Notation bpow e := (bpow beta e).

Theorem float_distribution_pos : forall m1 e1 m2 e2 : Z, (0 < m1)%Z -> (F2R (Float beta m1 e1) < F2R (Float beta m2 e2) < F2R (Float beta (m1 + 1) e1))%R -> (e2 < e1)%Z /\ (e1 + ln_beta beta (Z2R m1) = e2 + ln_beta beta (Z2R m2))%Z.