(** IEEE-754 roundings: up, down and to zero *)

Global Instance valid_rnd_DN : Valid_rnd Zfloor.Global Instance valid_rnd_UP : Valid_rnd Zceil.Global Instance valid_rnd_ZR : Valid_rnd Ztrunc.Global Instance valid_rnd_AW : Valid_rnd Zaway.

Theorem ln_beta_round : forall rnd {Hrnd : Valid_rnd rnd} x, (round rnd x <> 0)%R -> (ln_beta beta (round rnd x) = ln_beta beta x :> Z) \/ Rabs (round rnd x) = bpow (Zmax (ln_beta beta x) (fexp (ln_beta beta x))).