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(* 02110-1301 USA                                                     *)
(** modulo Modulo Arithmetic.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** (Mod a b n) means (a = b (mod n)) *)
Definition Mod (a b : Z) (n : nat) :=
exists q : Z, a = (b + Z_of_nat n * q)%Z.
(** ZMod, same as Mod but only uses Z. *)
Definition ZMod (a b n : Z) := exists q : Z, a = (b + n * q)%Z.

Lemma zmodmod : forall a b n : Z, ZMod a b n -> Mod a b (Z.abs_nat n).