Proof using.
intros.
rewrite collate_ls_not_in; auto.
apply not_in_not_in_filter_rel.
assumption.
Qed.

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