Proof using.
intros.
in_crush; destruct x; simpl in *; eauto.
Qed.

(* NB: generalizing this over R makes it useless to [eauto using] because the conclusion is too general. *)
Lemma nw_nw_distinct_pairs_and_elim :
forall xs a b ys l,
l = xs ++ a :: ys ->
In b (xs ++ ys) ->
distinct_pairs_and network_network_invariant l ->
network_network_invariant a b.