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).

Lemma round_NE_abs: forall x : R, round beta fexp ZnearestE (Rabs x) = Rabs (round beta fexp ZnearestE x).