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(* 02110-1301 USA                                                     *)
(** pock. Pocklington's criterion.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** For all numbers q in natlist qlist: 1 is a linear combination of a^(m*qlist/qi) and n modulo n. *)
Definition allLinCombMod (a : Z) (n m : nat) (qlist : natlist) :=
alllist nat
(fun q : nat =>
LinCombMod 1 (Exp a (m * multDrop q qlist) - 1) (Z_of_nat n) n) qlist.
(** Pocklington's theorem (finally). Proves Pocklington for natural numbers. *)
(** Below is an attempt to restate Pocklington's theorem using only numbers from Z. This will make concrete computations faster. *)
(** ZallLinCombMod is equivalent to AllLinCombMod but uses Z. *)
Definition ZallLinCombMod (a n m N : Z) (qlist : Zlist) :=
alllist Z
(fun q : Z => ZLinCombMod 1 (ZExp a (m * zmultDrop q qlist) - 1) n N)
qlist.
(** Zpocklington is equivalent to pocklington but only uses numbers in Z. *)

Theorem Zpocklington :
forall (n q m a : Z) (qlist : Zlist),
(n > 1)%Z ->
(0 <= q)%Z ->
(0 <= m)%Z ->
n = (q * m + 1)%Z ->
q = zproduct qlist ->
allZPrime qlist ->
ZMod (ZExp a (n - 1)) 1 n ->
ZallLinCombMod a n m n qlist -> (n <= q * q)%Z -> ZPrime n.