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).