diff options
Diffstat (limited to 'src')
| -rw-r--r-- | src/Util/ZUtil.v | 42 |
1 files changed, 42 insertions, 0 deletions
diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index 8177444a5..7b7daef67 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -34,12 +34,16 @@ Create HintDb push_Zpow discriminated. Create HintDb pull_Zpow discriminated. Create HintDb push_Zmul discriminated. Create HintDb pull_Zmul discriminated. +Create HintDb pull_Zmod discriminated. +Create HintDb push_Zmod discriminated. Hint Extern 1 => autorewrite with push_Zopp in * : push_Zopp. Hint Extern 1 => autorewrite with pull_Zopp in * : pull_Zopp. Hint Extern 1 => autorewrite with push_Zpow in * : push_Zpow. Hint Extern 1 => autorewrite with pull_Zpow in * : pull_Zpow. Hint Extern 1 => autorewrite with push_Zmul in * : push_Zmul. Hint Extern 1 => autorewrite with pull_Zmul in * : pull_Zmul. +Hint Extern 1 => autorewrite with pull_Zmod in * : pull_Zmod. +Hint Extern 1 => autorewrite with push_Zmod in * : push_Zmod. Hint Rewrite Z.div_opp_l_nz Z.div_opp_l_z using lia : pull_Zopp. Hint Rewrite Z.mul_opp_l : pull_Zopp. Hint Rewrite <- Z.opp_add_distr : pull_Zopp. @@ -50,6 +54,7 @@ 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. (** For the occasional lemma that can remove the use of [Z.div] *) Create HintDb zstrip_div. @@ -1140,6 +1145,43 @@ Module Z. reflexivity. Qed. + Lemma mul_mod_l a b n : n <> 0 -> (a * b) mod n = ((a mod n) * b) mod n. + Proof. + intros; rewrite (Z.mul_mod a b), (Z.mul_mod (a mod n) b) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- mul_mod_l using lia : pull_Zmod. + + Lemma mul_mod_r a b n : n <> 0 -> (a * b) mod n = (a * (b mod n)) mod n. + Proof. + intros; rewrite (Z.mul_mod a b), (Z.mul_mod a (b mod n)) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- mul_mod_r using lia : pull_Zmod. + + Definition NoZMod (x : Z) := True. + Ltac NoZMod := + lazymatch goal with + | [ |- NoZMod (?x mod ?y) ] => fail 0 "Goal has" x "mod" y + | [ |- NoZMod _ ] => constructor + end. + + Lemma mul_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 Z.mul_mod; assumption. Qed. + Hint Rewrite mul_mod_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma add_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 Z.add_mod; assumption. Qed. + Hint Rewrite add_mod_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma mul_mod_l_push a b n : n <> 0 -> NoZMod a -> (a * b) mod n = ((a mod n) * b) mod n. + Proof. intros; apply mul_mod_l; assumption. Qed. + Hint Rewrite mul_mod_l_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma mul_mod_r_push a b n : n <> 0 -> NoZMod b -> (a * b) mod n = (a * (b mod n)) mod n. + Proof. intros; apply mul_mod_r; assumption. Qed. + Hint Rewrite mul_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. |