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 | input stringlengths 73 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... | 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))... | Please generate a tactic in lean4 to solve the state.
STATE:
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 (f... |
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))... | 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) | Please generate a tactic in lean4 to solve the state.
STATE:
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.pro... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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.pro... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 ∈ (extC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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, ↑(ext... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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, ↑(e... |
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)... | 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
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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
⊢ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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 (... | Please generate a tactic in lean4 to solve the state.
STATE:
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, Holom... |
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 (... | 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 (uncurr... | Please generate a tactic in lean4 to solve the state.
STATE:
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 →... |
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 (uncurr... | 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' : Holomorphic... | Please generate a tactic in lean4 to solve the state.
STATE:
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 →... |
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' : Holomorphic... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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)) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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))
TACTIC:
|
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 _) (po... | 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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)) ^ (-(↑... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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, ∀ ... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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, ∀ ... | 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, ... | Please generate a tactic in lean4 to solve the state.
STATE:
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) ∘ fu... |
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, ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 ... | 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 (Compl... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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 (Compl... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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 ... |
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... | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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.bottc... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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), HasDeriv... | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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), HasDeriv... | 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), HasDeriv... | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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
ca... | 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), HasDeriv... | 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), HasDeriv... | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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), HasDeriv... | 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), HasDeriv... | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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), HasDeriv... | 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), HasDeriv... | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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), HasDeriv... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 /... |
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 (... | 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.... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)) ^ (-... |
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.... | 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) ^ (... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)... |
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) ^ (... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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.potentia... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 ... | 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 ... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 ... | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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_... | 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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 ... |
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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.... |
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... | 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)).l... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)).l... | 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)).... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)).... | 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)).... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)).... | 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)).... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)).... | 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)).... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)).... | 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)^[... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)^[... | 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)^[... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 (... |
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)^[... | 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)^[... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 (... |
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)^[... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 (... |
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) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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
TACTIC:
|
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)).... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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... |
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)^[... | 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)^[... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 (... |
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)^[... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 (... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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| <... | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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| <... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
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... | 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... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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... | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
⊢ |... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
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 - ... |
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : ℝ
⊢ HasDerivAt ene (-dene x) x
TACTIC:
|
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