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(* 02110-1301 USA                                                     *)
(** order. The order of elements in the multgroup modulo p.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** (Order b q p) means the order of b is q in multgroup(Z_p). *)
Definition Order (b : Z) (q p : nat) :=
0 < q /\
Mod (Exp b q) 1 p /\ (forall d : nat, 0 < d -> Mod (Exp b d) 1 p -> q <= d).

(* Specifications: *)
Lemma orderdec : forall (b : Z) (q p : nat), Order b q p \/ ~ Order b q p.
Lemma order_ex1 :
forall (b : Z) (p : nat),
Prime p ->
(exists d : nat, 0 < d /\ Mod (Exp b d) 1 p) -> exists x : nat, Order b x p.
Lemma order_ex :
forall (b : Z) (p : nat),
Prime p -> ~ Mod b 0 p -> exists x : nat, x < p /\ Order b x p.
Lemma order_div :
forall (b : Z) (x p : nat),
Order b x p -> forall y : nat, 0 < y -> Mod (Exp b y) 1 p -> Divides x y.
Lemma order_le_predp :
forall (b : Z) (q p : nat), Prime p -> Order b q p -> q <= pred p.