url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | simp only [s.bottcher_not_basin m, Complex.abs.map_one, s.potential_eq_one m'] | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z)... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | replace m := s.bottcher_ext m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) < 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, s.bottcher c z) ∈ s.ext
⊢ Complex.abs (s.bottcher c z) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | simp only [Super.ext, mem_setOf] at m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, s.bottcher c z) ∈ s.ext
⊢ Complex.abs (s.bottcher c z) < 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : Complex.abs (s.bottcher c z) < s.p c
⊢ Complex.abs (s.bottcher c z) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | exact lt_of_lt_of_le m s.p_le_one | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : Complex.abs (s.bottcher c z) < s.p c
⊢ Complex.abs (s.bottcher c z) < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.d0 | [68, 1] | [68, 73] | have h := s.d2 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
⊢ d ≠ 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
h : 2 ≤ d
⊢ d ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.d0 | [68, 1] | [68, 73] | omega | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
h : 2 ≤ d
⊢ d ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.dr2 | [73, 1] | [73, 101] | norm_num | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
⊢ 2 ≤ ↑2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | g0 | [82, 1] | [82, 92] | simp only [g, eq_self_iff_true, if_true] | f✝ : ℂ → ℂ
d✝ : ℕ
z : ℂ
t : Set ℂ
f : ℂ → ℂ
d : ℕ
⊢ g f d 0 = 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.approx | [85, 1] | [88, 10] | have a := s.fa0.leading_approx | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
⊢ (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
a : (fun z => f z - (z - 0) ^ orderAt f 0 • leadingCoeff f 0) =o[𝓝 0] fun z => (z - 0) ^ orderAt f 0
⊢ (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.approx | [85, 1] | [88, 10] | simp only [s.fd, s.fc, sub_zero, Pi.one_apply, Algebra.id.smul_eq_mul, mul_one] at a | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
a : (fun z => f z - (z - 0) ^ orderAt f 0 • leadingCoeff f 0) =o[𝓝 0] fun z => (z - 0) ^ orderAt f 0
⊢ (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
a : (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d
⊢ (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.approx | [85, 1] | [88, 10] | exact a | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
a : (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d
⊢ (fun z => f z - z ^ d) =o[𝓝 0] fun z => z ^ d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.f0 | [91, 1] | [93, 28] | simp [s.fd, s.dp] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
⊢ orderAt f 0 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | by_cases z0 : z = 0 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
⊢ f z = z ^ d * g f d z | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : z = 0
⊢ f z = z ^ d * g f d z
case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : ¬z = 0
⊢ f z = z ^ d * g f d z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | simp only [z0, zero_pow s.d0, s.f0, MulZeroClass.zero_mul] | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : z = 0
⊢ f z = z ^ d * g f d z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | simp only [g, z0, if_false] | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : ¬z = 0
⊢ f z = z ^ d * g f d z | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : ¬z = 0
⊢ f z = z ^ d * (f z / z ^ d) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.fg | [96, 1] | [99, 49] | field_simp [z0] | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
z : ℂ
z0 : ¬z = 0
⊢ f z = z ^ d * (f z / z ^ d) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rcases fa.exists_ball_analyticOn with ⟨r, rp, fa⟩ | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
c : ℂ
fa : AnalyticAt ℂ f c
⊢ AnalyticAt ℂ (g f d) c | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
⊢ AnalyticAt ℂ (g f d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have o : IsOpen (ball c r) := isOpen_ball | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
⊢ AnalyticAt ℂ (g f d) c | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
o : IsOpen (ball c r)
⊢ AnalyticAt ℂ (g f d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | generalize ht : ball c r = t | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
o : IsOpen (ball c r)
⊢ AnalyticAt ℂ (g f d) c | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
o : IsOpen (ball c r)
t : Set ℂ
ht : ball c r = t
⊢ AnalyticAt ℂ (g f d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [ht] at fa o | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
fa : AnalyticOn ℂ f (ball c r)
o : IsOpen (ball c r)
t : Set ℂ
ht : ball c r = t
⊢ AnalyticAt ℂ (g f d) c | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ AnalyticAt ℂ (g f d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | suffices h : AnalyticOn ℂ (g f d) t by rw [← ht] at h; exact h _ (mem_ball_self rp) | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ AnalyticAt ℂ (g f d) c | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ AnalyticOn ℂ (g f d) t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have ga : DifferentiableOn ℂ (g f d) (t \ {0}) := by
have e : ∀ z : ℂ, z ∈ t \ {0} → g f d z = f z / z ^ d := by
intro z zs; simp only [Set.mem_diff, Set.mem_singleton_iff] at zs
simp only [g, zs.2, if_false]
rw [differentiableOn_congr e]
apply DifferentiableOn.div (fa.mono (Set.diff_subset _ _)).differen... | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ AnalyticOn ℂ (g f d) t | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
⊢ AnalyticOn ℂ (g f d) t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [analyticOn_iff_differentiableOn o] | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
⊢ AnalyticOn ℂ (g f d) t | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
⊢ DifferentiableOn ℂ (g f d) t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | by_cases t0 : (0 : ℂ) ∉ t | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
⊢ DifferentiableOn ℂ (g f d) t | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∉ t
⊢ DifferentiableOn ℂ (g f d) t
case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Set.not_not_mem] at t0 | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : ¬0 ∉ t
⊢ DifferentiableOn ℂ (g f d) t | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
⊢ DifferentiableOn ℂ (g f d) t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact (Complex.differentiableOn_compl_singleton_and_continuousAt_iff (o.mem_nhds t0)).mp ⟨ga, gc⟩ | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
gc : ContinuousAt (g f d) 0
⊢ DifferentiableOn ℂ (g f d) t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [← ht] at h | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
h : AnalyticOn ℂ (g f d) t
⊢ AnalyticAt ℂ (g f d) c | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
h : AnalyticOn ℂ (g f d) (ball c r)
⊢ AnalyticAt ℂ (g f d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact h _ (mem_ball_self rp) | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
h : AnalyticOn ℂ (g f d) (ball c r)
⊢ AnalyticAt ℂ (g f d) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | have e : ∀ z : ℂ, z ∈ t \ {0} → g f d z = f z / z ^ d := by
intro z zs; simp only [Set.mem_diff, Set.mem_singleton_iff] at zs
simp only [g, zs.2, if_false] | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ DifferentiableOn ℂ (g f d) (t \ {0}) | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (g f d) (t \ {0}) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [differentiableOn_congr e] | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (g f d) (t \ {0}) | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (fun x => f x / x ^ d) (t \ {0}) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | apply DifferentiableOn.div (fa.mono (Set.diff_subset _ _)).differentiableOn | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (fun x => f x / x ^ d) (t \ {0}) | case hd
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (fun x => x ^ d) (t \ {0})
case hx
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact (Differentiable.pow differentiable_id _).differentiableOn | case hd
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ DifferentiableOn ℂ (fun x => x ^ d) (t \ {0})
case hx
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
f... | case hx
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ ∀ x ∈ t \ {0}, x ^ d ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | case hx
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
⊢ ∀ x ∈ t \ {0}, x ^ d ≠ 0 | case hx
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
z : ℂ
zs : z ∈ t \ {0}
⊢ z ^ d ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact pow_ne_zero _ (Set.mem_diff_singleton.mp zs).2 | case hx
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
e : ∀ z ∈ t \ {0}, g f d z = f z / z ^ d
z : ℂ
zs : z ∈ t \ {0}
⊢ z ^ d ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
⊢ ∀ z ∈ t \ {0}, g f d z = f z / z ^ d | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
z : ℂ
zs : z ∈ t \ {0}
⊢ g f d z = f z / z ^ d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Set.mem_diff, Set.mem_singleton_iff] at zs | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
z : ℂ
zs : z ∈ t \ {0}
⊢ g f d z = f z / z ^ d | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
z : ℂ
zs : z ∈ t ∧ ¬z = 0
⊢ g f d z = f z / z ^ d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [g, zs.2, if_false] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
z : ℂ
zs : z ∈ t ∧ ¬z = 0
⊢ g f d z = f z / z ^ d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Set.diff_singleton_eq_self t0] at ga | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∉ t
⊢ DifferentiableOn ℂ (g f d) t | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) t
t0 : 0 ∉ t
⊢ DifferentiableOn ℂ (g f d) t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact ga | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) t
t0 : 0 ∉ t
⊢ DifferentiableOn ℂ (g f d) t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Metric.continuousAt_iff] | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
⊢ ContinuousAt (g f d) 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, dist x 0 < δ → dist (g f d x) (g f d 0) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro e ep | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
⊢ ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, dist x 0 < δ → dist (g f d x) (g f d 0) < ε | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
e : ℝ
ep : e > 0
⊢ ∃ δ > 0, ∀ {x : ℂ}, dist x 0 < δ → dist (g f d x) (g f d 0) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rcases Metric.eventually_nhds_iff.mp
(Asymptotics.isBigOWith_iff.mp (s.approx.forall_isBigOWith (by linarith : e / 2 > 0))) with
⟨t, tp, h⟩ | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
e : ℝ
ep : e > 0
⊢ ∃ δ > 0, ∀ {x : ℂ}, dist x 0 < δ → dist (g f d x) (g f d 0) < e | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | use t, tp | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2... | case right
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2 * ‖y ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | intro z zs | case right
f : ℂ → ℂ
d : ℕ
z : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2 * ‖y ... | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2 * ‖y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | specialize h zs | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
h : ∀ ⦃y : ℂ⦄, dist y 0 < t → ‖f y - y ^ d‖ ≤ e / 2 * ‖y... | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : ‖f z - z ^ d‖ ≤ e / 2 * ‖z ^... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [Complex.norm_eq_abs] at h | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : ‖f z - z ^ d‖ ≤ e / 2 * ‖z ^... | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [g, Complex.dist_eq] | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ ... | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | by_cases z0 : z = 0 | case right
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ ... | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [z0, if_false, eq_self_iff_true, if_true] | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | calc
abs (f z / z ^ d - 1) = abs (f z * (z ^ d)⁻¹ - 1) := by rw [div_eq_mul_inv]
_ = abs ((f z - z ^ d) * (z ^ d)⁻¹) := by
rw [mul_sub_right_distrib, mul_inv_cancel (pow_ne_zero d z0)]
_ = abs (f z - z ^ d) * (abs (z ^ d))⁻¹ := by rw [Complex.abs.map_mul, map_inv₀]
_ ≤ e / 2 * abs (z ^ d) * (abs (z ^ d))⁻¹ ... | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | linarith | f : ℂ → ℂ
d : ℕ
z : ℂ
t✝ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t : Set ℂ
o : IsOpen t
fa : AnalyticOn ℂ f t
ht : ball c r = t
ga : DifferentiableOn ℂ (g f d) (t \ {0})
t0 : 0 ∈ t
e : ℝ
ep : e > 0
⊢ e / 2 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | simp only [z0, sub_self, AbsoluteValue.map_zero] | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | exact ep | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [div_eq_mul_inv] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [mul_sub_right_distrib, mul_inv_cancel (pow_ne_zero d z0)] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | rw [Complex.abs.map_mul, map_inv₀] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | bound | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | ring | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | bound | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.ga_of_fa | [102, 1] | [140, 100] | ring | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t✝¹ : Set ℂ
s : SuperAt f d
c : ℂ
fa✝ : AnalyticAt ℂ f c
r : ℝ
rp : 0 < r
t✝ : Set ℂ
o : IsOpen t✝
fa : AnalyticOn ℂ f t✝
ht : ball c r = t✝
ga : DifferentiableOn ℂ (g f d) (t✝ \ {0})
t0 : 0 ∈ t✝
e : ℝ
ep : e > 0
t : ℝ
tp : t > 0
z : ℂ
zs : dist z 0 < t
h : Complex.abs (f z - z ^ d) ≤ e / 2 * Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z z0 zs | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
⊢ ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
z0 : z ≠ 0
zs : z ∈ ball 0 r
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs, lt_min_iff] at zs | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
z0 : z ≠ 0
zs : z ∈ ball 0 r
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | specialize gs zs | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
gs : Complex.abs (g f d z - 1) < 1 / 4
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [g, z0, if_false, eq_self_iff_true, if_true] at gs | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
gs : Complex.abs (g f d z - 1) < 1 / 4
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
gs : Complex.abs (f z / z ^ d - 1) < 1 / 4
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | exact gs.le | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
z : ℂ
z0 : z ≠ 0
zs : Complex.abs z < r
gs : Complex.abs (f z / z ^ d - 1) < 1 / 4
⊢ Complex.abs (f z / z ^ d - 1) ≤ 1 / 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z zs | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
⊢ ∀ {z : ℂ}, z ∈ ball 0 r → Complex.abs z ≤ 1 / 2 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : z ∈ ball 0 r
⊢ Complex.abs z ≤ 1 / 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs] at zs | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : z ∈ ball 0 r
⊢ Complex.abs z ≤ 1 / 2 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : Complex.abs z < r
⊢ Complex.abs z ≤ 1 / 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | exact le_trans zs.le r2 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : Complex.abs z < r
⊢ Complex.abs z ≤ 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | intro z zs | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
⊢ MapsTo f (ball 0 r) (ball 0 r) | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : z ∈ ball 0 r
⊢ f z ∈ ball 0 r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs] at zs gs ⊢ | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
gs : ∀ {z : ℂ}, z ≠ 0 → z ∈ ball 0 r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z : ℂ
zs : z ∈ ball 0 r
⊢ f z ∈ ball 0 r | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
⊢ Complex.abs (f z) < r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | by_cases z0 : z = 0 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
⊢ Complex.abs (f z) < r | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : z = 0
⊢ Comp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | calc abs (f z)
_ = abs (f z / z ^ d * z ^ d) := by rw [div_mul_cancel₀ _ (pow_ne_zero d z0)]
_ = abs (f z / z ^ d - 1 + 1) * abs z ^ d := by
simp only [AbsoluteValue.map_mul, Complex.abs_pow, sub_add_cancel]
_ ≤ (abs (f z / z ^ d - 1) + abs (1 : ℂ)) * r ^ d := by bound
_ ≤ (1 / 4 + abs (1 : ℂ)) * r ^ d := b... | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [z0, s.f0, rp, AbsoluteValue.map_zero] | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : z = 0
⊢ Comp... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | rw [div_mul_cancel₀ _ (pow_ne_zero d z0)] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ Complex.abs ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | simp only [AbsoluteValue.map_mul, Complex.abs_pow, sub_add_cancel] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ Complex.abs ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ Complex.abs ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound [gs z0 zs] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ (Complex.abs... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | rw [mul_assoc, ← pow_succ, Nat.sub_add_cancel (le_trans one_le_two s.d2)] | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ (1 / 4 + Com... | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ (1 / 4 + Com... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | norm_num | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ (1 / 4 + Com... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ 5 / 4 * r ^ ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | bound | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ 5 / 4 * (1 /... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | norm_num | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ 5 / 4 * (1 /... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.super_on_ball | [148, 1] | [179, 31] | linarith | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperAt f d
r : ℝ
rp : 0 < r
r2 : r ≤ 1 / 2
fa : AnalyticOn ℂ f (ball 0 r)
gs✝ : ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g f d z - 1) < 1 / 4
z : ℂ
zs : Complex.abs z < r
gs : ∀ {z : ℂ}, z ≠ 0 → Complex.abs z < r → Complex.abs (f z / z ^ d - 1) ≤ 1 / 4
z0 : ¬z = 0
⊢ 5 / 8 * r < ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | rcases s.fa0.exists_ball_analyticOn with ⟨r0, r0p, fa⟩ | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
⊢ ∃ t, SuperNear f d t | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
⊢ ∃ t, SuperNear f d t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | rcases Metric.continuousAt_iff.mp (s.ga_of_fa (fa 0 (mem_ball_self r0p))).continuousAt (1 / 4)
(by norm_num) with
⟨r1, r1p, gs⟩ | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
⊢ ∃ t, SuperNear f d t | case intro.intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
⊢ ∃ t, SuperNear f d t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | set r := min r0 (min r1 (1 / 2)) | case intro.intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
⊢ ∃ t, SuperNear f d t | case intro.intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
⊢ ∃ t, SuperNear f d t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | use ball 0 r | case intro.intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
⊢ ∃ t, SuperNear f d t | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
⊢ SuperNear f d (ball 0 r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have rp : 0 < r := by bound | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
⊢ SuperNear f d (ball 0 r) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
⊢ SuperNear f d (ball 0 r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have r2 : r ≤ 1 / 2 := le_trans (min_le_right _ _) (min_le_right _ _) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
⊢ SuperNear f d (ball 0 r) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
⊢ SuperNear f d (ball 0 r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | have rr1 : r ≤ r1 := le_trans (min_le_right r0 _) (min_le_left r1 _) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
⊢ SuperNear f d (ball 0 r) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
rr1 : r ≤ r1
⊢ SuperNear f d (ball 0 r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | simp only [g0, dist_zero_right, Complex.norm_eq_abs, Complex.dist_eq, sub_zero] at gs | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
rr1 : r ≤ r1
⊢ SuperNear f d (ball 0 r) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
rr1 : r ≤ r1
gs : ∀ {x : ℂ}, Complex.abs x < r1 → Complex.abs (g f d x - 1) < 1 / 4
⊢ SuperNear f d (ball 0 r) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | exact s.super_on_ball rp r2 (fa.mono (Metric.ball_subset_ball (min_le_left r0 _))) (fun {z} zr ↦
gs (lt_of_lt_of_le zr rr1)) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
r : ℝ := min r0 (min r1 (1 / 2))
rp : 0 < r
r2 : r ≤ 1 / 2
rr1 : r ≤ r1
gs : ∀ {x : ℂ}, Complex.abs x < r1 → Complex.abs (g f d x - 1) < 1 / 4
⊢ SuperNear f d (ball 0 r) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | norm_num | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
⊢ 1 / 4 > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAt.superNear | [183, 1] | [195, 32] | bound | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperAt f d
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ f (ball 0 r0)
r1 : ℝ
r1p : r1 > 0
gs : ∀ {x : ℂ}, dist x 0 < r1 → dist (g f d x) (g f d 0) < 1 / 4
r : ℝ := min r0 (min r1 (1 / 2))
⊢ 0 < r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | by_cases z0 : z = 0 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
⊢ Complex.abs (g f d z - 1) ≤ 1 / 4 | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : z = 0
⊢ Complex.abs (g f d z - 1) ≤ 1 / 4
case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : ¬z = 0
⊢ Complex.abs (g f d z - 1) ≤ 1 / 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | simp only [z0, g0, sub_self, AbsoluteValue.map_zero, one_div, inv_nonneg, zero_le_bit0,
zero_le_one] | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : z = 0
⊢ Complex.abs (g f d z - 1) ≤ 1 / 4 | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : z = 0
⊢ 0 ≤ 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | norm_num | case pos
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : z = 0
⊢ 0 ≤ 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.gs | [198, 1] | [202, 45] | simp only [g, z0, if_false, s.gs' z0 zt] | case neg
f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
z0 : ¬z = 0
⊢ Complex.abs (g f d z - 1) ≤ 1 / 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | have h := s.gs zt | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
⊢ g f d z ≠ 0 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
h : Complex.abs (g f d z - 1) ≤ 1 / 4
⊢ g f d z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNear.g_ne_zero | [205, 1] | [206, 85] | contrapose h | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
h : Complex.abs (g f d z - 1) ≤ 1 / 4
⊢ g f d z ≠ 0 | f : ℂ → ℂ
d : ℕ
z✝ : ℂ
t : Set ℂ
s : SuperNear f d t
z : ℂ
zt : z ∈ t
h : ¬g f d z ≠ 0
⊢ ¬Complex.abs (g f d z - 1) ≤ 1 / 4 |
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