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(* as published by the Free Software Foundation; either version 2.1   *)
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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** gcd. Greatest common divisor.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Linear combinations. *)
Definition LinComb (c x y : Z) :=
exists a : Z, (exists b : Z, c = (x * a + y * b)%Z).
Definition LinCombMod (c x y : Z) (n : nat) :=
exists a : Z, (exists b : Z, Mod c (x * a + y * b) n).
Definition ZLinCombMod (c x y n : Z) :=
exists a : Z, (exists b : Z, ZMod c (x * a + y * b) n).

Lemma zlincombmodlincombmod :
forall c x y n : Z, ZLinCombMod c x y n -> LinCombMod c x y (Z.abs_nat n).