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.
intros x; try assumption.
elim x.
intros px; red in px.
simpl in px.
inversion px.
simpl in |- *.
auto with *.
intros p px; red in px.
simpl in px.
inversion px.
Qed.
intros a b c; try assumption.
unfold Z.lt, not in |- *.
intros H'; try assumption.
rewrite <- H'.
apply Zcompare_plus_compat; assumption.
Qed.
intros x H'; try assumption.
apply Z.lt_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.
simple induction n; simpl in |- *; auto with *.
apply Trans with r; auto with *.
intros n0 H'; try assumption.
apply Trans with (sgroup_law G (sgroup_law G r (nat_to_group n0)) r);
auto with *.
Qed.
Hint Resolve nat_to_group_com: algebra.

Lemma nat_to_group_add :
forall n m : nat,
Equal (nat_to_group (n + m))
(sgroup_law G (nat_to_group n) (nat_to_group m)).