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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).
(** * Greatest common divisor. *)
Definition common_div (x y : Z) (d : nat) :=
Divides d (Z.abs_nat x) /\ Divides d (Z.abs_nat y).
Definition gcd (x y : Z) (d : nat) :=
common_div x y d /\ (forall e : nat, common_div x y e -> e <= d).
(** * Euclid's theorem. *)
(** * Greatest common divisor can be written as linear combination. *)

Lemma gcd_lincomb_nat :
forall x y d : nat,
x > 0 ->
gcd (Z_of_nat x) (Z_of_nat y) d ->
LinComb (Z_of_nat d) (Z_of_nat x) (Z_of_nat y).