Section Float_prop.
Variable beta : radix.
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 + mag beta (IZR m1) = e2 + mag beta (IZR m2))%Z.