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(* 02110-1301 USA                                                     *)
(** * Greatest common divisor (gcd). *)
(** There is no unicity of the gcd; hence we define the predicate [gcd a b d] expressing that [d] is a gcd of [a] and [b]. (We show later that the [gcd] is actually unique if we discard its sign.) *)
Inductive gcd (a b d : Z) : Prop :=
gcd_intro :
(d | a)%Z ->
(d | b)%Z ->
(forall x : Z, (x | a)%Z -> (x | b)%Z -> (x | d)%Z) -> gcd a b d.
(** Trivial properties of [gcd] *)
Hint Resolve gcd_sym gcd_0 gcd_minus gcd_opp.
(** * Extended Euclid algorithm. *)
(** Euclid's algorithm to compute the [gcd] mainly relies on the following property. *)

Lemma gcd_for_euclid : forall a b d q : Z, gcd b (a - q * b) d -> gcd a b d.