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).