Proof using.
induction xs; intros; subst; simpl in *; intuition eauto; subst; eauto.
apply network_network_invariant_sym. eauto.
Qed.

Lemma distinct_pairs_and_app_cons :
forall A R xs p ys l,
l = (xs ++ p :: ys) ->
distinct_pairs_and (A:=A) R l ->
distinct_pairs_and R (xs ++ ys).