Proof using.
intros.
rewrite collate_ls_not_in; auto.
apply not_in_not_in_filter_rel.
intro.
contradict H.
eapply in_remove_all_was_in; eauto.
Qed.

(* Specifications: *)
Lemma collate_ls_in_remove_all :
forall mg n h ns (f : A -> A -> list B) 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.