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/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
refine holomorphicAt_snd.comp (i.he_symm_holomorphic.comp_of_eq ?_ ?_)
|
case gh
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => (↑i.he.symm (x.1, ↑(extChartAt I (f c z)) x.2)).2) (c, f c z)
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
case gh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) = (c, i.fz')
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
apply holomorphicAt_fst.prod
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun x => (x.1, ↑(extChartAt I (f c z)) x.2)) (c, f c z)
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
refine (HolomorphicAt.extChartAt ?_).comp holomorphicAt_snd
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ HolomorphicAt (I.prod I) I (fun x => ↑(extChartAt I (f c z)) x.2) (c, f c z)
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (c, f c z).2 ∈ (extChartAt I (f c z)).source
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
exact mem_extChartAt_source _ _
|
case gh.refine_1
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (c, f c z).2 ∈ (extChartAt I (f c z)).source
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
rfl
|
case gh.refine_2
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2) = (c, i.fz')
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
ComplexInverseFun.Cinv.ga
|
[296, 1]
|
[303, 19]
|
exact i.inv_at
|
case e
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
i : Cinv f c z
⊢ (↑i.he.symm ((c, f c z).1, ↑(extChartAt I (f c z)) (c, f c z).2)).2 = ↑(extChartAt I z) z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun
|
[311, 1]
|
[320, 41]
|
have i : ComplexInverseFun.Cinv f c z :=
{ fa
nc }
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
i : ComplexInverseFun.Cinv f c z
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun
|
[311, 1]
|
[320, 41]
|
use i.g, i.ga, i.left_inv, i.right_inv
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
i : ComplexInverseFun.Cinv f c z
⊢ ∃ g,
HolomorphicAt (I.prod I) I (uncurry g) (c, f c z) ∧
(∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2) ∧ ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun'
|
[324, 1]
|
[333, 60]
|
set f' : ℂ → S → T := fun _ z ↦ f z
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun'
|
[324, 1]
|
[333, 60]
|
have fa' : HolomorphicAt II I (uncurry f') (0, z) := fa.comp_of_eq holomorphicAt_snd rfl
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun'
|
[324, 1]
|
[333, 60]
|
rcases complex_inverse_fun fa' nc with ⟨g, ga, gf, fg⟩
|
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (0, f' 0 z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (0, z), g x.1 (f' x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (0, f' 0 z), f' x.1 (g x.1 x.2) = x.2
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/AnalyticManifold/Inverse.lean
|
complex_inverse_fun'
|
[324, 1]
|
[333, 60]
|
use g 0, ga.comp (holomorphicAt_const.prod holomorphicAt_id),
(continuousAt_const.prod continuousAt_id).eventually gf,
(continuousAt_const.prod continuousAt_id).eventually fg
|
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
f' : ℂ → S → T := fun x z => f z
fa' : HolomorphicAt (I.prod I) I (uncurry f') (0, z)
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (0, f' 0 z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (0, z), g x.1 (f' x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (0, f' 0 z), f' x.1 (g x.1 x.2) = x.2
⊢ ∃ g, HolomorphicAt I I g (f z) ∧ (∀ᶠ (x : S) in 𝓝 z, g (f x) = x) ∧ ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
set s := superF d
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (⋯.potential c ↑z))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
suffices h : Tendsto (fun n ↦ (abs ((f' d c)^[n] z) *
s.potential c ↑((f' d c)^[n] z)) ^ (-((d ^ n : ℕ) : ℝ)⁻¹))
atTop (𝓝 1) by
replace h := h.mul_const (s.potential c z)
simp only [div_mul_cancel₀ _ potential_pos.ne', one_mul, ← f_f'_iter, s.potential_eqn_iter,
Real.mul_rpow (Complex.abs.nonneg _) (pow_nonneg s.potential_nonneg _),
Real.pow_rpow_inv_natCast s.potential_nonneg (pow_ne_zero _ (d_ne_zero d)),
Real.rpow_neg (pow_nonneg s.potential_nonneg _), ← div_eq_mul_inv] at h
exact h
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [← s.abs_bottcher, ← Complex.abs.map_mul, mul_comm _ (s.bottcher _ _)]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [Metric.tendsto_atTop]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < ε
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
intro r rp
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < ε
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rcases Metric.tendsto_atTop.mp ((bottcher_large_approx d c).comp (tendsto_iter_atInf d z3 cz))
(min (1 / 2) (r / 4)) (by bound) with ⟨n, h⟩
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
|
case intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
use n
|
case intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∃ N, ∀ n ≥ N, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n] z) * (f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∀ n_1 ≥ n, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n_1] z) * (f' d c)^[n_1] z) ^ (-(↑(d ^ n_1))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
intro k nk
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
⊢ ∀ n_1 ≥ n, dist (Complex.abs (s.bottcher c ↑((f' d c)^[n_1] z) * (f' d c)^[n_1] z) ^ (-(↑(d ^ n_1))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
k : ℕ
nk : k ≥ n
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
specialize h k nk
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n : ℕ
h : ∀ n_1 ≥ n, dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) n_1) 1 < min (1 / 2) (r / 4)
k : ℕ
nk : k ≥ n
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
generalize hw : (f' d c)^[k] z = w
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
⊢ dist (Complex.abs (s.bottcher c ↑((f' d c)^[k] z) * (f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
⊢ dist (Complex.abs (s.bottcher c ↑w * w) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
generalize hp : s.bottcher c w * w = p
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
⊢ dist (Complex.abs (s.bottcher c ↑w * w) ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
⊢ dist (Complex.abs p ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [hw, hp, Function.comp, Complex.dist_eq, Real.dist_eq] at h ⊢
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
h : dist (((fun z => ⋯.bottcher c ↑z * z) ∘ fun n => (f' d c)^[n] z) k) 1 < min (1 / 2) (r / 4)
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
⊢ dist (Complex.abs p ^ (-(↑(d ^ k))⁻¹)) 1 < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
clear hp w hw nk n s cz z3
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
n k : ℕ
nk : k ≥ n
w : ℂ
hw : (f' d c)^[k] z = w
p : ℂ
hp : s.bottcher c ↑w * w = p
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
generalize ha : abs p = a
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
⊢ |Complex.abs p ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
⊢ |a ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
generalize hb : ((d ^ k : ℕ) : ℝ)⁻¹ = b
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
⊢ |a ^ (-(↑(d ^ k))⁻¹) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have a1 : |a - 1| < min (1 / 2) (r / 4) := by
rw [← ha]; refine lt_of_le_of_lt ?_ h
rw [← Complex.abs.map_one]; apply Complex.abs.abs_abv_sub_le_abv_sub
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have am : a ∈ ball (1 : ℝ) (1 / 2) := by
simp only [mem_ball, Real.dist_eq]; exact (lt_min_iff.mp a1).1
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have b0 : 0 ≤ b := by rw [← hb]; bound
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have b1 : b ≤ 1 := by rw [← hb]; bound
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have hd : ∀ x, x ∈ ball (1 : ℝ) (1 / 2) →
HasDerivAt (fun x ↦ x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * log x) x := by
intro x m; apply HasDerivAt.rpow (hasDerivAt_id _) (hasDerivAt_const _ _)
simp only [mem_ball, Real.dist_eq, id] at m ⊢; linarith [abs_lt.mp m]
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [MulZeroClass.zero_mul, add_zero, one_mul] at hd
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have bound : ∀ x, x ∈ ball (1 : ℝ) (1 / 2) → ‖deriv (fun x ↦ x ^ (-b)) x‖ ≤ 4 := by
intro x m
simp only [(hd x m).deriv, Real.norm_eq_abs, abs_mul, abs_neg, abs_of_nonneg b0]
simp only [mem_ball, Real.dist_eq, abs_lt, lt_sub_iff_add_lt, sub_lt_iff_lt_add] at m
norm_num at m
have x0 : 0 < x := by linarith
calc b * |x ^ (-b - 1)|
_ ≤ 1 * |x| ^ (-b - 1) := by bound
_ = (x ^ (b + 1))⁻¹ := by rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0]
_ ≤ ((1 / 2 : ℝ) ^ (b + 1))⁻¹ := by bound
_ = 2 ^ (b + 1) := by rw [one_div, Real.inv_rpow zero_le_two, inv_inv]
_ ≤ 2 ^ (1 + 1 : ℝ) := by bound
_ ≤ 4 := by norm_num
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have le := Convex.norm_image_sub_le_of_norm_deriv_le (fun x m ↦ (hd x m).differentiableAt) bound
(convex_ball _ _) (mem_ball_self (by norm_num)) am
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : ‖a ^ (-b) - 1 ^ (-b)‖ ≤ 4 * ‖a - 1‖
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [Real.norm_eq_abs, Real.one_rpow] at le
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : ‖a ^ (-b) - 1 ^ (-b)‖ ≤ 4 * ‖a - 1‖
⊢ |a ^ (-b) - 1| < r
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ |a ^ (-b) - 1| < r
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
calc |a ^ (-b) - 1|
_ ≤ 4 * |a - 1| := le
_ < 4 * (r / 4) := by linarith [(lt_min_iff.mp a1).2]
_ = r := by ring
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ |a ^ (-b) - 1| < r
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
replace h := h.mul_const (s.potential c z)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun n => (Complex.abs ((f' d c)^[n] z) * s.potential c ↑((f' d c)^[n] z)) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 1)
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h :
Tendsto
(fun k => (Complex.abs ((f' d c)^[k] z) * s.potential c ↑((f' d c)^[k] z)) ^ (-(↑(d ^ k))⁻¹) * s.potential c ↑z)
atTop (𝓝 (1 * s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [div_mul_cancel₀ _ potential_pos.ne', one_mul, ← f_f'_iter, s.potential_eqn_iter,
Real.mul_rpow (Complex.abs.nonneg _) (pow_nonneg s.potential_nonneg _),
Real.pow_rpow_inv_natCast s.potential_nonneg (pow_ne_zero _ (d_ne_zero d)),
Real.rpow_neg (pow_nonneg s.potential_nonneg _), ← div_eq_mul_inv] at h
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h :
Tendsto
(fun k => (Complex.abs ((f' d c)^[k] z) * s.potential c ↑((f' d c)^[k] z)) ^ (-(↑(d ^ k))⁻¹) * s.potential c ↑z)
atTop (𝓝 (1 * s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun k => Complex.abs ((f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) atTop (𝓝 (s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
exact h
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
h : Tendsto (fun k => Complex.abs ((f' d c)^[k] z) ^ (-(↑(d ^ k))⁻¹)) atTop (𝓝 (s.potential c ↑z))
⊢ Tendsto (fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (s.potential c ↑z))
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
r : ℝ
rp : r > 0
⊢ min (1 / 2) (r / 4) > 0
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [← ha]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |a - 1| < min (1 / 2) (r / 4)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| < min (1 / 2) (r / 4)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
refine lt_of_le_of_lt ?_ h
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| < min (1 / 2) (r / 4)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| ≤ Complex.abs (p - 1)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [← Complex.abs.map_one]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - 1| ≤ Complex.abs (p - 1)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - Complex.abs 1| ≤ Complex.abs (p - 1)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
apply Complex.abs.abs_abv_sub_le_abv_sub
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
⊢ |Complex.abs p - Complex.abs 1| ≤ Complex.abs (p - 1)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [mem_ball, Real.dist_eq]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ a ∈ ball 1 (1 / 2)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a - 1| < 1 / 2
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
exact (lt_min_iff.mp a1).1
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
⊢ |a - 1| < 1 / 2
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [← hb]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ b
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ (↑(d ^ k))⁻¹
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
⊢ 0 ≤ (↑(d ^ k))⁻¹
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [← hb]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ b ≤ 1
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ (↑(d ^ k))⁻¹ ≤ 1
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
⊢ (↑(d ^ k))⁻¹ ≤ 1
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
intro x m
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
⊢ ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
apply HasDerivAt.rpow (hasDerivAt_id _) (hasDerivAt_const _ _)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ HasDerivAt (fun x => x ^ (-b)) (1 * -b * x ^ (-b - 1) + 0 * x ^ (-b) * x.log) x
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ 0 < id x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [mem_ball, Real.dist_eq, id] at m ⊢
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ 0 < id x
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : |x - 1| < 1 / 2
⊢ 0 < x
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
linarith [abs_lt.mp m]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
x : ℝ
m : |x - 1| < 1 / 2
⊢ 0 < x
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
intro x m
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
⊢ ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [(hd x m).deriv, Real.norm_eq_abs, abs_mul, abs_neg, abs_of_nonneg b0]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ b * |x ^ (-b - 1)| ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
simp only [mem_ball, Real.dist_eq, abs_lt, lt_sub_iff_add_lt, sub_lt_iff_lt_add] at m
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : x ∈ ball 1 (1 / 2)
⊢ b * |x ^ (-b - 1)| ≤ 4
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : -(1 / 2) + 1 < x ∧ x < 1 / 2 + 1
⊢ b * |x ^ (-b - 1)| ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
norm_num at m
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : -(1 / 2) + 1 < x ∧ x < 1 / 2 + 1
⊢ b * |x ^ (-b - 1)| ≤ 4
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ b * |x ^ (-b - 1)| ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
have x0 : 0 < x := by linarith
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ b * |x ^ (-b - 1)| ≤ 4
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 4
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
calc b * |x ^ (-b - 1)|
_ ≤ 1 * |x| ^ (-b - 1) := by bound
_ = (x ^ (b + 1))⁻¹ := by rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0]
_ ≤ ((1 / 2 : ℝ) ^ (b + 1))⁻¹ := by bound
_ = 2 ^ (b + 1) := by rw [one_div, Real.inv_rpow zero_le_two, inv_inv]
_ ≤ 2 ^ (1 + 1 : ℝ) := by bound
_ ≤ 4 := by norm_num
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
linarith
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
⊢ 0 < x
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ b * |x ^ (-b - 1)| ≤ 1 * |x| ^ (-b - 1)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [← Real.rpow_neg x0.le, neg_add', one_mul, abs_of_pos x0]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 1 * |x| ^ (-b - 1) = (x ^ (b + 1))⁻¹
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ (x ^ (b + 1))⁻¹ ≤ ((1 / 2) ^ (b + 1))⁻¹
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
rw [one_div, Real.inv_rpow zero_le_two, inv_inv]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ ((1 / 2) ^ (b + 1))⁻¹ = 2 ^ (b + 1)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
bound
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (b + 1) ≤ 2 ^ (1 + 1)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
norm_num
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
x : ℝ
m : 1 / 2 < x ∧ x < 3 / 2
x0 : 0 < x
⊢ 2 ^ (1 + 1) ≤ 4
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
norm_num
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
⊢ 0 < 1 / 2
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
linarith [(lt_min_iff.mp a1).2]
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * |a - 1| < 4 * (r / 4)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_potential
|
[31, 1]
|
[86, 21]
|
ring
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
r : ℝ
rp : r > 0
k : ℕ
p : ℂ
h : Complex.abs (p - 1) < min (1 / 2) (r / 4)
a : ℝ
ha : Complex.abs p = a
b : ℝ
hb : (↑(d ^ k))⁻¹ = b
a1 : |a - 1| < min (1 / 2) (r / 4)
am : a ∈ ball 1 (1 / 2)
b0 : 0 ≤ b
b1 : b ≤ 1
hd : ∀ x ∈ ball 1 (1 / 2), HasDerivAt (fun x => x ^ (-b)) (-b * x ^ (-b - 1)) x
bound : ∀ x ∈ ball 1 (1 / 2), ‖deriv (fun x => x ^ (-b)) x‖ ≤ 4
le : |a ^ (-b) - 1| ≤ 4 * |a - 1|
⊢ 4 * (r / 4) = r
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
set s := superF d
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(⋯.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have zn1 : ∀ {n}, 1 < abs ((f' d c)^[n] z) := by
intro n; exact lt_of_lt_of_le (by norm_num) (le_trans z3 (le_self_iter d z3 cz _))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have zn0 : ∀ {n}, 0 < abs ((f' d c)^[n] z) := fun {_} ↦ lt_trans zero_lt_one zn1
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have ln0 : ∀ {n}, 0 < log (abs ((f' d c)^[n] z)) := fun {_} ↦ Real.log_pos zn1
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have dn0 : ∀ {n}, (d:ℝ)^n ≠ 0 := fun {_} ↦ pow_ne_zero _ (Nat.cast_ne_zero.mpr (d_ne_zero d))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have p0 : 0 < s.potential c z := potential_pos
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have p1 : s.potential c z < 1 := potential_lt_one_of_two_lt (by linarith) cz
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
set f := fun x ↦ log (log x⁻¹)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have fc : ContinuousAt f ((superF d).potential c z) := by
refine ((NormedField.continuousAt_inv.mpr p0.ne').log (inv_ne_zero p0.ne')).log ?_
exact Real.log_ne_zero_of_pos_of_ne_one (inv_pos.mpr p0) (inv_ne_one.mpr p1.ne)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
have t := Tendsto.comp fc (tendsto_potential d z3 cz)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
t : Tendsto (f ∘ fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (f (⋯.potential c ↑z)))
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
simpa only [Real.log_inv, Real.log_neg_eq_log, Nat.cast_pow, Function.comp_def, Real.log_rpow zn0,
neg_mul, ← div_eq_inv_mul, Real.log_div ln0.ne' dn0, Real.log_pow, f] using t
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
fc : ContinuousAt f (⋯.potential c ↑z)
t : Tendsto (f ∘ fun n => Complex.abs ((f' d c)^[n] z) ^ (-(↑(d ^ n))⁻¹)) atTop (𝓝 (f (⋯.potential c ↑z)))
⊢ Tendsto (fun n => (Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) atTop (𝓝 (-(s.potential c ↑z).log).log)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
intro n
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
⊢ ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < Complex.abs ((f' d c)^[n] z)
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
exact lt_of_lt_of_le (by norm_num) (le_trans z3 (le_self_iter d z3 cz _))
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < Complex.abs ((f' d c)^[n] z)
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
norm_num
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
n : ℕ
⊢ 1 < 3
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
linarith
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
⊢ 2 < Complex.abs z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
refine ((NormedField.continuousAt_inv.mpr p0.ne').log (inv_ne_zero p0.ne')).log ?_
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ ContinuousAt f (⋯.potential c ↑z)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ (s.potential c ↑z)⁻¹.log ≠ 0
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
tendsto_log_neg_log_potential
|
[88, 1]
|
[106, 82]
|
exact Real.log_ne_zero_of_pos_of_ne_one (inv_pos.mpr p0) (inv_ne_one.mpr p1.ne)
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
s : Super (_root_.f d) d OnePoint.infty := superF d
zn1 : ∀ {n : ℕ}, 1 < Complex.abs ((f' d c)^[n] z)
zn0 : ∀ {n : ℕ}, 0 < Complex.abs ((f' d c)^[n] z)
ln0 : ∀ {n : ℕ}, 0 < (Complex.abs ((f' d c)^[n] z)).log
dn0 : ∀ {n : ℕ}, ↑d ^ n ≠ 0
p0 : 0 < s.potential c ↑z
p1 : s.potential c ↑z < 1
f : ℝ → ℝ := fun x => x⁻¹.log.log
⊢ (s.potential c ↑z)⁻¹.log ≠ 0
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
apply le_of_forall_pos_lt_add
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| ≤ iter_error d c z
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (ε : ℝ), 0 < ε → |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + ε
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
intro e ep
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ ∀ (ε : ℝ), 0 < ε → |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + ε
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
rcases (Metric.tendsto_nhds.mp (tendsto_log_neg_log_potential d z3 cz) e ep).exists with ⟨n,t⟩
|
case h
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
have ie := iter_approx d z3 cz n
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
generalize log (-log ((superF d).potential c z)) = p at ie t
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) (-(⋯.potential c ↑z).log).log < e
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |(-(⋯.potential c ↑z).log).log - (Complex.abs z).log.log| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
p : ℝ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
generalize log (log (Complex.abs ((f' d c)^[n] z))) = x at ie t
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
ie : |(Complex.abs ((f' d c)^[n] z)).log.log - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
p : ℝ
t : dist ((Complex.abs ((f' d c)^[n] z)).log.log - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
ie : |x - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
t : dist (x - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
generalize log (log (Complex.abs z)) = y at ie t
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
ie : |x - (Complex.abs z).log.log - ↑n * (↑d).log| ≤ iter_error d c z
t : dist (x - ↑n * (↑d).log) p < e
⊢ |p - (Complex.abs z).log.log| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : dist (x - ↑n * (↑d).log) p < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
rw [Real.dist_eq, abs_sub_comm] at t
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : dist (x - ↑n * (↑d).log) p < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
rw [add_comm]
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < iter_error d c z + e
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < e + iter_error d c z
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
calc |p - y|
_ = |(p - (x - n * log d)) + (x - y - n * log d)| := by ring_nf
_ ≤ |p - (x - n * log d)| + |x - y - n * log d| := abs_add _ _
_ < e + _ := add_lt_add_of_lt_of_le t ie
|
case h.intro
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| < e + iter_error d c z
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
log_neg_log_potential_approx
|
[108, 1]
|
[122, 47]
|
ring_nf
|
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
e : ℝ
ep : 0 < e
n : ℕ
p x : ℝ
t : |p - (x - ↑n * (↑d).log)| < e
y : ℝ
ie : |x - y - ↑n * (↑d).log| ≤ iter_error d c z
⊢ |p - y| = |p - (x - ↑n * (↑d).log) + (x - y - ↑n * (↑d).log)|
|
no goals
|
https://github.com/girving/ray.git
|
0be790285dd0fce78913b0cb9bddaffa94bd25f9
|
Ray/Dynamics/Multibrot/Potential.lean
|
hasDerivAt_ene
|
[136, 1]
|
[140, 68]
|
have h : HasDerivAt (fun x ↦ exp (-exp x)) (exp (-exp x) * -exp x) x :=
HasDerivAt.exp (Real.hasDerivAt_exp x).neg
|
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
⊢ HasDerivAt ene (-dene x) x
|
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
|
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