| import Mathlib.Data.Int.ModEq |
| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticSupport |
|
|
| namespace OAI |
|
|
| /-! # The exact loss of frequency precision at a reversing node -/ |
|
|
| namespace Ostmann |
|
|
| theorem modEq_cancel_coprime_left (m u a b : ℤ) (hu : IsCoprime u m) |
| (h : u * a ≡ u * b [ZMOD m]) : a ≡ b [ZMOD m] := by |
| apply Int.modEq_iff_dvd.mpr |
| apply hu.symm.dvd_of_dvd_mul_left |
| have hd := h.dvd |
| rwa [← mul_sub] at hd |
|
|
| / |
| theorem reversal_residue_precision (R s u u' N N' p p' : ℤ) (k : ℕ) |
| (hs : s ≠ 0) (hsR : s ∣ R) (hu : IsCoprime u R) |
| (hN : N = s * u * p) (hN' : N' = s * u' * p') |
| (hmod : N ≡ N' [ZMOD R ^ (k + 1)]) (humod : u ≡ u' [ZMOD R ^ k]) : |
| p ≡ p' [ZMOD R ^ k] := by |
| have hdiv : s * R ^ k ∣ N' - N := by |
| apply dvd_trans _ hmod.dvd |
| simpa only [pow_succ, mul_comm s] using mul_dvd_mul_left (R ^ k) hsR |
| rw [hN', hN, show s * u' * p' - s * u * p = s * (u' * p' - u * p) by ring] at hdiv |
| have hproducts : u * p ≡ u' * p' [ZMOD R ^ k] := |
| Int.modEq_iff_dvd.mpr (Int.dvd_of_mul_dvd_mul_left hs hdiv) |
| apply modEq_cancel_coprime_left (R ^ k) u p p' hu.pow_right |
| exact hproducts.trans (humod.symm.mul_right p') |
|
|
| / |
| theorem reversal_chain_residue_precision (R : ℤ) (s u u' v v' : ℕ → ℤ) (k : ℕ) |
| (hs : ∀ j, s j ≠ 0) (hsR : ∀ j, s j ∣ R) (hu : ∀ j, IsCoprime (u j) R) |
| (hv : ∀ j, v j = s j * u j * v (j + 1)) |
| (hv' : ∀ j, v' j = s j * u' j * v' (j + 1)) |
| (hmod : v 0 ≡ v' 0 [ZMOD R ^ (k + 1)]) |
| (humod : ∀ j, u j ≡ u' j [ZMOD R ^ (k + 1)]) : |
| ∀ j ≤ k + 1, v j ≡ v' j [ZMOD R ^ (k + 1 - j)] := by |
| intro j |
| induction j with |
| | zero => intro _; simpa using hmod |
| | succ j ih => |
| intro hj |
| have hj' : j ≤ k := by omega |
| have hexp : k + 1 - j = (k - j) + 1 := by omega |
| have hprev := ih (by omega) |
| rw [hexp] at hprev |
| have hunit : u j ≡ u' j [ZMOD R ^ (k - j)] := |
| (humod j).of_dvd (pow_dvd_pow R (by omega : k - j ≤ k + 1)) |
| have h := reversal_residue_precision R (s j) (u j) (u' j) (v j) (v' j) |
| (v (j + 1)) (v' (j + 1)) (k - j) (hs j) (hsR j) (hu j) |
| (hv j) (hv' j) hprev hunit |
| simpa only [show k + 1 - (j + 1) = k - j by omega] using h |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
