Definition Znearest x :=
  match Rcompare (x - Z2R (Zfloor x)) (/2) with
  | Lt => Zfloor x
  | Eq => if choice (Zfloor x) then Zceil x else Zfloor x
  | Gt => Zceil x
  end.

Theorem Znearest_ge_floor : forall x, (Zfloor x <= Znearest x)%Z.