proposition: Theorem Zceil_floor_neq : forall x : R, (IZR (Zfloor x) <> x)%R -> (Zceil x = Zfloor x + 1)%Z.
proof: Proof. intros x Hx. apply Zceil_imp. split. ring_simplify (Zfloor x + 1 - 1)%Z. apply Rnot_le_lt. intros H. apply Hx. apply Rle_antisym. apply Zfloor_lb. exact H. apply Rlt_le. rewrite plus_IZR. apply Zfloor_ub. Qed.
imports: Coq(Psatz Reals ZArith) Zaux