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 generic_format_F2R :
forall m e,
( m <> 0 -> canonic_exp (F2R (Float beta m e)) <= e )%Z ->
generic_format (F2R (Float beta m e)).