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(* 02110-1301 USA                                                     *)
(** divides. The division predicate.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Division on nat *)
Definition Divides (n m : nat) : Prop := exists q : nat, m = n * q.
(** Divides with bounded ex quantifier. *)
Definition bDivides (d n : nat) :=
n = 0 \/ (exists q : nat, q < S n /\ n = d * q).
(** Divides is decidable. *)
(** If d|n, then either d<sqrt(n) or the other divisor x<sqrt(n). *)
(** Division with remainder. *)
(** Compatibility results. *)
(** * Division on Z. *)
Definition ZDivides (x y : Z) : Prop := exists q : Z, y = (x * q)%Z.

Theorem zdiv_rem :
forall d n : Z,
(d > 0)%Z ->
exists q : Z, (exists r : Z, (0 <= r < d)%Z /\ n = (q * d + r)%Z).