(** ensures a real number can always be rounded *)
Inductive satisfies_any (F : R -> Prop) :=
  Satisfies_any :
    F 0 -> ( forall x : R, F x -> F (-x) ) ->
    round_pred_total (Rnd_DN_pt F) -> satisfies_any F.(** ensures a real number can always be rounded *)
Inductive satisfies_any (F : R -> Prop) :=
  Satisfies_any :
    F 0 -> ( forall x : R, F x -> F (-x) ) ->
    round_pred_total (Rnd_DN_pt F) -> satisfies_any F.

Theorem satisfies_any_imp_DN : forall F : R -> Prop, satisfies_any F -> round_pred (Rnd_DN_pt F).