(**********************************************************************)
 
Inductive is_gcd (a b c : nat) : Prop :=
    is_gcd_intro :
      divides c a ->
      divides c b ->
      (forall d : nat, divides d a -> divides d b -> divides d c) ->
      is_gcd a b c.(**********************************************************************)
 
Inductive is_gcd (a b c : nat) : Prop :=
    is_gcd_intro :
      divides c a ->
      divides c b ->
      (forall d : nat, divides d a -> divides d b -> divides d c) ->
      is_gcd a b c.

Lemma prime_gcd : forall p n : nat, prime p -> ~ divides p n -> is_gcd n p 1.