ 1 - x + x = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have hd : ∀ x, 2 ≤ x → HasDerivAt (fun x ↦ log (x-1) / log x)
((1 / (x-1) * log x - log (x-1) * x⁻¹) / (log x)^2) x := by
intro x x2
have l0 : 0 < log x := Real.log_pos (by linarith)
refine HasDerivAt.div ?_ (Real.hasDerivAt_log (by positivity)) l0.ne'
exact HasDerivAt.log ((hasDerivAt_id _).sub_const _) (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have d : DifferentiableOn ℝ (fun x ↦ log (x-1) / log x) (Ici 2) :=
fun x m ↦ (hd x m).differentiableAt.differentiableWithinAt | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | apply monotoneOn_of_deriv_nonneg (convex_Ici _) | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2) | case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2)
case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2))
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ MonotoneOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | intro x x2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have l0 : 0 < log x := Real.log_pos (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | refine HasDerivAt.div ?_ (Real.hasDerivAt_log (by positivity)) l0.ne' | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact HasDerivAt.log ((hasDerivAt_id _).sub_const _) (by linarith) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ HasDerivAt (fun x => (x - 1).log) (1 / (x - 1)) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x - 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
x2 : 2 ≤ x
l0 : 0 < x.log
⊢ x - 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact d.continuousOn | case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ContinuousOn (fun x => (x - 1).log / x.log) (Ici 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact d.mono interior_subset | case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (interior (Ici 2))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | intro x m | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
⊢ ∀ x ∈ interior (Ici 2), 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : x ∈ interior (Ici 2)
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | have l0 : 0 < log x := Real.log_pos (by linarith) | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [(hd x m.le).deriv, one_div] | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ deriv (fun x => (x - 1).log / x.log) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | refine div_nonneg ?_ (by positivity) | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2 | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ ((x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹) / x.log ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | simp only [sub_nonneg, mul_comm] | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹ | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ * x.log - (x - 1).log * x⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | apply mul_le_mul | case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log | case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹
case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log
case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log
case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ * (x - 1).log ≤ (x - 1)⁻¹ * x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | positivity | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x.log ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x.log ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact inv_le_inv_of_le (by linarith) (by linarith) | case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.h₁
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x⁻¹ ≤ (x - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x - 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ x - 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact Real.log_le_log (by linarith) (by linarith) | case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.h₂
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ (x - 1).log ≤ x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact Real.log_nonneg (by linarith) | case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.c0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 1 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 1 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | exact inv_nonneg.mpr (by linarith) | case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'_nonneg.b0
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ (x - 1)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_ratio_mono | [41, 1] | [64, 41] | linarith | c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝ : Fact (2 ≤ d✝)
hd :
∀ (x : ℝ), 2 ≤ x → HasDerivAt (fun x => (x - 1).log / x.log) ((1 / (x - 1) * x.log - (x - 1).log * x⁻¹) / x.log ^ 2) x
d : DifferentiableOn ℝ (fun x => (x - 1).log / x.log) (Ici 2)
x : ℝ
m : 2 < x
l0 : 0 < x.log
⊢ 0 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | generalize abs (f' d c z) = w at zw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
⊢ (Complex.abs z).log.log + 0.548 ≤ (Complex.abs (f' d c z)).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | generalize abs z = x at zw cz z4 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
w : ℝ
zw : (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ w
⊢ (Complex.abs z).log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | clear z c cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
cz : Complex.abs c ≤ x
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have x0 : 0 < x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have x1 : 1 < x := by linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [pow_one] at zw | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have lx1 : 1 < log (x-1) :=
lt_trans (by norm_num) (lt_of_lt_of_le lt_log_3 (Real.log_le_log (by norm_num) (by linarith))) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have lx : 1 < log x := lt_trans lx1 (Real.log_lt_log (by linarith) (by linarith)) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have ll0 : 0 ≤ log (x-1) / log x := by positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | have ll1 : log (x-1) / log x ≤ 1 :=
div_le_one_of_le (Real.log_le_log (by linarith) (by linarith)) (by positivity) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | calc log (log w)
_ ≥ log (log ((x-1)*x)) := log_log_mono (by nlinarith) zw
_ = log (log x + log (x-1)) := by rw [mul_comm, Real.log_mul (by positivity) (by linarith)]
_ = log (log x) + log (1 + log (x-1) / log x) := by rw [log_add _ _ (by linarith) (by linarith)]
_ ≥ log (log x) + log 2 * (log (x-1) / log x) := by bound [le_log_one_add ll0 ll1]
_ ≥ log (log x) + 0.693 * 0.791 := by bound [lt_log_2]
_ ≥ log (log x) + 0.548 := by bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.548 ≤ w.log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | refine iter_large (d := d) (n := 1) ?_ ?_ ?_ cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z) | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ (Complex.abs z - 1) ^ 1 * Complex.abs z ≤ Complex.abs (f' d c z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ 2 ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | exact le_refl _ | case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Complex.abs z ≤ Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ 0 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
⊢ 0 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
zw : (x - 1) ^ 1 * x ≤ w
z4 : 4 ≤ x
x0 : 0 < x
⊢ 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 1 < 1.098 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 1 < 1.098
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 3 ≤ x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
⊢ 3 ≤ x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x - 1 < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
⊢ x - 1 < x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | apply log_ratio_mono ?_ ?_ z4 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (x - 1).log / x.log ≥ (4 - 1).log / log 4 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (x - 1).log / x.log ≥ (4 - 1).log / log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [mem_Ici] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 4 ∈ Ici 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | simp only [mem_Ici] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2 | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ x ∈ Ici 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 2 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (4 - 1).log / log 4 = log 3 / log 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ (4 - 1).log / log 4 = log 3 / log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 ≤ log 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 ≤ log 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 < log 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 0 < log 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | norm_num | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 1.098 / 1.387 ≥ 0.791 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
⊢ 1.098 / 1.387 ≥ 0.791
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ 0 ≤ (x - 1).log / x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
⊢ 0 ≤ (x - 1).log / x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 < x - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 < x - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x - 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ x - 1 ≤ x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 ≤ x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
⊢ 0 ≤ x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | nlinarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 1 < (x - 1) * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 1 < (x - 1) * x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | rw [mul_comm, Real.log_mul (by positivity) (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ ((x - 1) * x).log.log = (x.log + (x - 1).log).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ ((x - 1) * x).log.log = (x.log + (x - 1).log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | positivity | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x - 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x - 1 ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | rw [log_add _ _ (by linarith) (by linarith)] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ (x.log + (x - 1).log).log = x.log.log + (1 + (x - 1).log / x.log).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ (x.log + (x - 1).log).log = x.log.log + (1 + (x - 1).log / x.log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | linarith | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log + (x - 1).log | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ 0 < x.log + (x - 1).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound [le_log_one_add ll0 ll1] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + (1 + (x - 1).log / x.log).log ≥ x.log.log + log 2 * ((x - 1).log / x.log) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + (1 + (x - 1).log / x.log).log ≥ x.log.log + log 2 * ((x - 1).log / x.log)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound [lt_log_2] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + log 2 * ((x - 1).log / x.log) ≥ x.log.log + 0.693 * 0.791 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + log 2 * ((x - 1).log / x.log) ≥ x.log.log + 0.693 * 0.791
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | log_log_iter | [66, 1] | [100, 40] | bound | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.693 * 0.791 ≥ x.log.log + 0.548 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
w x : ℝ
z4 : 4 ≤ x
x0 : 0 < x
x1 : 1 < x
zw : (x - 1) * x ≤ w
lx1 : 1 < (x - 1).log
lx : 1 < x.log
ll : 0.791 ≤ (x - 1).log / x.log
ll0 : 0 ≤ (x - 1).log / x.log
ll1 : (x - 1).log / x.log ≤ 1
⊢ x.log.log + 0.693 * 0.791 ≥ x.log.log + 0.548
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have d0 : 0 < d := d_pos d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have lcz : log (log (abs c)) ≤ log (log (abs z)) := log_log_mono (by linarith) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [Postcritical, multibrot_p] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ Postcritical ⋯ c ↑z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | set s := superF d | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
⊢ ⋯.potential c ↑z < ⋯.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [← Real.pow_rpow_inv_natCast s.potential_nonneg d0.ne', ←
Real.pow_rpow_inv_natCast (s.potential_nonneg : 0 ≤ s.potential c 0) d0.ne'] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c ↑z < s.potential c 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [← s.potential_eqn] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ (s.potential c ↑z ^ d) ^ (↑d)⁻¹ < (s.potential c 0 ^ d) ^ (↑d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine Real.rpow_lt_rpow s.potential_nonneg ?_ (by bound) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) ^ (↑d)⁻¹ < s.potential c (f d c 0) ^ (↑d)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | generalize hw : f' d c z = w | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have e : f d c z = w := by rw [f, lift_coe', hw] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | simp only [f_0, e] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c (f d c ↑z) < s.potential c (f d c 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | clear e | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
e : f d c ↑z = ↑w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have zw : abs z ≤ abs w := by rw [←hw]; exact le_self_iter d (by linarith) cz 1 | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have cw : abs c ≤ abs w := le_trans cz zw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have pc1 : s.potential c c < 1 := potential_lt_one_of_two_lt (by linarith) (le_refl _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have pw1 : s.potential c w < 1 := potential_lt_one_of_two_lt (by linarith) (by linarith) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | rw [←log_neg_log_strict_anti potential_pos potential_pos pw1 pc1] | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ s.potential c ↑w < s.potential c ↑c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine lt_of_lt_of_le ?_ (le_sub_iff_add_le.mp (abs_le.mp
(log_neg_log_potential_approx d (by linarith) cw)).1) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < (-(⋯.potential c ↑w).log).log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | refine lt_of_le_of_lt (sub_le_iff_le_add.mp (abs_le.mp
(log_neg_log_potential_approx d (by linarith) (le_refl _))).2) ?_ | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ (-(⋯.potential c ↑c).log).log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have lzw : log (log (abs z)) + 0.548 ≤ log (log (abs w)) := by
rw [←hw]; exact log_log_iter (by linarith) cz | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have ie : ∀ z : ℂ, 4 ≤ abs z → abs c ≤ abs z → iter_error d c z ≤ 0.15 := by
intro z z4 cz
refine le_trans (iter_error_le_of_z4 d z4 cz) ?_
calc 0.8095 / (abs z * log (abs z))
_ ≤ 0.8095 / (4 * log 4) := by bound
_ ≤ 0.8095 / (4 * 1.386) := by bound [lt_log_4]
_ ≤ 0.15 := by norm_num | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have iec := ie c c4 (le_refl _) | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Postcritical.lean | postcritical_large | [103, 1] | [142, 54] | have iew := ie w (by linarith) cw | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
iew : iter_error d c w ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
c4 : 4 ≤ Complex.abs c
cz : Complex.abs c ≤ Complex.abs z
d0 : 0 < d
lcz : (Complex.abs c).log.log ≤ (Complex.abs z).log.log
s : Super (f d) d ∞ := superF d
w : ℂ
hw : f' d c z = w
zw : Complex.abs z ≤ Complex.abs w
cw : Complex.abs c ≤ Complex.abs w
pc1 : s.potential c ↑c < 1
pw1 : s.potential c ↑w < 1
lzw : (Complex.abs z).log.log + 0.548 ≤ (Complex.abs w).log.log
ie : ∀ (z : ℂ), 4 ≤ Complex.abs z → Complex.abs c ≤ Complex.abs z → iter_error d c z ≤ 0.15
iec : iter_error d c c ≤ 0.15
⊢ iter_error d c c + (Complex.abs c).log.log < -iter_error d c w + (Complex.abs w).log.log
TACTIC:
|
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