Class Exists_NE :=
  exists_NE : Zeven beta = false \/ forall e,
  ((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\ ((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e).Class Exists_NE :=
  exists_NE : Zeven beta = false \/ forall e,
  ((fexp e < e)%Z -> (fexp (e + 1) < e)%Z) /\ ((e <= fexp e)%Z -> fexp (fexp e + 1) = fexp e).

(* Specifications: *)
Theorem DN_UP_parity_generic_pos : DN_UP_parity_pos_prop.
Theorem DN_UP_parity_generic : DN_UP_parity_prop.
Theorem Rnd_NE_pt_total : round_pred_total Rnd_NE_pt.
Theorem Rnd_NE_pt_monotone : round_pred_monotone Rnd_NE_pt.
Theorem Rnd_NE_pt_round : round_pred Rnd_NE_pt.
Lemma round_NE_pt_pos : forall x, (0 < x)%R -> Rnd_NE_pt x (round beta fexp ZnearestE x).
Theorem round_NE_opp : forall x, round beta fexp ZnearestE (-x) = (- round beta fexp ZnearestE x)%R.
Lemma round_NE_abs: forall x : R, round beta fexp ZnearestE (Rabs x) = Rabs (round beta fexp ZnearestE x).
Theorem round_NE_pt : forall x, Rnd_NE_pt x (round beta fexp ZnearestE x).
Theorem DN_UP_parity_generic_pos : DN_UP_parity_pos_prop.
Theorem DN_UP_parity_generic : DN_UP_parity_prop.
Theorem Rnd_NE_pt_total : round_pred_total Rnd_NE_pt.
Theorem Rnd_NE_pt_monotone : round_pred_monotone Rnd_NE_pt.
Theorem Rnd_NE_pt_round : round_pred Rnd_NE_pt.
Lemma round_NE_pt_pos : forall x, (0 < x)%R -> Rnd_NE_pt x (round beta fexp ZnearestE x).
Theorem round_NE_opp : forall x, round beta fexp ZnearestE (-x) = (- round beta fexp ZnearestE x)%R.
Lemma round_NE_abs: forall x : R, round beta fexp ZnearestE (Rabs x) = Rabs (round beta fexp ZnearestE x).
Theorem round_NE_pt : forall x, Rnd_NE_pt x (round beta fexp ZnearestE x).