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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.

Lemma zexp_pred_succ : forall a x : Z, ZExp a (x + 1 - 1) = ZExp a x.