Proof using.
intros.
induction ns; simpl in *; [ intuition | idtac ].
break_if; simpl in *.
- break_or_hyp; auto.
concludes.
break_and.
auto.
- concludes.
break_and.
auto.
Qed.

Lemma related_filter_rel :
forall n n' ns,
In n' ns ->
R n n' ->
In n' (filter_rel R_dec n ns).