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(* as published by the Free Software Foundation; either version 2.1   *)
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(*                                                                    *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
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(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 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).

(* Specifications: *)
Lemma lincombmodzlincombmod :
forall (c x y : Z) (n : nat),
LinCombMod c x y n -> ZLinCombMod c x y (Z_of_nat n).
Lemma zlincombmodlincombmod :
forall c x y n : Z, ZLinCombMod c x y n -> LinCombMod c x y (Zabs_nat n).