Definition canonic_exp x :=
fexp (ln_beta beta x).
Definition canonic (f : float beta) :=
Fexp f = canonic_exp (F2R f).
Definition scaled_mantissa x :=
(x * bpow (- canonic_exp x))%R.
Definition generic_format (x : R) :=
x = F2R (Float beta (Ztrunc (scaled_mantissa x)) (canonic_exp x)).

Theorem mantissa_UP_small_pos :
forall x ex,
(bpow (ex - 1) <= x < bpow ex)%R ->
(ex <= fexp ex)%Z ->
Zceil (x * bpow (- fexp ex)) = 1%Z.