Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files
xet
 Community
1
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
f_converges
[263, 1]
[277, 33]
intro zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ z ∈ t → Complex.abs (f z) ≤ 5 / 8 * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f z) ≤ 5 / 8 * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
rw [s.fg]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f z) ≤ 5 / 8 * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (z ^ d * g f d z) ≤ 5 / 8 * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
simp
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (z ^ d * g f d z) ≤ 5 / 8 * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
have gs : abs (g f d z) ≤ 5 / 4 := by calc abs (g f d z) _ = abs (g f d z - 1 + 1) := by ring_nf _ ≤ abs (g f d z - 1) + abs (1 : ℂ) := by bound _ ≤ 1 / 4 + abs (1 : ℂ) := by linarith [s.gs zt] _ ≤ 5 / 4 := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
have az1 : abs z ≤ 1 := le_trans (s.t2 zt) (by norm_num)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
calc abs z ^ d * abs (g f d z) _ ≤ abs z ^ 2 * (5 / 4) := by bound _ = abs z * abs z * (5 / 4) := by ring_nf _ ≤ 1 / 2 * abs z * (5 / 4) := by bound [s.t2 zt] _ = 5 / 8 * abs z := by ring
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ 5 / 8 * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
calc abs (g f d z) _ = abs (g f d z - 1 + 1) := by ring_nf _ ≤ abs (g f d z - 1) + abs (1 : ℂ) := by bound _ ≤ 1 / 4 + abs (1 : ℂ) := by linarith [s.gs zt] _ ≤ 5 / 4 := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) ≤ 5 / 4
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
ring_nf
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z) = Complex.abs (g f d z - 1 + 1)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1 + 1) ≤ Complex.abs (g f d z - 1) + Complex.abs 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
linarith [s.gs zt]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (g f d z - 1) + Complex.abs 1 ≤ 1 / 4 + Complex.abs 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ 1 / 4 + Complex.abs 1 ≤ 5 / 4
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 ⊢ 1 / 2 ≤ 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ d * Complex.abs (g f d z) ≤ Complex.abs z ^ 2 * (5 / 4)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
ring_nf
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z ^ 2 * (5 / 4) = Complex.abs z * Complex.abs z * (5 / 4)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
bound [s.t2 zt]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ Complex.abs z * Complex.abs z * (5 / 4) ≤ 1 / 2 * Complex.abs z * (5 / 4)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
f_converges
[263, 1]
[277, 33]
ring
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t gs : Complex.abs (g f d z) ≤ 5 / 4 az1 : Complex.abs z ≤ 1 ⊢ 1 / 2 * Complex.abs z * (5 / 4) = 5 / 8 * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_le
[279, 1]
[280, 78]
intro rp
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → (5 / 8) ^ n * r ≤ r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ r
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_le
[279, 1]
[280, 78]
trans (1:ℝ) ^ n * r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ 1 ^ n * r f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_le
[279, 1]
[280, 78]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ (5 / 8) ^ n * r ≤ 1 ^ n * r f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_le
[279, 1]
[280, 78]
simp only [one_pow, one_mul, le_refl]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 ⊢ 1 ^ n * r ≤ r
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
intro rp np
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ ⊢ r > 0 → n ≠ 0 → (5 / 8) ^ n * r < r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ (5 / 8) ^ n * r < r
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
have h : (5 / 8 : ℝ) ^ n < 1 := pow_lt_one (by norm_num) (by norm_num) np
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ (5 / 8) ^ n * r < r
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ (5 / 8) ^ n * r < r
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
exact lt_of_lt_of_le (mul_lt_mul_of_pos_right h rp) (by simp only [one_mul, le_refl])
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ (5 / 8) ^ n * r < r
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 0 ≤ 5 / 8
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 ⊢ 5 / 8 < 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
five_eights_pow_lt
[282, 1]
[285, 88]
simp only [one_mul, le_refl]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ n : ℕ r : ℝ rp : r > 0 np : n ≠ 0 h : (5 / 8) ^ n < 1 ⊢ 1 * r ≤ r
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNear.mapsTo
[288, 1]
[290, 49]
induction' n with n h
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ MapsTo f^[n] t t
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ MapsTo f^[0] t t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNear.mapsTo
[288, 1]
[290, 49]
simp only [Set.mapsTo_id, Function.iterate_zero]
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ MapsTo f^[0] t t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNear.mapsTo
[288, 1]
[290, 49]
rw [Function.iterate_succ']
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo f^[n + 1] t t
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo (f ∘ f^[n]) t t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
SuperNear.mapsTo
[288, 1]
[290, 49]
exact s.ft.comp h
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : MapsTo f^[n] t t ⊢ MapsTo (f ∘ f^[n]) t t
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
intro n zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
induction' n with n nh
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f^[0] z) ≤ (5 / 8) ^ 0 * Complex.abs z case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs (f^[n + 1] z) ≤ (5 / 8) ^ (n + 1) * Comple...
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
simp only [Function.iterate_zero, id, pow_zero, one_mul, Nat.cast_one, le_refl]
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ Complex.abs (f^[0] z) ≤ (5 / 8) ^ 0 * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
rw [Function.iterate_succ']
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs (f^[n + 1] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
trans (5/8 : ℝ) * abs (f^[n] z)
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ 5 / 8 * Complex.abs (f^[n] z) f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 /...
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
exact f_converges s (s.mapsTo n zt)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ Complex.abs ((f ∘ f^[n]) z) ≤ 5 / 8 * Complex.abs (f^[n] z)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
calc (5/8 : ℝ) * abs (f^[n] z) _ ≤ (5/8 : ℝ) * ((5/8 : ℝ) ^ n * abs z) := by bound _ = 5/8 * (5/8 : ℝ) ^ n * abs z := by ring _ = (5/8 : ℝ) ^ (n + 1) * abs z := by rw [← pow_succ'] _ = (5/8 : ℝ) ^ n.succ * abs z := rfl
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ (5 / 8) ^ (n + 1) * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * Complex.abs (f^[n] z) ≤ 5 / 8 * ((5 / 8) ^ n * Complex.abs z)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
ring
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * ((5 / 8) ^ n * Complex.abs z) = 5 / 8 * (5 / 8) ^ n * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_converge
[293, 1]
[305, 46]
rw [← pow_succ']
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ nh : Complex.abs (f^[n] z) ≤ (5 / 8) ^ n * Complex.abs z ⊢ 5 / 8 * (5 / 8) ^ n * Complex.abs z = (5 / 8) ^ (n + 1) * Complex.abs z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
intro n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ f^[n] t
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ AnalyticOn ℂ f^[n] t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
induction' n with n h
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ AnalyticOn ℂ f^[n] t
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ f^[0] t case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ f^[n + 1] t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
simp only [Function.iterate_zero]
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ f^[0] t
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ id t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
exact analyticOn_id _
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ AnalyticOn ℂ id t
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
rw [Function.iterate_succ']
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ f^[n + 1] t
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ (f ∘ f^[n]) t
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
intro z zt
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t ⊢ AnalyticOn ℂ (f ∘ f^[n]) t
case succ f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (f ∘ f^[n]) z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_analytic
[308, 1]
[310, 90]
exact (s.fa _ (s.mapsTo n zt)).comp (h z zt)
case succ f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ h : AnalyticOn ℂ f^[n] t z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (f ∘ f^[n]) z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_analytic
[313, 1]
[317, 90]
intro n z zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), AnalyticOn ℂ (term f d n) t
f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (term f d n) z
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_analytic
[313, 1]
[317, 90]
refine AnalyticAt.cpow ?_ analyticAt_const ?_
f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (term f d n) z
case refine_1 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (fun z => g f d (f^[n] z)) z case refine_2 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ g f d (f^[n] z) ∈ Complex.slitPlane
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_analytic
[313, 1]
[317, 90]
exact (s.ga _ (s.mapsTo n zt)).comp (iterates_analytic s n z zt)
case refine_1 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ AnalyticAt ℂ (fun z => g f d (f^[n] z)) z
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_analytic
[313, 1]
[317, 90]
exact mem_slitPlane_of_near_one (lt_of_le_of_lt (s.gs (s.mapsTo n zt)) (by norm_num))
case refine_2 f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ g f d (f^[n] z) ∈ Complex.slitPlane
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_analytic
[313, 1]
[317, 90]
norm_num
f : ℂ → ℂ d : ℕ z✝ : ℂ t : Set ℂ s : SuperNear f d t n : ℕ z : ℂ zt : z ∈ t ⊢ 1 / 4 < 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
intro n zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → 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 n : ℕ zt : z ∈ t ⊢ Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
rw [term]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 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 n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 1 / 2 * (1 / 2) ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
trans 4 * abs (g f d (f^[n] z) - 1) * abs (1 / (d ^ (n + 1) : ℕ) : ℂ)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 1 / 2 * (1 / 2) ^ n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * C...
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
apply pow_small
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) ^ (1 / ↑(d ^ (n + 1))) - 1) ≤ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1)))
case zs f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 2 case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
exact le_trans (s.gs (s.mapsTo n zt)) (by norm_num)
case zs f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 / 4 ≤ 1 / 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
simp only [one_div, map_inv₀, Complex.abs_pow, Complex.abs_natCast, Nat.cast_pow]
case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1
case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ (↑d ^ (n + 1))⁻¹ ≤ 1
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
apply inv_le_one
case ws f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ (↑d ^ (n + 1))⁻¹ ≤ 1
case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ ↑d ^ (n + 1)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
have hd : 1 ≤ (d : ℝ) := le_trans (by norm_num) s.dr2
case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ ↑d ^ (n + 1)
case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t hd : 1 ≤ ↑d ⊢ 1 ≤ ↑d ^ (n + 1)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
exact one_le_pow_of_one_le hd _
case ws.ha f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t hd : 1 ≤ ↑d ⊢ 1 ≤ ↑d ^ (n + 1)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 1 ≤ 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
have gs : abs (g f d (f^[n] z) - 1) ≤ 1 / 4 := s.gs (s.mapsTo n zt)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
have ps : abs (1 / (d:ℂ) ^ (n + 1) : ℂ) ≤ 1/2 * (1/2 : ℝ) ^ n := by have nn : (1/2:ℝ) * (1/2 : ℝ) ^ n = (1/2 : ℝ) ^ (n + 1) := (pow_succ' _ _).symm rw [nn] simp only [one_div, map_inv₀, map_pow, Complex.abs_natCast, inv_pow, ge_iff_le] bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
calc (4:ℝ) * abs (g f d (f^[n] z) - 1) * abs ((1:ℂ) / (d ^ (n + 1) : ℕ) : ℂ) _ = (4:ℝ) * abs (g f d (f^[n] z) - 1) * abs ((1:ℂ) / (d:ℂ) ^ (n + 1) : ℂ) := by rw [Nat.cast_pow] _ ≤ 4 * (1 / 4) * (1 / 2 * (1 / 2 : ℝ) ^ n) := by bound _ = 1 / 2 * (1 / 2 : ℝ) ^ n := by ring
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) ≤ 1 / 2 * (1 / 2) ^ n
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
have nn : (1/2:ℝ) * (1/2 : ℝ) ^ n = (1/2 : ℝ) ^ (n + 1) := (pow_succ' _ _).symm
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
rw [nn]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ (1 / 2) ^ (n + 1)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
simp only [one_div, map_inv₀, map_pow, Complex.abs_natCast, inv_pow, ge_iff_le]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ Complex.abs (1 / ↑d ^ (n + 1)) ≤ (1 / 2) ^ (n + 1)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 nn : 1 / 2 * (1 / 2) ^ n = (1 / 2) ^ (n + 1) ⊢ (↑d ^ (n + 1))⁻¹ ≤ (2 ^ (n + 1))⁻¹
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
rw [Nat.cast_pow]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑(d ^ (n + 1))) = 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n ...
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
bound
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * Complex.abs (g f d (f^[n] z) - 1) * Complex.abs (1 / ↑d ^ (n + 1)) ≤ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_converges
[322, 1]
[341, 45]
ring
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t gs : Complex.abs (g f d (f^[n] z) - 1) ≤ 1 / 4 ps : Complex.abs (1 / ↑d ^ (n + 1)) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 4 * (1 / 4) * (1 / 2 * (1 / 2) ^ n) = 1 / 2 * (1 / 2) ^ n
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
intro n zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), z ∈ t → term f d n z ≠ 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ term f d n z ≠ 0
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
have h := term_converges s n zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t ⊢ term f d n z ≠ 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ term f d n z ≠ 0
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
have o : 1 / 2 * (1 / 2 : ℝ) ^ n < 1 := by have p : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one n (by norm_num) (by linarith) calc 1 / 2 * (1 / 2 : ℝ) ^ n ≤ 1 / 2 * 1 := by linarith _ < 1 := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ term f d n z ≠ 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n o : 1 / 2 * (1 / 2) ^ n < 1 ⊢ term f d n z ≠ 0
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
exact near_one_avoids_zero (lt_of_le_of_lt h o)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n o : 1 / 2 * (1 / 2) ^ n < 1 ⊢ term f d n z ≠ 0
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
have p : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one n (by norm_num) (by linarith)
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 * (1 / 2) ^ n < 1
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n < 1
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
calc 1 / 2 * (1 / 2 : ℝ) ^ n ≤ 1 / 2 * 1 := by linarith _ < 1 := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n < 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 0 ≤ 1 / 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
linarith
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n ⊢ 1 / 2 ≤ 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
linarith
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * (1 / 2) ^ n ≤ 1 / 2 * 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_nonzero
[344, 1]
[352, 50]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ zt : z ∈ t h : Complex.abs (term f d n z - 1) ≤ 1 / 2 * (1 / 2) ^ n p : (1 / 2) ^ n ≤ 1 ⊢ 1 / 2 * 1 < 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
have c12 : (1 / 2 : ℝ) ≤ 1 / 2 := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
have a0 : 0 ≤ (1 / 2 : ℝ) := by norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
exact fast_products_converge' s.o c12 a0 (by linarith) (term_analytic s) fun n z ↦ term_converges s n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ ProdExistsOn (term f d) t ∧ AnalyticOn ℂ (tprodOn (term f d)) t ∧ ∀ z ∈ t, tprodOn (term f d) z ≠ 0
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ 1 / 2 ≤ 1 / 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
norm_num
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 ⊢ 0 ≤ 1 / 2
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
term_prod
[355, 1]
[361, 33]
linarith
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t c12 : 1 / 2 ≤ 1 / 2 a0 : 0 ≤ 1 / 2 ⊢ 1 / 2 < 1
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
simp_rw [bottcherNear]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f z) = bottcherNear f d z ^ d
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
have pe := (term_prod_exists s) z zt
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
simp only [mul_pow, product_pow' pe]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = (z * ∏' (n : ℕ), term f d n z) ^ d
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
have pe : ProdExists fun n ↦ term f d n z ^ d := by rcases pe with ⟨g, hg⟩; exact ⟨_, product_pow d hg⟩
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe✝ : ProdExists fun n => term f d n z pe : ProdExists fun n => term f d n z ^ d ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
simp only [product_split pe, ← term_eqn s, ← mul_assoc, ← mul_pow, ← term_base s]
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe✝ : ProdExists fun n => term f d n z pe : ProdExists fun n => term f d n z ^ d ⊢ f z * ∏' (n : ℕ), term f d n (f z) = z ^ d * ∏' (n : ℕ), term f d n z ^ d
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
rcases pe with ⟨g, hg⟩
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t pe : ProdExists fun n => term f d n z ⊢ ProdExists fun n => term f d n z ^ d
case intro f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t g : ℂ hg : HasProd (fun n => term f d n z) g ⊢ ProdExists fun n => term f d n z ^ d
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn
[376, 1]
[383, 84]
exact ⟨_, product_pow d hg⟩
case intro f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t g : ℂ hg : HasProd (fun n => term f d n z) g ⊢ ProdExists fun n => term f d n z ^ d
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn_iter
[386, 1]
[390, 43]
induction' n with n h
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ ⊢ bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f^[0] z) = bottcherNear f d z ^ d ^ 0 case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f...
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn_iter
[386, 1]
[390, 43]
simp only [Function.iterate_zero, id, pow_zero, pow_one]
case zero f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t ⊢ bottcherNear f d (f^[0] z) = bottcherNear f d z ^ d ^ 0 case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f...
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
bottcherNear_eqn_iter
[386, 1]
[390, 43]
simp only [Function.comp, Function.iterate_succ', pow_succ, pow_mul, bottcherNear_eqn s (s.mapsTo n zt), h]
case succ f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t zt : z ∈ t n : ℕ h : bottcherNear f d (f^[n] z) = bottcherNear f d z ^ d ^ n ⊢ bottcherNear f d (f^[n + 1] z) = bottcherNear f d z ^ d ^ (n + 1)
no goals
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/BottcherNear.lean
iterates_at_zero
[393, 1]
[395, 67]
intro n
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t ⊢ ∀ (n : ℕ), f^[n] 0 = 0
f : ℂ → ℂ d : ℕ z : ℂ t : Set ℂ s : SuperNear f d t n : ℕ ⊢ f^[n] 0 = 0