(** 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 scaled_mantissa_DN : forall x, (0 < round Zfloor x)%R -> scaled_mantissa (round Zfloor x) = Z2R (Zfloor (scaled_mantissa x)).