(** 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 round_DN_small_pos : forall x ex, (bpow (ex - 1) <= x < bpow ex)%R -> (ex <= fexp ex)%Z -> round Zfloor x = 0%R.