| 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 | |