Definition Zpos_div_eucl_aux (a b : positive) :=
  match Pos.compare a b with
  | Lt => (Z0, Zpos a)
  | Eq => (1%Z, Z0)
  | Gt => Zpos_div_eucl_aux1 a b
  end.

Lemma Zpos_div_eucl_aux_correct : forall a b, Zpos_div_eucl_aux a b = Z.pos_div_eucl a (Zpos b).