internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	hasDerivAt_ene	[136, 1]	[140, 68]	simp only [mul_neg, ← Real.exp_add, neg_add_eq_sub] at h	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x ⊢ HasDerivAt ene (-dene x) x	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x ⊢ HasDerivAt ene (-dene x) x
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	hasDerivAt_ene	[136, 1]	[140, 68]	exact h	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x ⊢ HasDerivAt ene (-dene x) x	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	refine Real.exp_le_exp.mpr ?_	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ dene x ≥ dene y	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ y - y.exp ≤ x - x.exp
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	exact a x0 (le_trans x0 xy) xy	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y a : AntitoneOn (fun x => x - x.exp) (Ici 0) ⊢ y - y.exp ≤ x - x.exp	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	have hd : ∀ x, HasDerivAt (fun x ↦ x - exp x) (1 - exp x) x := fun x ↦ (hasDerivAt_id x).sub (Real.hasDerivAt_exp x)	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	have d : Differentiable ℝ (fun x ↦ x - exp x) := fun x ↦ (hd x).differentiableAt	c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)	c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	apply antitoneOn_of_deriv_nonpos (convex_Ici _)	c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)	case hf c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ContinuousOn (fun x => x - x.exp) (Ici 0) case hf' c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0)) case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	exact d.continuous.continuousOn	case hf c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	exact d.differentiableOn	case hf' c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	intro x m	case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0	case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : x ∈ interior (Ici 0) ⊢ deriv (fun x => x - x.exp) x ≤ 0
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m	case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : x ∈ interior (Ici 0) ⊢ deriv (fun x => x - x.exp) x ≤ 0	case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : 0 < x ⊢ deriv (fun x => x - x.exp) x ≤ 0
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_anti	[148, 1]	[160, 33]	simp only [(hd x).deriv, sub_nonpos, Real.one_le_exp m.le]	case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : 0 < x ⊢ deriv (fun x => x - x.exp) x ≤ 0	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_eq	[162, 1]	[169, 60]	have l0 : 0 < log z := Real.log_pos z1	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_eq	[162, 1]	[169, 60]	simp only [dene, sub_eq_add_neg, Real.exp_add, Real.exp_log l0, mul_comm _ (exp (-k)), ←neg_mul]	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_eq	[162, 1]	[169, 60]	apply congr_arg₂ _ rfl	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_eq	[162, 1]	[169, 60]	rw [mul_comm, Real.exp_mul, Real.exp_log (by positivity)]	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	dene_eq	[162, 1]	[169, 60]	positivity	c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ 0 < z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have z1 : 1 < abs z := by linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have z4 : 4 ≤ abs z := by linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have l1 : 1.386 < log (abs z) := lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) z4)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [potential_error, dene_eq z1]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have ie := iter_error_le_of_z4 d z4 cz	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have ie0 := iter_error_nonneg d (by linarith) cz	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ie0 : 0 ≤ iter_error d c z ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	generalize hk : (0.8095 : ℝ) = k at ie	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ie0 : 0 ≤ iter_error d c z ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have k0 : 0 < k := by rw [← hk]; norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) k0 : 0 < k ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	generalize hx : abs z = x at ie z1 l0 l1 z4	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) k0 : 0 < k ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x ie : iter_error d c z ≤ k / (x * x.log) z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x ⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	generalize iter_error d c z = i at ie ie0	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x ie : iter_error d c z ≤ k / (x * x.log) z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x ⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / x ^ (1 + (-k / (x * x.log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	have x0 : 0 < x := by rw [←hx]; linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	refine le_trans (mul_le_mul_of_nonneg_left ie (by positivity)) ?_	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [←mul_assoc, div_eq_inv_mul, mul_inv, mul_comm _ (log x)]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [←mul_assoc, mul_comm _ (log x)⁻¹, inv_mul_cancel l0.ne', one_mul]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [mul_assoc]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	refine le_trans (mul_le_of_le_one_left (by positivity) (Real.exp_le_one_iff.mpr (by linarith))) ?_	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [←mul_assoc, ←Real.rpow_neg_one x, ←Real.rpow_add (lt_trans zero_lt_one z1), ←Real.rpow_neg x0.le]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	refine mul_le_mul_of_nonneg_right (Real.rpow_le_rpow_of_exponent_le z1.le ?_) k0.le	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	simp only [Real.rpow_neg_one, mul_neg, neg_add_rev, le_add_neg_iff_add_le, neg_add_cancel_right, neg_le_neg_iff, Real.exp_le_exp, ←mul_inv, ←div_eq_inv_mul k, mul_comm _ x, ie]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1 < Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ 4 ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ 0 < 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ 0 < 1.386	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ 3 ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	rw [← hk]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < k	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < 0.8095
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < 0.8095	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	rw [←hx]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < x	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < Complex.abs z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ 0 ≤ (-i).exp * x.log * x ^ (-(-i).exp)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ 0 ≤ x ^ (-(-i).exp) * (x⁻¹ * k)	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le	[179, 1]	[204, 84]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -i ≤ 0	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	have z0 : 0 < abs z := by linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	have l1 : 1.386 < log (abs z) := lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) (by linarith))	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	refine le_trans (potential_error_le d b4 bz cz) (div_le_div_of_nonneg_left (by norm_num) (by positivity) ?_)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	refine Real.rpow_le_rpow_of_exponent_le (by linarith) ?_	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	simp only [add_comm (1:ℝ), ←sub_le_iff_le_add]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	refine le_trans ij (Real.exp_le_exp.mpr ?_)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	simp only [neg_div, neg_le_neg_iff, div_mul_eq_div_div]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	rw [div_le_iff l0, mul_comm j, ←div_le_iff j0]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	trans 0.8095 / b / j	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / b / j ≤ (Complex.abs z).log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b ⊢ 0 < Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ 0 < 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ 4 ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ 0 < 1.386	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0 ≤ 0.8095	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0 < Complex.abs z ^ i	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	linarith	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 1 ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	gcongr	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	rw [Real.le_log_iff_exp_le z0]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / b / j ≤ (Complex.abs z).log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (0.8095 / b / j).exp ≤ Complex.abs z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le'	[206, 1]	[227, 25]	exact le_trans bj bz	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (0.8095 / b / j).exp ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	apply potential_error_le' d _ (j := 0.146) (b := 4) (by norm_num) z4 cz (by norm_num)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.864	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.864 - 1 ≤ (-0.146).exp case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 4 / 0.146).exp ≤ 4
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 4 ≤ 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < 0.146	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	norm_num	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.864 - 1 ≤ (-0.146).exp	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 108 / 125 ≤ (-(73 / 500)).exp
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	exact (lt_exp_neg_div).le	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 108 / 125 ≤ (-(73 / 500)).exp	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	norm_num	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 4 / 0.146).exp ≤ 4	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (1619 / 1168).exp ≤ 4
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z4	[229, 1]	[235, 36]	exact (exp_div_lt).le	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (1619 / 1168).exp ≤ 4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	apply potential_error_le' d _ (j := 0.0753) (b := 6) (by norm_num) z6 cz (by norm_num)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.927	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.927 - 1 ≤ (-753e-4).exp case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 6 / 753e-4).exp ≤ 6
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 4 ≤ 6	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < 753e-4	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	norm_num	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.927 - 1 ≤ (-753e-4).exp	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 927 / 1000 ≤ (-(753 / 10000)).exp
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	exact (lt_exp_neg_div).le	case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 927 / 1000 ≤ (-(753 / 10000)).exp	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	norm_num	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 6 / 753e-4).exp ≤ 6	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (8095 / 4518).exp ≤ 6
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	potential_error_le_of_z6	[237, 1]	[243, 36]	exact (exp_div_lt).le	case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (8095 / 4518).exp ≤ 6	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	have hl : 1.38 ≤ log (abs z) := by rw [Real.le_log_iff_exp_le (by positivity)] norm_num exact le_trans (exp_div_lt).le z4	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ iter_error d c z ≤ (Complex.abs z).log.log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ iter_error d c z ≤ (Complex.abs z).log.log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	have hll : 0.32 ≤ log (log (abs z)) := by rw [Real.le_log_iff_exp_le (by positivity)] norm_num exact le_trans (exp_div_lt).le hl	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ iter_error d c z ≤ (Complex.abs z).log.log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ (Complex.abs z).log.log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	refine le_trans ?_ hll	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ (Complex.abs z).log.log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ 0.32
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	apply le_trans (iter_error_le_of_z4 d z4 cz)	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ 0.32	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	rw [div_le_iff' (by positivity), ←div_le_iff (by norm_num)]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	refine le_trans (by norm_num) (mul_le_mul z4 hl (by positivity) (by positivity))	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	rw [Real.le_log_iff_exp_le (by positivity)]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.38 ≤ (Complex.abs z).log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ exp 1.38 ≤ Complex.abs z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ exp 1.38 ≤ Complex.abs z	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (69 / 50).exp ≤ Complex.abs z
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	exact le_trans (exp_div_lt).le z4	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (69 / 50).exp ≤ Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < Complex.abs z	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	rw [Real.le_log_iff_exp_le (by positivity)]	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ 0.32 ≤ (Complex.abs z).log.log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ exp 0.32 ≤ (Complex.abs z).log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ exp 0.32 ≤ (Complex.abs z).log	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ (8 / 25).exp ≤ (Complex.abs z).log
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	exact le_trans (exp_div_lt).le hl	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ (8 / 25).exp ≤ (Complex.abs z).log	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ 0 < (Complex.abs z).log	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	positivity	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 < Complex.abs z * (Complex.abs z).log	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 < 0.32	no goals
https://github.com/girving/ray.git	0be790285dd0fce78913b0cb9bddaffa94bd25f9	Ray/Dynamics/Multibrot/Potential.lean	iter_error_le_log_log_abs	[245, 1]	[259, 83]	norm_num	c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ 4 * 1.38	no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

hasDerivAt_ene

[136, 1]

[140, 68]

simp only [mul_neg, ← Real.exp_add, neg_add_eq_sub] at h

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) ((-x.exp).exp * -x.exp) x ⊢ HasDerivAt ene (-dene x) x

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x ⊢ HasDerivAt ene (-dene x) x

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

hasDerivAt_ene

[136, 1]

[140, 68]

exact h

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x : ℝ h : HasDerivAt (fun x => (-x.exp).exp) (-(x - x.exp).exp) x ⊢ HasDerivAt ene (-dene x) x

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

refine Real.exp_le_exp.mpr ?_

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ dene x ≥ dene y

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ y - y.exp ≤ x - x.exp

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

exact a x0 (le_trans x0 xy) xy

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y a : AntitoneOn (fun x => x - x.exp) (Ici 0) ⊢ y - y.exp ≤ x - x.exp

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

have hd : ∀ x, HasDerivAt (fun x ↦ x - exp x) (1 - exp x) x := fun x ↦ (hasDerivAt_id x).sub (Real.hasDerivAt_exp x)

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

have d : Differentiable ℝ (fun x ↦ x - exp x) := fun x ↦ (hd x).differentiableAt

c z : ℂ d : ℕ inst✝ : Fact (2 ≤ d) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)

c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

apply antitoneOn_of_deriv_nonpos (convex_Ici _)

c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ AntitoneOn (fun x => x - x.exp) (Ici 0)

case hf c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ContinuousOn (fun x => x - x.exp) (Ici 0) case hf' c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0)) case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

exact d.continuous.continuousOn

case hf c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ContinuousOn (fun x => x - x.exp) (Ici 0)

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

exact d.differentiableOn

case hf' c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ DifferentiableOn ℝ (fun x => x - x.exp) (interior (Ici 0))

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

intro x m

case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x y : ℝ x0 : 0 ≤ x xy : x ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp ⊢ ∀ x ∈ interior (Ici 0), deriv (fun x => x - x.exp) x ≤ 0

case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : x ∈ interior (Ici 0) ⊢ deriv (fun x => x - x.exp) x ≤ 0

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

simp only [nonempty_Iio, interior_Ici', mem_Ioi] at m

case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : x ∈ interior (Ici 0) ⊢ deriv (fun x => x - x.exp) x ≤ 0

case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : 0 < x ⊢ deriv (fun x => x - x.exp) x ≤ 0

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_anti

[148, 1]

[160, 33]

simp only [(hd x).deriv, sub_nonpos, Real.one_le_exp m.le]

case hf'_nonpos c z : ℂ d✝ : ℕ inst✝ : Fact (2 ≤ d✝) x✝ y : ℝ x0 : 0 ≤ x✝ xy : x✝ ≤ y hd : ∀ (x : ℝ), HasDerivAt (fun x => x - x.exp) (1 - x.exp) x d : Differentiable ℝ fun x => x - x.exp x : ℝ m : 0 < x ⊢ deriv (fun x => x - x.exp) x ≤ 0

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_eq

[162, 1]

[169, 60]

have l0 : 0 < log z := Real.log_pos z1

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_eq

[162, 1]

[169, 60]

simp only [dene, sub_eq_add_neg, Real.exp_add, Real.exp_log l0, mul_comm _ (exp (-k)), ←neg_mul]

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ dene (z.log.log - k) = (-k).exp * z.log * z ^ (-(-k).exp)

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_eq

[162, 1]

[169, 60]

apply congr_arg₂ _ rfl

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-k).exp * z.log * (-(-k).exp * z.log).exp = (-k).exp * z.log * z ^ (-(-k).exp)

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_eq

[162, 1]

[169, 60]

rw [mul_comm, Real.exp_mul, Real.exp_log (by positivity)]

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ (-(-k).exp * z.log).exp = z ^ (-(-k).exp)

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

dene_eq

[162, 1]

[169, 60]

positivity

c z✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℝ z1 : 1 < z k : ℝ l0 : 0 < z.log ⊢ 0 < z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have z1 : 1 < abs z := by linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have z4 : 4 ≤ abs z := by linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have l1 : 1.386 < log (abs z) := lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) z4)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [potential_error, dene_eq z1]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have ie := iter_error_le_of_z4 d z4 cz

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have ie0 := iter_error_nonneg d (by linarith) cz

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ie0 : 0 ≤ iter_error d c z ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

generalize hk : (0.8095 : ℝ) = k at ie

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ie0 : 0 ≤ iter_error d c z ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ 0.8095 / Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have k0 : 0 < k := by rw [← hk]; norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) k0 : 0 < k ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

generalize hx : abs z = x at ie z1 l0 l1 z4

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) k0 : 0 < k ⊢ (-iter_error d c z).exp * (Complex.abs z).log * Complex.abs z ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / Complex.abs z ^ (1 + (-k / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x ie : iter_error d c z ≤ k / (x * x.log) z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x ⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

generalize iter_error d c z = i at ie ie0

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x ie : iter_error d c z ≤ k / (x * x.log) z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x ⊢ (-iter_error d c z).exp * x.log * x ^ (-(-iter_error d c z).exp) * iter_error d c z ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

have x0 : 0 < x := by rw [←hx]; linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

refine le_trans (mul_le_mul_of_nonneg_left ie (by positivity)) ?_

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * i ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [←mul_assoc, div_eq_inv_mul, mul_inv, mul_comm _ (log x)]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x.log * x ^ (-(-i).exp) * (k / (x * x.log)) ≤ k / x ^ (1 + (-k / (x * x.log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [←mul_assoc, mul_comm _ (log x)⁻¹, inv_mul_cancel l0.ne', one_mul]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x.log * (-i).exp * x ^ (-(-i).exp) * x.log⁻¹ * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [mul_assoc]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * x ^ (-(-i).exp) * x⁻¹ * k ≤ (x ^ (1 + (x.log⁻¹ * x⁻¹ * -k).exp))⁻¹ * k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

refine le_trans (mul_le_of_le_one_left (by positivity) (Real.exp_le_one_iff.mpr (by linarith))) ?_

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ (-i).exp * (x ^ (-(-i).exp) * (x⁻¹ * k)) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [←mul_assoc, ←Real.rpow_neg_one x, ←Real.rpow_add (lt_trans zero_lt_one z1), ←Real.rpow_neg x0.le]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp) * (x⁻¹ * k) ≤ (x ^ (1 + (x.log⁻¹ * (x⁻¹ * -k)).exp))⁻¹ * k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

refine mul_le_mul_of_nonneg_right (Real.rpow_le_rpow_of_exponent_le z1.le ?_) k0.le

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ x ^ (-(-i).exp + -1) * k ≤ x ^ (-(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)) * k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

simp only [Real.rpow_neg_one, mul_neg, neg_add_rev, le_add_neg_iff_add_le, neg_add_cancel_right, neg_le_neg_iff, Real.exp_le_exp, ←mul_inv, ←div_eq_inv_mul k, mul_comm _ x, ie]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -(-i).exp + -1 ≤ -(1 + (x.log⁻¹ * x ^ (-1) * -k).exp)

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1 < Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z ⊢ 4 ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z ⊢ 0 < 4

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ 0 < 1.386

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie : iter_error d c z ≤ 0.8095 / (Complex.abs z * (Complex.abs z).log) ⊢ 3 ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

rw [← hk]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < k

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < 0.8095

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z z1 : 1 < Complex.abs z z4 : 4 ≤ Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ie0 : 0 ≤ iter_error d c z k : ℝ hk : 0.8095 = k ie : iter_error d c z ≤ k / (Complex.abs z * (Complex.abs z).log) ⊢ 0 < 0.8095

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

rw [←hx]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < x

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < Complex.abs z

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i ⊢ 0 < Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ 0 ≤ (-i).exp * x.log * x ^ (-(-i).exp)

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ 0 ≤ x ^ (-(-i).exp) * (x⁻¹ * k)

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le

[179, 1]

[204, 84]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z k : ℝ hk : 0.8095 = k k0 : 0 < k x : ℝ hx : Complex.abs z = x z1 : 1 < x l0 : 0 < x.log l1 : 1.386 < x.log z4 : 4 ≤ x i : ℝ ie : i ≤ k / (x * x.log) ie0 : 0 ≤ i x0 : 0 < x ⊢ -i ≤ 0

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

have z0 : 0 < abs z := by linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

have l1 : 1.386 < log (abs z) := lt_of_lt_of_le lt_log_4 (Real.log_le_log (by linarith) (by linarith))

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

have l0 : 0 < log (abs z) := lt_trans (by norm_num) l1

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

refine le_trans (potential_error_le d b4 bz cz) (div_le_div_of_nonneg_left (by norm_num) (by positivity) ?_)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ i

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

refine Real.rpow_le_rpow_of_exponent_le (by linarith) ?_

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ Complex.abs z ^ i ≤ Complex.abs z ^ (1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

simp only [add_comm (1:ℝ), ←sub_le_iff_le_add]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i ≤ 1 + (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

refine le_trans ij (Real.exp_le_exp.mpr ?_)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ i - 1 ≤ (-0.8095 / (Complex.abs z * (Complex.abs z).log)).exp

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

simp only [neg_div, neg_le_neg_iff, div_mul_eq_div_div]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ -j ≤ -0.8095 / (Complex.abs z * (Complex.abs z).log)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

rw [div_le_iff l0, mul_comm j, ←div_le_iff j0]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / (Complex.abs z).log ≤ j

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

trans 0.8095 / b / j

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ (Complex.abs z).log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / b / j ≤ (Complex.abs z).log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b ⊢ 0 < Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ 0 < 4

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z ⊢ 4 ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log ⊢ 0 < 1.386

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0 ≤ 0.8095

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0 < Complex.abs z ^ i

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

linarith

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 1 ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

gcongr

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / Complex.abs z / j ≤ 0.8095 / b / j

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

rw [Real.le_log_iff_exp_le z0]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ 0.8095 / b / j ≤ (Complex.abs z).log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (0.8095 / b / j).exp ≤ Complex.abs z

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le'

[206, 1]

[227, 25]

exact le_trans bj bz

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) i j b : ℝ c z : ℂ b4 : 4 ≤ b bz : b ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z j0 : 0 < j ij : i - 1 ≤ (-j).exp bj : (0.8095 / b / j).exp ≤ b z0 : 0 < Complex.abs z l1 : 1.386 < (Complex.abs z).log l0 : 0 < (Complex.abs z).log ⊢ (0.8095 / b / j).exp ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

apply potential_error_le' d _ (j := 0.146) (b := 4) (by norm_num) z4 cz (by norm_num)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.864

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.864 - 1 ≤ (-0.146).exp case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 4 / 0.146).exp ≤ 4

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 4 ≤ 4

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < 0.146

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

norm_num

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.864 - 1 ≤ (-0.146).exp

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 108 / 125 ≤ (-(73 / 500)).exp

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

exact (lt_exp_neg_div).le

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 108 / 125 ≤ (-(73 / 500)).exp

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

norm_num

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 4 / 0.146).exp ≤ 4

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (1619 / 1168).exp ≤ 4

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z4

[229, 1]

[235, 36]

exact (exp_div_lt).le

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (1619 / 1168).exp ≤ 4

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

apply potential_error_le' d _ (j := 0.0753) (b := 6) (by norm_num) z6 cz (by norm_num)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ potential_error d c z ≤ 0.8095 / Complex.abs z ^ 1.927

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.927 - 1 ≤ (-753e-4).exp case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 6 / 753e-4).exp ≤ 6

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 4 ≤ 6

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < 753e-4

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

norm_num

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.927 - 1 ≤ (-753e-4).exp

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 927 / 1000 ≤ (-(753 / 10000)).exp

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

exact (lt_exp_neg_div).le

case ij c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 927 / 1000 ≤ (-(753 / 10000)).exp

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

norm_num

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (0.8095 / 6 / 753e-4).exp ≤ 6

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (8095 / 4518).exp ≤ 6

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

potential_error_le_of_z6

[237, 1]

[243, 36]

exact (exp_div_lt).le

case bj c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z6 : 6 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (8095 / 4518).exp ≤ 6

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

have hl : 1.38 ≤ log (abs z) := by rw [Real.le_log_iff_exp_le (by positivity)] norm_num exact le_trans (exp_div_lt).le z4

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ iter_error d c z ≤ (Complex.abs z).log.log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ iter_error d c z ≤ (Complex.abs z).log.log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

have hll : 0.32 ≤ log (log (abs z)) := by rw [Real.le_log_iff_exp_le (by positivity)] norm_num exact le_trans (exp_div_lt).le hl

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ iter_error d c z ≤ (Complex.abs z).log.log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ (Complex.abs z).log.log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

refine le_trans ?_ hll

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ (Complex.abs z).log.log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ 0.32

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

apply le_trans (iter_error_le_of_z4 d z4 cz)

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ iter_error d c z ≤ 0.32

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

rw [div_le_iff' (by positivity), ←div_le_iff (by norm_num)]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / (Complex.abs z * (Complex.abs z).log) ≤ 0.32

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

refine le_trans (by norm_num) (mul_le_mul z4 hl (by positivity) (by positivity))

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ Complex.abs z * (Complex.abs z).log

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

rw [Real.le_log_iff_exp_le (by positivity)]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 1.38 ≤ (Complex.abs z).log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ exp 1.38 ≤ Complex.abs z

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ exp 1.38 ≤ Complex.abs z

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (69 / 50).exp ≤ Complex.abs z

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

exact le_trans (exp_div_lt).le z4

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ (69 / 50).exp ≤ Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z ⊢ 0 < Complex.abs z

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

rw [Real.le_log_iff_exp_le (by positivity)]

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ 0.32 ≤ (Complex.abs z).log.log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ exp 0.32 ≤ (Complex.abs z).log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ exp 0.32 ≤ (Complex.abs z).log

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ (8 / 25).exp ≤ (Complex.abs z).log

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

exact le_trans (exp_div_lt).le hl

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ (8 / 25).exp ≤ (Complex.abs z).log

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log ⊢ 0 < (Complex.abs z).log

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

positivity

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 < Complex.abs z * (Complex.abs z).log

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0 < 0.32

no goals

https://github.com/girving/ray.git

0be790285dd0fce78913b0cb9bddaffa94bd25f9

Ray/Dynamics/Multibrot/Potential.lean

iter_error_le_log_log_abs

[245, 1]

[259, 83]

norm_num

c✝ z✝ : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z4 : 4 ≤ Complex.abs z cz : Complex.abs c ≤ Complex.abs z hl : 1.38 ≤ (Complex.abs z).log hll : 0.32 ≤ (Complex.abs z).log.log ⊢ 0.8095 / 0.32 ≤ 4 * 1.38

no goals