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Diffstat (limited to 'src/Util/ZUtil.v')
| -rw-r--r-- | src/Util/ZUtil.v | 12 |
1 files changed, 12 insertions, 0 deletions
diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index 973c2685d..2ff1ef834 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -964,6 +964,18 @@ Module Z. Qed. Hint Resolve div_sub_mod_cond : zarith. + Lemma div_add_mod_cond_l : forall x y d, d <> 0 -> (x + y) / d = (x mod d + y) / d + x / d. + Proof. + intros. replace (x + y) with ((x - x mod d) + (x mod d + y)) by lia. + rewrite Z.div_add_exact by auto with zarith. + rewrite <- Z.div_sub_mod_exact by lia; lia. + Qed. + + Lemma div_add_mod_cond_r : forall x y d, d <> 0 -> (x + y) / d = (x + y mod d) / d + y / d. + Proof. + intros. rewrite Z.add_comm, div_add_mod_cond_l by auto. repeat (f_equal; try ring). + Qed. + Lemma div_between n a b : 0 <= n -> b <> 0 -> n * b <= a < (1 + n) * b -> a / b = n. Proof. intros; Z.div_mod_to_quot_rem; nia. Qed. Hint Rewrite div_between using zutil_arith : zsimplify. |