(** * 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.