Proof using.
induction ns; intros; auto.
assert (H_neq: a <> n).
intro.
subst.
auto with datatypes.
assert (H_not_in: ~ In n ns).
intro.
intuition.
simpl.
break_if; auto.
simpl.
intro.
break_or_hyp; auto.
intuition eauto.
Qed.

Lemma NoDup_filter_rel:
forall h ns,
NoDup ns ->
NoDup (filter_rel R_dec h ns).