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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. *)

Theorem pocklington :
forall (n q m : nat) (a : Z) (qlist : natlist),
n > 1 ->
n = S (q * m) ->
q = product qlist ->
allPrime qlist ->
Mod (Exp a (pred n)) 1 n ->
allLinCombMod a n m qlist -> n <= q * q -> Prime n.