diff options
| -rw-r--r-- | src/Util/ZUtil.v | 37 |
1 files changed, 36 insertions, 1 deletions
diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index b7abb2023..319e018a9 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -55,7 +55,8 @@ Hint Rewrite Z.pow_sub_r Z.pow_div_l Z.pow_twice_r Z.pow_mul_l Z.pow_add_r using Hint Rewrite <- Z.pow_sub_r Z.pow_div_l Z.pow_mul_l Z.pow_add_r Z.pow_twice_r using lia : pull_Zpow. Hint Rewrite Z.mul_add_distr_l Z.mul_add_distr_r Z.mul_sub_distr_l Z.mul_sub_distr_r : push_Zmul. Hint Rewrite <- Z.mul_add_distr_l Z.mul_add_distr_r Z.mul_sub_distr_l Z.mul_sub_distr_r : pull_Zmul. -Hint Rewrite <- Z.mul_mod Z.add_mod using lia : pull_Zmod. +Hint Rewrite <- Z.mul_mod Z.add_mod Zminus_mod using lia : pull_Zmod. +Hint Rewrite Zminus_mod_idemp_l Zminus_mod_idemp_r : pull_Zmod. (** For the occasional lemma that can remove the use of [Z.div] *) Create HintDb zstrip_div. @@ -1160,6 +1161,20 @@ Module Z. Qed. Hint Rewrite <- mul_mod_r using lia : pull_Zmod. + Lemma add_mod_l a b n : n <> 0 -> (a + b) mod n = ((a mod n) + b) mod n. + Proof. + intros; rewrite (Z.add_mod a b), (Z.add_mod (a mod n) b) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- add_mod_l using lia : pull_Zmod. + + Lemma add_mod_r a b n : n <> 0 -> (a + b) mod n = (a + (b mod n)) mod n. + Proof. + intros; rewrite (Z.add_mod a b), (Z.add_mod a (b mod n)) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- add_mod_r using lia : pull_Zmod. + Definition NoZMod (x : Z) := True. Ltac NoZMod := lazymatch goal with @@ -1183,6 +1198,26 @@ Module Z. Proof. intros; apply mul_mod_r; assumption. Qed. Hint Rewrite mul_mod_r_push using solve [ NoZMod | lia ] : push_Zmod. + Lemma add_mod_l_push a b n : n <> 0 -> NoZMod a -> (a + b) mod n = ((a mod n) + b) mod n. + Proof. intros; apply add_mod_l; assumption. Qed. + Hint Rewrite add_mod_l_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma add_mod_r_push a b n : n <> 0 -> NoZMod b -> (a + b) mod n = (a + (b mod n)) mod n. + Proof. intros; apply add_mod_r; assumption. Qed. + Hint Rewrite add_mod_r_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma sub_mod_push a b n : n <> 0 -> NoZMod a -> NoZMod b -> (a - b) mod n = ((a mod n) - (b mod n)) mod n. + Proof. intros; apply Zminus_mod; assumption. Qed. + Hint Rewrite sub_mod_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma sub_mod_l_push a b n : n <> 0 -> NoZMod a -> (a - b) mod n = ((a mod n) - b) mod n. + Proof. intros; symmetry; apply Zminus_mod_idemp_l; assumption. Qed. + Hint Rewrite sub_mod_l_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma sub_mod_r_push a b n : n <> 0 -> NoZMod b -> (a - b) mod n = (a - (b mod n)) mod n. + Proof. intros; symmetry; apply Zminus_mod_idemp_r; assumption. Qed. + Hint Rewrite sub_mod_r_push using solve [ NoZMod | lia ] : push_Zmod. + Section equiv_modulo. Context (N : Z). Definition equiv_modulo x y := x mod N = y mod N. |