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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.

(* Specifications: *)
Lemma nodoubles_nil : forall p : nat, nodoubles p (Nil Z).
Lemma nodoubles_drop :
forall (p : nat) (l : Zlist) (x : Z),
nodoubles p l -> nodoubles p (zdrop x l).
Lemma nodoubles_ind :
forall (p : nat) (h : Z) (t : Zlist),
(forall x : Z, inlist Z x t -> ~ Mod h x p) ->
nodoubles p t -> nodoubles p (Cons Z h t).