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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. *)

Theorem notdiv_rem :
forall d n : nat,
0 < d ->
~ Divides d n ->
exists q : nat, (exists r : nat, 0 < r /\ r < d /\ n = q * d + r).