(** * 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 until_mapmult_permmod :
forall (p : nat) (a : Z),
Prime p ->
~ Mod a 0 p -> permmod p (until (pred p)) (mapmult a (until (pred p))).