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

(* Specifications: *)
Lemma not_in_not_in_filter_rel :
forall ns n h,
~ In n ns ->
~ In n (filter_rel R_dec h ns).