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/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h _ i.mem) with ⟨_, _, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := f... | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact s | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | rcases s.superNearC' isOpen_univ fun _ _ ↦ Set.mem_univ _ with ⟨t, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
⊢ ∃ t, SuperNearC f d u t | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
t : Set (ℂ × ℂ)
left✝ : t ⊆ univ
s : SuperNearC f d u t
⊢ ∃ t, SuperNearC f d u t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | exact ⟨t, s⟩ | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
t : Set (ℂ × ℂ)
left✝ : t ⊆ univ
s : SuperNearC f d u t
⊢ ∃ t, SuperNearC f d u t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | induction' n with n nh | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[n] z) c | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[0] z) c
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.iterate_zero, id] | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[0] z) c | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact analyticAt_const | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => z) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp_rw [Function.iterate_succ'] | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c)^[n + 1] z) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c ∘ (f c)^[n]) z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.comp_apply] | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c ∘ (f c)^[n]) z) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => f c ((f c)^[n] z)) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | refine (s.fa _ ?_).comp ((analyticAt_id _ _).prod nh) | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => f c ((f c)^[n] z)) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ (id c, (f c)^[n] z) ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact (s.ts m).mapsTo n m | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ (id c, (f c)^[n] z) ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine AnalyticAt.cpow ?_ analyticAt_const ?_ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => term (f c) d n z) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c
case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ g (f c) d ((f c)^[n] z) ∈ Complex.slitPla... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | have e : (fun c ↦ g (f c) d ((f c)^[n] z)) = fun c ↦ g2 f d (c, (f c)^[n] z) := rfl | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | rw [e] | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g2 f d (c, (f c)^[n] z)) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine (s.ga _ ?_).comp ?_ | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g2 f d (c, (f c)^[n] z)) c | case refine_1.refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ (c, (f c)^[n] z) ∈ t
case refine_1.refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | exact (s.ts m).mapsTo n m | case refine_1.refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ (c, (f c)^[n] z) ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | apply (analyticAt_id _ _).prod (iterates_analytic_c s n m) | case refine_1.refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => (c, (f c)^[n] z)) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine mem_slitPlane_of_near_one ?_ | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ g (f c) d ((f c)^[n] z) ∈ Complex.slitPlane | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ Complex.abs (g (f c) d ((f c)^[n] z) - 1) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | exact lt_of_le_of_lt ((s.ts m).gs ((s.ts m).mapsTo n m)) (by norm_num) | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ Complex.abs (g (f c) d ((f c)^[n] z) - 1) < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ 1 / 4 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have c12 : (1 / 2 : ℝ) ≤ 1 / 2 := by norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have a0 : 0 ≤ (1 / 2 : ℝ) := by norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | set t' := {c | (c, z) ∈ t} | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have o' : IsOpen t' := s.o.preimage (by continuity) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
o' : IsOpen t'
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | refine (fast_products_converge' o' c12 a0 (by linarith) ?_
fun n c m ↦ term_converges (s.ts m) n m).2.1 _ m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
o' : IsOpen t'
⊢ AnalyticAt ℂ (fun c => ∏' (n : ℕ), term (f c) d n z) c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
o' : IsOpen t'
⊢ ∀ (n : ℕ), AnalyticOn ℂ (fun c => term (f (c, z).1) d n z) t' |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | exact fun n c m ↦ term_analytic_c s n m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
o' : IsOpen t'
⊢ ∀ (n : ℕ), AnalyticOn ℂ (fun c => term (f (c, z).1) d n z) t' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ 1 / 2 ≤ 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
⊢ 0 ≤ 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | continuity | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
⊢ Continuous fun c => (c, z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | linarith | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
c12 : 1 / 2 ≤ 1 / 2
a0 : 0 ≤ 1 / 2
t' : Set ℂ := {c | (c, z) ∈ t}
o' : IsOpen t'
⊢ 1 / 2 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | refine Pair.hartogs s.o ?_ ?_ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ AnalyticOn ℂ (fun p => ∏' (n : ℕ), term (f p.1) d n p.2) t | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f (z0, c1).1) d n (z0, c1).2) c0
case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | intro c z m | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f (z0, c1).1) d n (z0, c1).2) c0 | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f (z0, z).1) d n (z0, z).2) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | simp only | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f (z0, z).1) d n (z0, z).2) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f z0) d n z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | exact term_prod_analytic_c s m | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => ∏' (n : ℕ), term (f z0) d n z) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | intro c z m | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z1 => ∏' (n : ℕ), term (f (c0, z1).1) d n (c0, z1).2) c1 | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => ∏' (n : ℕ), term (f (c, z1).1) d n (c, z1).2) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | simp only | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => ∏' (n : ℕ), term (f (c, z1).1) d n (c, z1).2) z | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => ∏' (n : ℕ), term (f c) d n z1) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | exact term_prod_analytic_z (s.ts m) _ m | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => ∏' (n : ℕ), term (f c) d n z1) z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have df : ∀ e z, (e, z) ∈ t →
deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z := by
intro e z m; apply HasDerivAt.deriv
have fg : f e = fun z ↦ z ^ d * g (f e) d z := by funext; rw [(s.ts m).fg]
nth_rw 1 [fg]
apply HasDerivAt.mul; apply hasDerivAt_pow
rw [hasDerivAt_deriv_iff]... | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), deriv (f p.1) p.2 = 0 ↔ p.2 = 0 | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), deriv (f p.1) p.2 = 0 ↔ p.2 = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply small.mp | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢ ∀ᶠ (p :... | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢ ∀ᶠ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply (s.o.eventually_mem (s.s m).t0).mp | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢ ∀ᶠ (x :... | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢ ∀ᶠ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply Filter.eventually_of_forall | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢ ∀ᶠ (x :... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | clear small | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2)
⊢... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
⊢ ∀ x ∈ t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (↑d * g (f x.1) d x.2) → (deriv (f x.1) x.2 = ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | intro ⟨e, w⟩ m' small | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
⊢ ∀ x ∈ t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (↑d * g (f x.1) d x.2) → (deriv (f x.1) x.2 = ... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs ((e, w).2 * deriv (g (f (e, w).1) d) (e, w).2) < Complex.abs (↑d * g (f (e,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [df _ _ m'] at small ⊢ | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs ((e, w).2 * deriv (g (f (e, w).1) d) (e, w).2) < Complex.abs (↑d * g (f (e,... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ ↑d * w ^ (d - ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | nth_rw 4 [← Nat.sub_add_cancel (Nat.succ_le_of_lt (s.s m).dp)] | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ ↑d * w ^ (d - ... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ ↑d * w ^ (d - ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [pow_add, pow_one, mul_comm _ (w ^ (d - 1)), mul_assoc (w ^ (d - 1)) _ _, ←
left_distrib, mul_eq_zero, pow_eq_zero_iff (Nat.sub_pos_of_lt (s.s m).d2).ne'] | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ ↑d * w ^ (d - ... | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ w = 0 ∨ ↑d * g... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact or_iff_left (add_ne_zero_of_abs_lt small) | case hp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : ℂ
m' : (e, w) ∈ t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (↑d * g (f e) d w)
⊢ w = 0 ∨ ↑d * g... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | intro e z m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
⊢ ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
⊢ deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply HasDerivAt.deriv | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
⊢ deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
⊢ HasDerivAt (f e) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have fg : f e = fun z ↦ z ^ d * g (f e) d z := by funext; rw [(s.ts m).fg] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
⊢ HasDerivAt (f e) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (f e) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | nth_rw 1 [fg] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (f e) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun z => z ^ d * g (f e) d z) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply HasDerivAt.mul | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun z => z ^ d * g (f e) d z) (↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h.hc
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun y => y ^ d) (↑d * z ^ (d - 1)) z
case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply hasDerivAt_pow | case h.hc
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun y => y ^ d) (↑d * z ^ (d - 1)) z
case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (... | case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | rw [hasDerivAt_deriv_iff] | case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z | case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ DifferentiableAt ℂ (fun y => g (f e) d y) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact ((s.ts m).ga _ m).differentiableAt | case h.hd
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
fg : f e = fun z => z ^ d * g (f e) d z
⊢ DifferentiableAt ℂ (fun y => g (f e) d y) z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | funext | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
⊢ f e = fun z => z ^ d * g (f e) d z | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
x✝ : ℂ
⊢ f e x✝ = x✝ ^ d * g (f e) d x✝ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | rw [(s.ts m).fg] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m✝ : c ∈ u
e z : ℂ
m : (e, z) ∈ t
x✝ : ℂ
⊢ f e x✝ = x✝ ^ d * g (f e) d x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have ga : AnalyticAt ℂ (uncurry fun c z ↦ g (f c) d z) (c, 0) := s.ga _ (s.s m).t0 | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (↑d * g (f p.1) d p.2) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply ContinuousAt.eventually_lt | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ ∀ᶠ (p : ℂ × ℂ) in 𝓝 (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d)... | case hf
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ ContinuousAt (fun x => Complex.abs (x.2 * deriv (g (f x.1) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact Complex.continuous_abs.continuousAt.comp (continuousAt_snd.mul ga.deriv2.continuousAt) | case hf
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ ContinuousAt (fun x => Complex.abs (x.2 * deriv (g (f x.1) ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact Complex.continuous_abs.continuousAt.comp (continuousAt_const.mul ga.continuousAt) | case hg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ ContinuousAt (fun x => Complex.abs (↑d * g (f x.1) d x.2)) ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [g0, MulZeroClass.zero_mul, Complex.abs.map_zero, Complex.abs.map_mul,
Complex.abs_natCast, Complex.abs.map_one, mul_one, Nat.cast_pos] | case hfg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ Complex.abs ((c, 0).2 * deriv (g (f (c, 0).1) d) (c, 0).2)... | case hfg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ 0 < d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact (s.s m).dp | case hfg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
df : ∀ (e z : ℂ), (e, z) ∈ t → deriv (f e) z = ↑d * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt ℂ (uncurry fun c z => g (f c) d z) (c, 0)
⊢ 0 < d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_contra bad | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ InjOn (bottcher d) (multibrotExt d) | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
bad : ¬InjOn (bottcher d) (multibrotExt d)
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [InjOn, not_forall, ← ne_eq] at bad | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
bad : ¬InjOn (bottcher d) (multibrotExt d)
⊢ False | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
bad : ∃ x, ∃ (_ : x ∈ multibrotExt d), ∃ x_1, ∃ (_ : x_1 ∈ multibrotExt d) (_ : bottcher d x = bottcher d x_1), x ≠ x_1
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases bad with ⟨x, xm, y, ym, bxy, xy⟩ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
bad : ∃ x, ∃ (_ : x ∈ multibrotExt d), ∃ x_1, ∃ (_ : x_1 ∈ multibrotExt d) (_ : bottcher d x = bottcher d x_1), x ≠ x_1
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize hb : potential d x = b | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have b1 : b < 1 := by rwa [← hb, potential_lt_one] | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set u := {c | potential d c ≤ b} | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t0 := u ×ˢ u | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t1 := {q : 𝕊 × 𝕊 | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0} | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t2 := {q : 𝕊 × 𝕊 | q.1 ≠ q.2 ∧ q ∈ t1} | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear x xm y ym bxy xy hb | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ue : u ⊆ multibrotExt d := by intro c m; rw [← potential_lt_one]; exact lt_of_le_of_lt m b1 | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
⊢ False |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t01 : t1 ⊆ t0 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
⊢ False | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12 : t2 ⊆ t1 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have uc : IsClosed u := isClosed_le potential_continuous continuous_const | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t0c : IsClosed t0 := uc.prod uc | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12' : closure t2 ⊆ t1 := by rw [← t1c.closure_eq]; exact closure_mono t12 | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2c' : IsCompact (closure t2) := isClosed_closure.isCompact | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2ne' : (closure t2).Nonempty := t2ne.closure | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pc : Continuous fun q : 𝕊 × 𝕊 ↦ potential d q.1 := potential_continuous.comp continuous_fst | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases t2c'.exists_isMinOn t2ne' pc.continuousOn with ⟨⟨x, y⟩, m2, min⟩ | case intro.intro.intro.intro.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [isMinOn_iff] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize xp : potential d x = p | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [xp] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m1 := t12' m2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pb : p ≤ b := by rw [← xp]; exact m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have xm : x ∈ multibrotExt d := ue m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ym : y ∈ multibrotExt d := ue m1.2.2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have yp : potential d y = p := by rw [← abs_bottcher, ← m1.1, abs_bottcher, xp] | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have p0i : p = 0 → x = ∞ ∧ y = ∞ := by intro p0; rw [p0, potential_eq_zero] at xp yp; use xp, yp | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases xy : x ≠ y | case intro.intro.intro.intro.intro.intro.mk.intro
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrot... | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_not] at xy | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [← xy] at m1 m2 p0i | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear xy ym yp y | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have db : mfderiv I I (bottcher d) x = 0 := by
contrapose m2; simp only [mem_closure_iff_frequently, Filter.not_frequently]
refine ((bottcherHolomorphic d _ xm).local_inj m2).mp (eventually_of_forall ?_)
intro ⟨x, y⟩ inj ⟨xy, e, _⟩; simp only at xy e inj; exact xy (inj e) | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases p0 : p ≠ 0 | case neg
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... | case pos
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
b : ℝ
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 : Set (𝕊 × 𝕊) := {q | q.1 ≠ q.2 ∧ q ∈ t1}
t2ne : t2.Nonempty
ue : u ⊆ multibrotExt d
t01 : t1 ⊆ t0
t12 : t2 ⊆ t1
uc : Is... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rwa [← hb, potential_lt_one] | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
⊢ b < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine ⟨⟨x, y⟩, xy, bxy, ?_, ?_⟩ | c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ∧ q ∈ t0}
t2 :... | case refine_1
c : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
x : 𝕊
xm : x ∈ multibrotExt d
y : 𝕊
ym : y ∈ multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x ≠ y
b : ℝ
hb : potential d x = b
b1 : b < 1
u : Set 𝕊 := {c | potential d c ≤ b}
t0 : Set (𝕊 × 𝕊) := u ×ˢ u
t1 : Set (𝕊 × 𝕊) := {q | bottcher d q.1 = bottcher d q.2 ... |
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