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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import Mathlib
namespace OAI
namespace PiExponent
noncomputable def approximatePeriod (p : ℤ) (q : ℕ) : ℂ :=
2 * Complex.I * (p : ℂ) / (q : ℂ)
theorem approximatePeriod_ne_zero {p : ℤ} {q : ℕ} (hp : p ≠ 0) (hq : q ≠ 0) :
approximatePeriod p q ≠ 0 := by
unfold approximatePeriod
exact div_ne_zero
(mul_ne_zero (mul_ne_zero (by norm_num) Complex.I_ne_zero) (by exact_mod_cast hp))
(by exact_mod_cast hq)
theorem approximation_centers_injective {p : ℤ} {q : ℕ}
(hp : p ≠ 0) (hq : q ≠ 0) :
Function.Injective (fun j : ℕ => (j : ℂ) * approximatePeriod p q) := by
intro j k h
have heq : (j : ℂ) = (k : ℂ) := mul_right_cancel₀ (approximatePeriod_ne_zero hp hq) h
exact_mod_cast heq
noncomputable def logarithmicPeriod : ℂ := 2 * (Real.pi : ℂ) * Complex.I
theorem norm_period_error (p : ℤ) (q : ℕ) :
‖approximatePeriod p q - logarithmicPeriod‖ =
2 * |Real.pi - (p : ℝ) / q| := by
have heq : approximatePeriod p q - logarithmicPeriod =
(2 * Complex.I) * (((p : ℝ) / q - Real.pi : ℝ) : ℂ) := by
unfold approximatePeriod logarithmicPeriod
push_cast
ring
rw [heq, norm_mul, Complex.norm_real, Real.norm_eq_abs]
norm_num [norm_mul, abs_sub_comm]
theorem norm_center_error_le (nu : ℝ) {p : ℤ} {q j K : ℕ}
(hj : j ≤ K)
(happrox : |Real.pi - (p : ℝ) / q| ≤ (q : ℝ) ^ (-nu)) :
‖(j : ℂ) * (approximatePeriod p q - logarithmicPeriod)‖ ≤
2 * K * (q : ℝ) ^ (-nu) := by
rw [norm_mul, norm_period_error]
have hjR : (j : ℝ) ≤ (K : ℝ) := by exact_mod_cast hj
have hqpow : 0 ≤ (q : ℝ) ^ (-nu) := Real.rpow_nonneg (by positivity) _
simp only [Complex.norm_natCast]
nlinarith [abs_nonneg (Real.pi - (p : ℝ) / q)]
theorem ceil_log_bounds {q : ℕ} (hq : 1 ≤ q) :
0 ≤ (⌈Real.log q⌉₊ : ℝ) - Real.log q ∧
(⌈Real.log q⌉₊ : ℝ) - Real.log q < 1 := by
have hlog : 0 ≤ Real.log (q : ℝ) := Real.log_nonneg (by exact_mod_cast hq)
constructor
· exact sub_nonneg.mpr (Nat.le_ceil _)
· have := Nat.ceil_lt_add_one hlog
linarith
theorem rpow_neg_le_exp_ceil_log (nu : ℝ) {q : ℕ} (hnu : 0 ≤ nu) (hq : 1 ≤ q) :
(q : ℝ) ^ (-nu) ≤
Real.exp nu * Real.exp (-nu * (⌈Real.log q⌉₊ : ℝ)) := by
have hqpos : (0 : ℝ) < q := by exact_mod_cast (by omega : 0 < q)
rw [Real.rpow_def_of_pos hqpos, ← Real.exp_add]
apply Real.exp_le_exp.mpr
have hw := (ceil_log_bounds hq).2
nlinarith
theorem norm_center_error_exp (nu : ℝ) {p : ℤ} {q j K : ℕ}
(hnu : 0 ≤ nu) (hq : 1 ≤ q) (hj : j ≤ K)
(happrox : |Real.pi - (p : ℝ) / q| ≤ (q : ℝ) ^ (-nu)) :
‖(j : ℂ) * (approximatePeriod p q - logarithmicPeriod)‖ ≤
(2 * K * Real.exp nu) * Real.exp (-nu * (⌈Real.log q⌉₊ : ℝ)) := by
calc
_ ≤ 2 * K * (q : ℝ) ^ (-nu) := norm_center_error_le nu hj happrox
_ ≤ 2 * K * (Real.exp nu * Real.exp (-nu * (⌈Real.log q⌉₊ : ℝ))) :=
mul_le_mul_of_nonneg_left (rpow_neg_le_exp_ceil_log nu hnu hq) (by positivity)
_ = _ := by ring
end PiExponent
end OAI
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