Definition NG_existence_prop (F : R -> Prop) (P : R -> R -> Prop) :=
  forall x d u, ~F x -> Rnd_DN_pt F x d -> Rnd_UP_pt F x u -> P x u \/ P x d.

(* Specifications: *)
Theorem satisfies_any_imp_NG : forall (F : R -> Prop) (P : R -> R -> Prop), satisfies_any F -> NG_existence_prop F P -> round_pred_total (Rnd_NG_pt F P).
Theorem satisfies_any_imp_NA : forall F : R -> Prop, satisfies_any F -> round_pred (Rnd_NA_pt F).
Theorem satisfies_any_imp_NG : forall (F : R -> Prop) (P : R -> R -> Prop), satisfies_any F -> NG_existence_prop F P -> round_pred_total (Rnd_NG_pt F P).
Theorem satisfies_any_imp_NA : forall F : R -> Prop, satisfies_any F -> round_pred (Rnd_NA_pt F).