Section RND_prop.
Open Scope R_scope.
Definition Rnd_DN (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_DN_pt F x (rnd x).
Definition Rnd_UP (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_UP_pt F x (rnd x).
Definition Rnd_ZR (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_ZR_pt F x (rnd x).
Definition Rnd_N (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_N_pt F x (rnd x).
Definition Rnd_NG (F : R -> Prop) (P : R -> R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_NG_pt F P x (rnd x).
Definition Rnd_NA (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_NA_pt F x (rnd x).

Theorem round_fun_of_pred :
forall rnd : R -> R -> Prop,
round_pred rnd ->
{ f : R -> R | forall x, rnd x (f x) }.