Proof using.
induction ns; intros; simpl in *; auto.
case (A_eq_dec n a); intro.
- subst.
break_if; auto.
intuition.
- break_if; auto.
simpl.
rewrite IHns; auto.
unfold update2.
break_if; auto.
break_and.
subst.
intuition.
Qed.

Lemma collate_ls_not_in_related :
forall ns (f : A -> A -> list B) n h mg,
~ In n ns ->
collate_ls A_eq_dec (filter_rel R_dec h ns) f h mg n h = f n h.