(** Rounding x + small epsilon *)

(* Specifications: *)
Theorem ln_beta_plus_eps:
forall x, (0 < x)%R -> F x ->
forall eps, (0 <= eps < ulp x)%R ->
ln_beta beta (x + eps) = ln_beta beta x :> Z.
Theorem round_DN_plus_eps_pos:
forall x, (0 <= x)%R -> F x ->
forall eps, (0 <= eps < ulp x)%R ->
round beta fexp Zfloor (x + eps) = x.
Theorem round_UP_plus_eps_pos :
forall x, (0 <= x)%R -> F x ->
forall eps, (0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R.
Theorem ln_beta_plus_eps:
forall x, (0 < x)%R -> F x ->
forall eps, (0 <= eps < ulp x)%R ->
ln_beta beta (x + eps) = ln_beta beta x :> Z.
Theorem round_DN_plus_eps_pos:
forall x, (0 <= x)%R -> F x ->
forall eps, (0 <= eps < ulp x)%R ->
round beta fexp Zfloor (x + eps) = x.
Theorem round_UP_plus_eps_pos :
forall x, (0 <= x)%R -> F x ->
forall eps, (0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R.