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 | iterates_at_zero | [393, 1] | [395, 67] | induction' n with n h | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
⊢ f^[n] 0 = 0 | case zero
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
⊢ f^[0] 0 = 0
case succ
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
h : f^[n] 0 = 0
⊢ f^[n + 1] 0 = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_at_zero | [393, 1] | [395, 67] | simp only [Function.iterate_zero, id] | case zero
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
⊢ f^[0] 0 = 0
case succ
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
h : f^[n] 0 = 0
⊢ f^[n + 1] 0 = 0 | case succ
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
h : f^[n] 0 = 0
⊢ f^[n + 1] 0 = 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_at_zero | [393, 1] | [395, 67] | simp only [Function.iterate_succ', Function.comp_apply, h, s.f0] | case succ
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
h : f^[n] 0 = 0
⊢ f^[n + 1] 0 = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_at_zero | [398, 1] | [399, 61] | simp only [term, iterates_at_zero s, g0, Complex.one_cpow] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
n : ℕ
⊢ term f d n 0 = 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_at_zero | [402, 1] | [403, 47] | simp_rw [tprodOn, term_at_zero s, tprod_one] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
⊢ tprodOn (term f d) 0 = 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_monic | [406, 1] | [410, 43] | have dz : HasDerivAt (fun z : ℂ ↦ z) 1 0 := hasDerivAt_id 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
⊢ HasDerivAt (bottcherNear f d) 1 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
⊢ HasDerivAt (bottcherNear f d) 1 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_monic | [406, 1] | [410, 43] | have db := HasDerivAt.mul dz (term_prod_analytic_z s 0 s.t0).differentiableAt.hasDerivAt | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
⊢ HasDerivAt (bottcherNear f d) 1 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) (1 * tprodOn (term f d) 0 + 0 * deriv (tprodOn (term f d)) 0) 0
⊢ HasDerivAt (bottcherNear f d) 1 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_monic | [406, 1] | [410, 43] | simp only [one_mul, MulZeroClass.zero_mul, add_zero] at db | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) (1 * tprodOn (term f d) 0 + 0 * deriv (tprodOn (term f d)) 0) 0
⊢ HasDerivAt (bottcherNear f d) 1 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) (tprodOn (term f d) 0) 0
⊢ HasDerivAt (bottcherNear f d) 1 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_monic | [406, 1] | [410, 43] | rw [term_prod_at_zero s] at db | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) (tprodOn (term f d) 0) 0
⊢ HasDerivAt (bottcherNear f d) 1 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) 1 0
⊢ HasDerivAt (bottcherNear f d) 1 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_monic | [406, 1] | [410, 43] | exact db | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
dz : HasDerivAt (fun z => z) 1 0
db : HasDerivAt (fun y => y * tprodOn (term f d) y) 1 0
⊢ HasDerivAt (bottcherNear f d) 1 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_zero | [413, 1] | [414, 50] | simp only [bottcherNear, MulZeroClass.zero_mul] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
⊢ bottcherNear f d 0 = 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | by_cases z0 : z = 0 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0) | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | simp only [z0, iterates_at_zero s, tendsto_const_nhds] | case pos
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0)
case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0) | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | rw [Metric.tendsto_atTop] | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ Tendsto (fun n => f^[n] z) atTop (𝓝 0) | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (f^[n] z) 0 < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | intro e ep | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (f^[n] z) 0 < ε | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, dist (f^[n] z) 0 < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | simp only [Complex.dist_eq, sub_zero] | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, dist (f^[n] z) 0 < e | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | have xp : e / abs z > 0 := div_pos ep (Complex.abs.pos z0) | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | rcases exists_pow_lt_of_lt_one xp (by norm_num : (5 / 8 : ℝ) < 1) with ⟨N, Nb⟩ | case neg
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e | case neg.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N < e / Complex.abs z
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | simp only [lt_div_iff (Complex.abs.pos z0)] at Nb | case neg.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N < e / Complex.abs z
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e | case neg.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | use N | case neg.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
⊢ ∃ N, ∀ n ≥ N, Complex.abs (f^[n] z) < e | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
⊢ ∀ n ≥ N, Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | intro n nN | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
⊢ ∀ n ≥ N, Complex.abs (f^[n] z) < e | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
n : ℕ
nN : n ≥ N
⊢ Complex.abs (f^[n] z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | refine lt_of_le_of_lt (iterates_converge s n zt) (lt_of_le_of_lt ?_ Nb) | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
n : ℕ
nN : n ≥ N
⊢ Complex.abs (f^[n] z) < e | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
n : ℕ
nN : n ≥ N
⊢ (5 / 8) ^ n * Complex.abs z ≤ (5 / 8) ^ N * Complex.abs z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | bound | case h
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
N : ℕ
Nb : (5 / 8) ^ N * Complex.abs z < e
n : ℕ
nN : n ≥ N
⊢ (5 / 8) ^ n * Complex.abs z ≤ (5 / 8) ^ N * Complex.abs z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_tendsto | [425, 1] | [435, 8] | norm_num | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
z0 : ¬z = 0
e : ℝ
ep : e > 0
xp : e / Complex.abs z > 0
⊢ 5 / 8 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | rcases Metric.continuousAt_iff.mp (bottcherNear_analytic_z s _ s.t0).continuousAt 1 zero_lt_one
with ⟨r, rp, rs⟩ | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (bottcherNear f d z) < 1 | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, dist x 0 < r → dist (bottcherNear f d x) (bottcherNear f d 0) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | simp only [Complex.dist_eq, sub_zero, bottcherNear_zero] at rs | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, dist x 0 < r → dist (bottcherNear f d x) (bottcherNear f d 0) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | have b' : ∀ᶠ n in atTop, abs (bottcherNear f d (f^[n] z)) < 1 := by
refine (Metric.tendsto_nhds.mp (iterates_tendsto s zt) r rp).mp
(Filter.eventually_of_forall fun n h ↦ ?_)
rw [Complex.dist_eq, sub_zero] at h; exact rs h | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | rcases b'.exists with ⟨n, b⟩ | case intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
⊢ Complex.abs (bottcherNear f d z) < 1 | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : Complex.abs (bottcherNear f d (f^[n] z)) < 1
⊢ Complex.abs (bot... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | contrapose b | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : Complex.abs (bottcherNear f d (f^[n] z)) < 1
⊢ Complex.abs (bot... | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : ¬Complex.abs (bottcherNear f d z) < 1
⊢ ¬Complex.abs (bottcherN... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | simp only [not_lt] at b ⊢ | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : ¬Complex.abs (bottcherNear f d z) < 1
⊢ ¬Complex.abs (bottcherN... | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : 1 ≤ Complex.abs (bottcherNear f d z)
⊢ 1 ≤ Complex.abs (bottche... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | simp only [bottcherNear_eqn_iter s zt n, Complex.abs.map_pow, one_le_pow_of_one_le b] | case intro.intro.intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
b' : ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1
n : ℕ
b : 1 ≤ Complex.abs (bottcherNear f d z)
⊢ 1 ≤ Complex.abs (bottche... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | refine (Metric.tendsto_nhds.mp (iterates_tendsto s zt) r rp).mp
(Filter.eventually_of_forall fun n h ↦ ?_) | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
⊢ ∀ᶠ (n : ℕ) in atTop, Complex.abs (bottcherNear f d (f^[n] z)) < 1 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
n : ℕ
h : dist (f^[n] z) 0 < r
⊢ Complex.abs (bottcherNear f d (f^[n] z)) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | rw [Complex.dist_eq, sub_zero] at h | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
n : ℕ
h : dist (f^[n] z) 0 < r
⊢ Complex.abs (bottcherNear f d (f^[n] z)) < 1 | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
n : ℕ
h : Complex.abs (f^[n] z) < r
⊢ Complex.abs (bottcherNear f d (f^[n] z)) < 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_lt_one | [438, 1] | [448, 88] | exact rs h | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
r : ℝ
rp : r > 0
rs : ∀ {x : ℂ}, Complex.abs x < r → Complex.abs (bottcherNear f d x) < 1
n : ℕ
h : Complex.abs (f^[n] z) < r
⊢ Complex.abs (bottcherNear f d (f^[n] z)) < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | simp only [bottcherNear, Complex.abs.map_mul] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (bottcherNear f d z) ≤ 3 * Complex.abs z | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs z * Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 * Complex.abs z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | rw [mul_comm] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs z * Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 * Complex.abs z | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (∏' (n : ℕ), term f d n z) * Complex.abs z ≤ 3 * Complex.abs z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | refine mul_le_mul_of_nonneg_right ?_ (Complex.abs.nonneg _) | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (∏' (n : ℕ), term f d n z) * Complex.abs z ≤ 3 * Complex.abs z | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | rcases term_prod_exists s _ zt with ⟨p, h⟩ | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
⊢ Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : HasProd (fun n => term f d n z) p
⊢ Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | rw [h.tprod_eq] | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : HasProd (fun n => term f d n z) p
⊢ Complex.abs (∏' (n : ℕ), term f d n z) ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : HasProd (fun n => term f d n z) p
⊢ Complex.abs p ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | simp only [HasProd] at h | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : HasProd (fun n => term f d n z) p
⊢ Complex.abs p ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
⊢ Complex.abs p ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | apply le_of_tendsto' (Filter.Tendsto.comp Complex.continuous_abs.continuousAt h) | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
⊢ Complex.abs p ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
⊢ ∀ (c : Finset ℕ), (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) c ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | intro A | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
⊢ ∀ (c : Finset ℕ), (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) c ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | clear h | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
h : Tendsto (fun s => s.prod fun x => term f d x z) atTop (𝓝 p)
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | simp only [Function.comp, Complex.abs.map_prod] | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
⊢ (⇑Complex.abs ∘ fun s => s.prod fun x => term f d x z) A ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | have tb : ∀ n, abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by
intro n
calc abs (term f d n z)
_ = abs (1 + (term f d n z - 1)) := by ring_nf
_ ≤ Complex.abs 1 + abs (term f d n z - 1) := by bound
_ = 1 + abs (term f d n z - 1) := by norm_num
_ ≤ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by bound [term_... | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | have p : ∀ n : ℕ, 0 < (1 : ℝ) + 1 / 2 * (1 / 2 : ℝ) ^ n := fun _ ↦ add_pos (by bound) (by bound) | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | have lb : ∀ n : ℕ, Real.log ((1 : ℝ) + 1 / 2 * (1 / 2 : ℝ) ^ n) ≤ 1 / 2 * (1 / 2 : ℝ) ^ n :=
fun n ↦ le_trans (Real.log_le_sub_one_of_pos (p n)) (le_of_eq (by ring)) | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z)) ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z))... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | refine le_trans (Finset.prod_le_prod (fun _ _ ↦ Complex.abs.nonneg _) fun n _ ↦ tb n) ?_ | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => Complex.abs (term f d i z))... | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => 1 + 1 / 2 * (1 / 2) ^ i) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | rw [← Real.exp_log (Finset.prod_pos fun n _ ↦ p n), Real.log_prod _ _ fun n _ ↦ (p n).ne'] | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.prod fun i => 1 + 1 / 2 * (1 / 2) ^ i) ≤ ... | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => (1 + 1 / 2 * (1 / 2) ^ i).lo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | refine le_trans (Real.exp_le_exp.mpr (Finset.sum_le_sum fun n _ ↦ lb n)) ?_ | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => (1 + 1 / 2 * (1 / 2) ^ i).lo... | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => 1 / 2 * (1 / 2) ^ i).exp ≤ 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | refine le_trans (Real.exp_le_exp.mpr ?_) Real.exp_one_lt_3.le | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => 1 / 2 * (1 / 2) ^ i).exp ≤ 3 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≤ 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | have geom := partial_scaled_geometric_bound (1 / 2) A one_half_pos.le one_half_lt_one | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
⊢ (A.sum fun i => 1 / 2 * (1 / 2) ^ i) ≤ 1 | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
geom : (A.sum fun n => ↑(1 / 2) * (1 / 2) ^ n)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | simp only [NNReal.coe_div, NNReal.coe_one, NNReal.coe_two] at geom | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
geom : (A.sum fun n => ↑(1 / 2) * (1 / 2) ^ n)... | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≤ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | exact le_trans geom (by norm_num) | case intro
f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≤ ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | intro n | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
⊢ ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | calc abs (term f d n z)
_ = abs (1 + (term f d n z - 1)) := by ring_nf
_ ≤ Complex.abs 1 + abs (term f d n z - 1) := by bound
_ = 1 + abs (term f d n z - 1) := by norm_num
_ ≤ 1 + 1 / 2 * (1 / 2 : ℝ) ^ n := by bound [term_converges s n zt] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | ring_nf | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ Complex.abs (term f d n z) = Complex.abs (1 + (term f d n z - 1)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | bound | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ Complex.abs (1 + (term f d n z - 1)) ≤ Complex.abs 1 + Complex.abs (term f d n z - 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | norm_num | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ Complex.abs 1 + Complex.abs (term f d n z - 1) = 1 + Complex.abs (term f d n z - 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | bound [term_converges s n zt] | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
n : ℕ
⊢ 1 + Complex.abs (term f d n z - 1) ≤ 1 + 1 / 2 * (1 / 2) ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | bound | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
x✝ : ℕ
⊢ 0 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | bound | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
x✝ : ℕ
⊢ 0 < 1 / 2 * (1 / 2) ^ x✝ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | ring | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
n : ℕ
⊢ 1 + 1 / 2 * (1 / 2) ^ n - 1 = 1 / 2 * (1 / 2) ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | bottcherNear_le | [451, 1] | [474, 36] | norm_num | f : ℂ → ℂ
d : ℕ
z : ℂ
t : Set ℂ
s : SuperNear f d t
zt : z ∈ t
p✝ : ℂ
A : Finset ℕ
tb : ∀ (n : ℕ), Complex.abs (term f d n z) ≤ 1 + 1 / 2 * (1 / 2) ^ n
p : ∀ (n : ℕ), 0 < 1 + 1 / 2 * (1 / 2) ^ n
lb : ∀ (n : ℕ), (1 + 1 / 2 * (1 / 2) ^ n).log ≤ 1 / 2 * (1 / 2) ^ n
geom : (A.sum fun x => 1 / 2 * (1 / 2) ^ x) ≤ 1 / 2 * (1 ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | have e : u = Prod.fst '' t := by
ext c; simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right]
exact ⟨fun m ↦ ⟨0, (s.s m).t0⟩, fun h ↦ Exists.elim h fun z m ↦ s.tc m⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ IsOpen u | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
e : u = Prod.fst '' t
⊢ IsOpen u |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | rw [e] | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
e : u = Prod.fst '' t
⊢ IsOpen u | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
e : u = Prod.fst '' t
⊢ IsOpen (Prod.fst '' t) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | exact isOpenMap_fst _ s.o | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
e : u = Prod.fst '' t
⊢ IsOpen (Prod.fst '' t) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | ext c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ u = Prod.fst '' t | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
⊢ c ∈ u ↔ c ∈ Prod.fst '' t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | simp only [Set.mem_image, Prod.exists, exists_and_right, exists_eq_right] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
⊢ c ∈ u ↔ c ∈ Prod.fst '' t | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
⊢ c ∈ u ↔ ∃ x, (c, x) ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.ou | [505, 1] | [509, 36] | exact ⟨fun m ↦ ⟨0, (s.s m).t0⟩, fun h ↦ Exists.elim h fun z m ↦ s.tc m⟩ | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
⊢ c ∈ u ↔ ∃ x, (c, x) ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.superAtC | [512, 1] | [521, 42] | intro c m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
⊢ ∀ {c : ℂ}, c ∈ u → SuperAt (f c) d | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c : ℂ
m : c ∈ u
⊢ SuperAt (f c) d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.superAtC | [512, 1] | [521, 42] | have s := s.s 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
⊢ SuperAt (f c) d | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s✝ : SuperNearC f d u t
c : ℂ
m : c ∈ u
s : SuperNear (f c) d {z | (c, z) ∈ t}
⊢ SuperAt (f c) d |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.superAtC | [512, 1] | [521, 42] | exact
{ d2 := s.d2
fa0 := s.fa0
fd := s.fd
fc := s.fc } | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s✝ : SuperNearC f d u t
c : ℂ
m : c ∈ u
s : SuperNear (f c) d {z | (c, z) ∈ t}
⊢ SuperAt (f c) d | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | refine Pair.hartogs o ?_ ?_ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
⊢ AnalyticOn ℂ (g2 f d) t | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z0 => g2 f d (z0, c1)) c0
case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : S... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | intro c z m | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z0 => g2 f d (z0, c1)) c0 | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => g2 f d (z0, z)) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | simp only [g2, g] | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => g2 f d (z0, z)) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | by_cases zero : z = 0 | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c | case pos
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : z = 0
⊢ AnalyticAt ℂ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c
case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Se... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | simp only [zero, eq_self_iff_true, if_true] | case pos
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : z = 0
⊢ AnalyticAt ℂ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c | case pos
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : z = 0
⊢ AnalyticAt ℂ (fun z0 => 1) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | exact analyticAt_const | case pos
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : z = 0
⊢ AnalyticAt ℂ (fun z0 => 1) c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | simp only [zero, if_false] | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ AnalyticAt ℂ (fun z0 => if z = 0 then 1 else f z0 z / z ^ d) c | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ AnalyticAt ℂ (fun z0 => f z0 z / z ^ d) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | refine AnalyticAt.div ?_ analyticAt_const (pow_ne_zero _ zero) | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ AnalyticAt ℂ (fun z0 => f z0 z / z ^ d) c | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ AnalyticAt ℂ (fun z0 => f z0 z) c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | refine (fa _ ?_).comp₂ (analyticAt_id _ _) analyticAt_const | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ AnalyticAt ℂ (fun z0 => f z0 z) c | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ (c, z) ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | exact m | case neg
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
zero : ¬z = 0
⊢ (c, z) ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | intro c z m | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ t → AnalyticAt ℂ (fun z1 => g2 f d (c0, z1)) c1 | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => g2 f d (c, z1)) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | apply (s.s (tc m)).ga_of_fa | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun z1 => g2 f d (c, z1)) z | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (f (c, z).1) z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | refine (fa _ ?_).comp₂ analyticAt_const (analyticAt_id _ _) | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (f (c, z).1) z | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ (c, id z) ∈ t |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.ga_of_fa | [527, 1] | [537, 73] | exact m | case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
t : Set (ℂ × ℂ)
o : IsOpen t
fa : AnalyticOn ℂ (uncurry f) t
tc : ∀ {p : ℂ × ℂ}, p ∈ t → p.1 ∈ u
c z : ℂ
m : (c, z) ∈ t
⊢ (c, id z) ∈ t | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | set tu := ⋃ i, t i | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
⊢ SuperNearC f d (⋃ i, u i) (⋃ i, t i) | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
⊢ SuperNearC f d (⋃ i, u i) tu |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | have o : IsOpen tu := isOpen_iUnion fun i ↦ (s i).o | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
⊢ SuperNearC f d (⋃ i, u i) tu | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
⊢ SuperNearC f d (⋃ i, u i) tu |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact
{ o
tc := by
intro p m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩
exact Set.subset_iUnion _ i ((s i).tc m)
fa := by intro p m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩; exact (s i).fa _ m
s := by
intro c m; rcases Set.mem_iUnion.mp m with ⟨i, m⟩; have s := (s i).s m
exact
... | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ SuperNearC f d (⋃ i, u i) tu | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro c z m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
⊢ ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m : (c, z) ∈ tu
⊢ ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases Set.mem_iUnion.mp m with ⟨i, m⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m : (c, z) ∈ tu
⊢ ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | use{z | (c, z) ∈ t i} | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u | case h
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ z ∈ {z | (c, z) ∈ t i} ∧ {z | (c, z) ∈ t i} ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d {z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | simp only [Set.mem_setOf_eq, m, Set.mem_iUnion, Set.setOf_subset_setOf, true_and, tu] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ z ∈ {z | (c, z) ∈ t i} ∧ {z | (c, z) ∈ t i} ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d {z ... | case h
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ (∀ (a : ℂ), (c, a) ∈ t i → ∃ i, (c, a) ∈ t i) ∧ SuperNear (f c) d {z | (c, z) ∈ t i} |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | constructor | case h
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ (∀ (a : ℂ), (c, a) ∈ t i → ∃ i, (c, a) ∈ t i) ∧ SuperNear (f c) d {z | (c, z) ∈ t i} | case h.left
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ ∀ (a : ℂ), (c, a) ∈ t i → ∃ i, (c, a) ∈ t i
case h.right
f : ℂ → ℂ → ℂ
d : ℕ
u✝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact fun z m ↦ ⟨i, m⟩ | case h.left
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ ∀ (a : ℂ), (c, a) ∈ t i → ∃ i, (c, a) ∈ t i | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact (s i).s ((s i).tc m) | case h.right
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
c z : ℂ
m✝ : (c, z) ∈ tu
i : I
m : (c, z) ∈ t i
⊢ SuperNear (f c) d {z | (c, z) ∈ t i} | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro p m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ ∀ {p : ℂ × ℂ}, p ∈ tu → p.1 ∈ ⋃ i, u i | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m : p ∈ tu
⊢ p.1 ∈ ⋃ i, u i |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases Set.mem_iUnion.mp m with ⟨i, m⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m : p ∈ tu
⊢ p.1 ∈ ⋃ i, u i | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m✝ : p ∈ tu
i : I
m : p ∈ t i
⊢ p.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact Set.subset_iUnion _ i ((s i).tc m) | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m✝ : p ∈ tu
i : I
m : p ∈ t i
⊢ p.... | no goals |
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