apply Zlt_gt.
replace (- x)%Z with (- x + 0)%Z.
pattern 0%Z at 1 in |- *.
replace 0%Z with (- x + x)%Z.
apply Zlt_reg_l.
auto with *.
apply Zplus_opp_l.
apply Zplus_0_r.
Qed.
Comments "The powers of" r ".".
Definition Z_to_group_nat_fun : ZZ -> G.
intros x.
case (Z_gt_le_dec x 0); intros z.
exact (nat_to_group (nat_of_P (pos_abs z))).
case (Z_le_lt_eq_dec _ _ z); intros.
cut (- x > 0)%Z.
intros H'.
exact (group_inverse G (nat_to_group (nat_of_P (pos_abs H')))).
apply Zlemma1.
auto with *.
exact (monoid_unit G).
Defined.

(* Specifications: *)
Lemma nat_to_group_com :
forall n : nat,
Equal (sgroup_law G (nat_to_group n) r) (sgroup_law G r (nat_to_group n)).