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 | 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/Hartogs/FubiniBall.lean | realCircleMap_eq_circleMap | [35, 1] | [39, 66] | simp only [realCircleMap, circleMap, Complex.equivRealProd_apply, Complex.add_re, Complex.mul_re,
Complex.ofReal_re, Complex.exp_ofReal_mul_I_re, Complex.ofReal_im, Complex.exp_ofReal_mul_I_im,
zero_mul, sub_zero, Complex.add_im, Complex.mul_im, add_zero] | c : ℂ
x : ℝ × ℝ
⊢ realCircleMap c x = Complex.equivRealProd (circleMap c x.1 x.2) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | realCircleMap.fderiv | [50, 1] | [54, 73] | simp_rw [realCircleMap] | c : ℂ
x : ℝ × ℝ
⊢ HasFDerivAt (fun x => realCircleMap c x) (rcmDeriv x) x | c : ℂ
x : ℝ × ℝ
⊢ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | realCircleMap.fderiv | [50, 1] | [54, 73] | apply_rules [hasFDerivAt_const, hasFDerivAt_fst, hasFDerivAt_snd, HasFDerivAt.cos,
HasFDerivAt.sin, HasFDerivAt.add, HasFDerivAt.mul, HasFDerivAt.prod] | c : ℂ
x : ℝ × ℝ
⊢ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm00 | [59, 1] | [60, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : ℝ × ℝ
⊢ rcmMatrix x 0 0 = x.2.cos | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm01 | [61, 1] | [62, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : ℝ × ℝ
⊢ rcmMatrix x 0 1 = -x.1 * x.2.sin | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm10 | [63, 1] | [64, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : ℝ × ℝ
⊢ rcmMatrix x 1 0 = x.2.sin | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm11 | [65, 1] | [66, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : ℝ × ℝ
⊢ rcmMatrix x 1 1 = x.1 * x.2.cos | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | rw [ContinuousLinearMap.det, ← LinearMap.det_toMatrix (Basis.finTwoProd ℝ), ←rcmMatrix] | x : ℝ × ℝ
⊢ (rcmDeriv x).det = x.1 | x : ℝ × ℝ
⊢ (rcmMatrix x).det = x.1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | rw [Matrix.det_fin_two, rcm00, rcm01, rcm10, rcm11] | x : ℝ × ℝ
⊢ (rcmMatrix x).det = x.1 | x : ℝ × ℝ
⊢ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | ring_nf | x : ℝ × ℝ
⊢ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1 | x : ℝ × ℝ
⊢ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | calc cos x.2 ^ 2 * x.1 + x.1 * sin x.2 ^ 2
_ = x.1 * (cos x.2 ^ 2 + sin x.2 ^ 2) := by ring
_ = x.1 := by simp only [Real.cos_sq_add_sin_sq, mul_one] | x : ℝ × ℝ
⊢ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | ring | x : ℝ × ℝ
⊢ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | simp only [Real.cos_sq_add_sin_sq, mul_one] | x : ℝ × ℝ
⊢ x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) = x.1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | constructor | S : Type
inst✝ : RCLike S
c : S
r : ℝ
⊢ sphere c r = ∅ ↔ r < 0 | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
⊢ sphere c r = ∅ → r < 0
case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
⊢ r < 0 → sphere c r = ∅ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | intro rp | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
⊢ sphere c r = ∅ → r < 0 | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : sphere c r = ∅
⊢ r < 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | contrapose rp | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : sphere c r = ∅
⊢ r < 0 | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : ¬r < 0
⊢ ¬sphere c r = ∅ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simp at rp | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : ¬r < 0
⊢ ¬sphere c r = ∅ | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : 0 ≤ r
⊢ ¬sphere c r = ∅ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | refine Nonempty.ne_empty ⟨c + r, ?_⟩ | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : 0 ≤ r
⊢ ¬sphere c r = ∅ | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : 0 ≤ r
⊢ c + ↑r ∈ sphere c r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simpa only [mem_sphere_iff_norm, add_sub_cancel_left, RCLike.norm_ofReal, abs_eq_self] | case mp
S : Type
inst✝ : RCLike S
c : S
r : ℝ
rp : 0 ≤ r
⊢ c + ↑r ∈ sphere c r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | intro n | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
⊢ r < 0 → sphere c r = ∅ | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : r < 0
⊢ sphere c r = ∅ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | contrapose n | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : r < 0
⊢ sphere c r = ∅ | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : ¬sphere c r = ∅
⊢ ¬r < 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | rw [← not_nonempty_iff_eq_empty] at n | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : ¬sphere c r = ∅
⊢ ¬r < 0 | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : ¬¬(sphere c r).Nonempty
⊢ ¬r < 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simpa only [not_lt, NormedSpace.sphere_nonempty, not_le] using n | case mpr
S : Type
inst✝ : RCLike S
c : S
r : ℝ
n : ¬¬(sphere c r).Nonempty
⊢ ¬r < 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | by_cases rp : r < 0 | c z : ℂ
r : ℝ
zs : z ∈ sphere c r
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | case pos
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : r < 0
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t
case neg
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : ¬r < 0
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [not_lt] at rp | case neg
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : ¬r < 0
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | case neg
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : 0 ≤ r
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [←abs_of_nonneg rp, ← range_circleMap, mem_range] at zs | case neg
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : 0 ≤ r
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | case neg
c z : ℂ
r : ℝ
zs : ∃ y, circleMap c r y = z
rp : 0 ≤ r
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rcases zs with ⟨t, ht⟩ | case neg
c z : ℂ
r : ℝ
zs : ∃ y, circleMap c r y = z
rp : 0 ≤ r
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | generalize ha : 2 * π = a | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | have ap : a > 0 := by rw [←ha]; bound | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | generalize hs : t + a - a * ⌈t / a⌉ = s | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | use s | case neg.intro
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ ∃ t ∈ Ioc 0 a, z = circleMap c r t | case h
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ s ∈ Ioc 0 a ∧ z = circleMap c r s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | constructor | case h
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ s ∈ Ioc 0 a ∧ z = circleMap c r s | case h.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ s ∈ Ioc 0 a
case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ z = circleMap c r s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [Metric.sphere_eq_empty.mpr rp, mem_empty_iff_false] at zs | case pos
c z : ℂ
r : ℝ
zs : z ∈ sphere c r
rp : r < 0
⊢ ∃ t ∈ Ioc 0 (2 * π), z = circleMap c r t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [←ha] | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
⊢ a > 0 | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
⊢ 2 * π > 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
⊢ 2 * π > 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [mem_Ioc, sub_pos, tsub_le_iff_right, ← hs] | case h.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ s ∈ Ioc 0 a | case h.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * ↑⌈t / a⌉ < t + a ∧ t + a ≤ a + a * ↑⌈t / a⌉ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | constructor | case h.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * ↑⌈t / a⌉ < t + a ∧ t + a ≤ a + a * ↑⌈t / a⌉ | case h.left.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * ↑⌈t / a⌉ < t + a
case h.left.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ t + a ≤ a +... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | calc a * ⌈t / a⌉
_ < a * (t / a + 1) := by bound
_ = a / a * t + a := by ring
_ = t + a := by field_simp [ap.ne'] | case h.left.left
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * ↑⌈t / a⌉ < t + a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * ↑⌈t / a⌉ < a * (t / a + 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | ring | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a * (t / a + 1) = a / a * t + a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | field_simp [ap.ne'] | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a / a * t + a = t + a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | calc a + a * ⌈t / a⌉
_ ≥ a + a * (t / a) := by bound
_ = a / a * t + a := by ring
_ = t + a := by field_simp [ap.ne'] | case h.left.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ t + a ≤ a + a * ↑⌈t / a⌉ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a + a * ↑⌈t / a⌉ ≥ a + a * (t / a) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | ring | c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ a + a * (t / a) = a / a * t + a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [←ht, circleMap, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_mul,
Complex.ofReal_intCast, add_right_inj, mul_eq_mul_left_iff, Complex.ofReal_eq_zero, ← hs] | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ z = circleMap c r s | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = ((↑t + ↑a - ↑a * ↑⌈t / a⌉) * I).exp ∨ r = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [mul_sub_right_distrib, right_distrib, Complex.exp_sub, Complex.exp_add] | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = ((↑t + ↑a - ↑a * ↑⌈t / a⌉) * I).exp ∨ r = 0 | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = (↑t * I).exp * (↑a * I).exp / (↑a * ↑⌈t / a⌉ * I).exp ∨ r = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [mul_comm _ (⌈_⌉:ℂ), mul_assoc, Complex.exp_int_mul, ← ha] | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = (↑t * I).exp * (↑a * I).exp / (↑a * ↑⌈t / a⌉ * I).exp ∨ r = 0 | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = (↑t * I).exp * (↑(2 * π) * I).exp / (↑(2 * π) * I).exp ^ ⌈t / (2 * π)⌉ ∨ r = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [Complex.ofReal_mul, Complex.ofReal_ofNat, Complex.exp_two_pi_mul_I, mul_one,
one_zpow, div_one, true_or] | case h.right
c z : ℂ
r : ℝ
rp : 0 ≤ r
t : ℝ
ht : circleMap c r t = z
a : ℝ
ha : 2 * π = a
ap : a > 0
s : ℝ
hs : t + a - a * ↑⌈t / a⌉ = s
⊢ (↑t * I).exp = (↑t * I).exp * (↑(2 * π) * I).exp / (↑(2 * π) * I).exp ^ ⌈t / (2 * π)⌉ ∨ r = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | simp only [square, gt_iff_lt, not_lt, ge_iff_le, zero_lt_two, mul_pos_iff_of_pos_left,
mem_prod, mem_Ioc, and_imp] | r0 r1 : ℝ
x : ℝ × ℝ
r0p : 0 ≤ r0
⊢ x ∈ square r0 r1 → 0 < x.1 | r0 r1 : ℝ
x : ℝ × ℝ
r0p : 0 ≤ r0
⊢ r0 < x.1 → x.1 ≤ r1 → 0 < x.2 → x.2 ≤ 2 * π → 0 < x.1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | intro h _ _ _ | r0 r1 : ℝ
x : ℝ × ℝ
r0p : 0 ≤ r0
⊢ r0 < x.1 → x.1 ≤ r1 → 0 < x.2 → x.2 ≤ 2 * π → 0 < x.1 | r0 r1 : ℝ
x : ℝ × ℝ
r0p : 0 ≤ r0
h : r0 < x.1
a✝² : x.1 ≤ r1
a✝¹ : 0 < x.2
a✝ : x.2 ≤ 2 * π
⊢ 0 < x.1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | linarith | r0 r1 : ℝ
x : ℝ × ℝ
r0p : 0 ≤ r0
h : r0 < x.1
a✝² : x.1 ≤ r1
a✝¹ : 0 < x.2
a✝ : x.2 ≤ 2 * π
⊢ 0 < x.1 | no goals |
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