Fixpoint Zpos_div_eucl_aux1 (a b : positive) {struct b} :=
match b with
| xO b' =>
match a with
| xO a' => let (q, r) := Zpos_div_eucl_aux1 a' b' in (q, 2 * r)%Z
| xI a' => let (q, r) := Zpos_div_eucl_aux1 a' b' in (q, 2 * r + 1)%Z
| xH => (Z0, Zpos a)
end
| xH => (Zpos a, Z0)
| xI _ => Z.pos_div_eucl a (Zpos b)
end.

Lemma Zpos_div_eucl_aux1_correct :
forall a b,
Zpos_div_eucl_aux1 a b = Z.pos_div_eucl a (Zpos b).