Proof using.
intros.
induction ns; auto.
invc_NoDup.
concludes.
simpl in *.
break_if; auto.
apply NoDup_cons; auto.
apply not_in_not_in_filter_rel.
assumption.
Qed.

Lemma filter_rel_self_eq {irreflexive_R : Irreflexive R} :
forall ns0 ns1 h,
filter_rel R_dec h (ns0 ++ h :: ns1) = filter_rel R_dec h (ns0 ++ ns1).