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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.
(** * Permutations modulo p. *)
Fixpoint permmod (p : nat) (l1 : Zlist) {struct l1} :
Zlist -> Prop :=
fun l2 : Zlist =>
match l1 with
| Nil => l2 = Nil Z
| Cons x t =>
exists y : Z, inlist Z y l2 /\ Mod x y p /\ permmod p t (zdrop y l2)
end.

Lemma allex_permmod :
forall (p : nat) (l0 l1 : Zlist),
length Z l0 = length Z l1 ->
(forall x0 : Z,
inlist Z x0 l0 ->
exists x1 : Z,
inlist Z x1 l1 /\ Mod x0 x1 p /\ permmod p (zdrop x0 l0) (zdrop x1 l1)) ->
permmod p l0 l1.