(** * 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 permmod_cons_extend : forall (p : nat) (x1 x2 : Z) (l1 l2 : Zlist), permmod p l1 l2 -> Mod x1 x2 p -> permmod p (Cons Z x1 l1) (Cons Z x2 l2).