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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** exp. Exponential function on Z.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Exponential function with exponent in nat. *)
Fixpoint Exp (a : Z) (n : nat) {struct n} : Z :=
match n with
| O => 1%Z
| S m => (a * Exp a m)%Z
end.
(** Convenience lemma for changing exponent. *)
(** * Exponential function with exponent in Z. *)
Definition ZExp (a n : Z) : Z :=
match n with
| Z0 => 1%Z
| Zpos p => Exp a (nat_of_P p)
| Zneg p => Exp a (nat_of_P p)
end.
(** Convenience lemma for changing exponent. *)

Lemma zexp_S1 :
forall a n : Z, (0 <= n)%Z -> ZExp a (n + 1) = (a * ZExp a n)%Z.