Section Zup1.
Variable G : GROUP.
Variable r : G.
Fixpoint nat_to_group (n : nat) : G :=
match n with
| O => monoid_unit G
| S n' => sgroup_law G (nat_to_group n') r
end.
Definition pos_abs : forall x : Z, (x > 0)%Z -> positive.
intros x; try assumption.
case x.
intros H'; red in H'.
simpl in H'.
inversion H'.
intros p H'.
exact p.
intros p H'; red in H'.
simpl in H'.
inversion H'.
Defined.

Lemma pos_abs_ok : forall (x : Z) (px : (x > 0)%Z), x = Zpos (pos_abs px).