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(* 02110-1301 USA                                                     *)
(** fermat. Fermat's little theorem.  @author Martijn Oostdijk @version $Revision$ *)
(** * No double elements modulo p. *)
Definition nodoubles (p : nat) (l : Zlist) : Prop :=
forall x : Z,
inlist Z x l -> forall y : Z, inlist Z y (zdrop x l) -> ~ Mod x y p.
(** * All elements from l1 occur in l2 modulo p. *)
Definition allex (p : nat) (l0 l1 : Zlist) : Prop :=
forall x : Z, inlist Z x l0 -> exists y : Z, inlist Z y l1 /\ Mod x y p.
(** * The list of integers between 1 and n (inclusive) *)
Fixpoint until (n : nat) : Zlist :=
match n with
| O => Nil Z
| S n => Cons Z (Z_of_nat (S n)) (until n)
end.

Lemma until_mod_all :
forall (p : nat) (x : Z),
0 < p ->
~ Mod x 0 p -> exists y : Z, inlist Z y (until (pred p)) /\ Mod x y p.