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).
Definition Rnd_NG_pt_unique_prop F P :=
forall x d u,
Rnd_DN_pt F x d -> Rnd_N_pt F x d ->
Rnd_UP_pt F x u -> Rnd_N_pt F x u ->
P x d -> P x u -> d = u.

Theorem Rnd_NA_NG_pt :
forall F : R -> Prop,
F 0 ->
forall x f,
Rnd_NA_pt F x f <-> Rnd_NG_pt F (fun x f => Rabs x <= Rabs f) x f.