Definition exists_arbitrarily_large (P: DS_set I -> Prop) :=
  forall i:DS_set I, exists j:DS_set I,
  DS_ord i j /\ P j.

(* Specifications: *)
Lemma not_eal_eventually_not: forall (P: DS_set I -> Prop), ~ exists_arbitrarily_large P -> eventually (fun i:DS_set I => ~ P i).
Lemma not_eventually_eal_not: forall (P: DS_set I -> Prop), ~ eventually P -> exists_arbitrarily_large (fun i:DS_set I => ~ P i).