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Diffstat (limited to 'theories/Numbers/Cyclic/Int63/Int63.v')
| -rw-r--r-- | theories/Numbers/Cyclic/Int63/Int63.v | 17 |
1 files changed, 17 insertions, 0 deletions
diff --git a/theories/Numbers/Cyclic/Int63/Int63.v b/theories/Numbers/Cyclic/Int63/Int63.v index 7bb725538b..a3ebe67325 100644 --- a/theories/Numbers/Cyclic/Int63/Int63.v +++ b/theories/Numbers/Cyclic/Int63/Int63.v @@ -205,6 +205,7 @@ Qed. Corollary to_Z_bounded : forall x, (0 <= φ x < wB)%Z. Proof. apply to_Z_rec_bounded. Qed. + (* =================================================== *) Local Open Scope Z_scope. (* General arithmetic results *) @@ -1904,6 +1905,22 @@ Qed. Lemma lxor0_r i : i lxor 0 = i. Proof. rewrite lxorC; exact (lxor0 i). Qed. +Lemma opp_to_Z_opp (x : int) : + φ x mod wB <> 0 -> + (- φ (- x)) mod wB = (φ x) mod wB. +Proof. + intros neqx0. + rewrite opp_spec. + rewrite (Z_mod_nz_opp_full (φ x%int63)) by assumption. + rewrite (Z.mod_small (φ x%int63)) by apply to_Z_bounded. + rewrite <- Z.add_opp_l. + rewrite Z.opp_add_distr, Z.opp_involutive. + replace (- wB) with (-1 * wB) by easy. + rewrite Z_mod_plus by easy. + now rewrite Z.mod_small by apply to_Z_bounded. +Qed. + + Module Export Int63Notations. Local Open Scope int63_scope. #[deprecated(since="8.13",note="use infix mod instead")] |