Definition Rnd_NG_pt_unicity_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 round_pred_lt_0 : forall P : R -> R -> Prop, round_pred_monotone P -> P 0 0 -> forall x f, P x f -> f < 0 -> x < 0.