proposition: Theorem Znearest_ge_floor : forall x, (Zfloor x <= Znearest x)%Z.
proof: Proof. intros x. destruct (Znearest_DN_or_UP x) as [Hx|Hx] ; rewrite Hx. apply Z.le_refl. apply le_IZR. apply Rle_trans with x. apply Zfloor_lb. apply Zceil_ub. Qed.
imports: Coq(ZArith Reals Lia) Zaux Raux Defs Round_pred Float_prop