Section Fcore_ulp.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable fexp : Z -> Z.
(** Definition and basic properties about the minimal exponent, when it exists *)
(** [negligible_exp] is either none (as in FLX) or Some n such that n <= fexp n. *)
Definition negligible_exp: option Z :=
match (LPO_Z _ (fun z => Z_le_dec_aux z (fexp z))) with
| inleft N => Some (proj1_sig N)
| inright _ => None
end.
Inductive negligible_exp_prop: option Z -> Prop :=
| negligible_None: (forall n, (fexp n < n)%Z) -> negligible_exp_prop None
| negligible_Some: forall n, (n <= fexp n)%Z -> negligible_exp_prop (Some n).
Context { valid_exp : Valid_exp fexp }.
(** Definition and basic properties about the ulp *)
(** Now includes a nice ulp(0): ulp(0) is now 0 when there is no minimal exponent, such as in FLX, and beta^(fexp n) when there is a n such that n <= fexp n. For instance, the value of ulp(O) is then beta^emin in FIX and FLT. The main lemma to use is ulp_neq_0 that is equivalent to the previous "unfold ulp" provided the value is not zero. *)
Definition ulp x := match Req_bool x 0 with
| true   => match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end
| false  => bpow (cexp beta fexp x)
end.
Notation F := (generic_format beta fexp).

Theorem round_UP_DN_ulp :
forall x, ~ F x ->
round beta fexp Zceil x = (round beta fexp Zfloor x + ulp x)%R.