Definition split_bits x :=
let mm := Zpower 2 mw in
let em := Zpower 2 ew in
(Zle_bool (mm * em) x, Zmod x mm, Zmod (Zdiv x mm) em)%Z.

Theorem join_split_bits :
forall x,
(0 <= x < Zpower 2 (mw + ew + 1))%Z ->
let '(s, m, e) := split_bits x in
join_bits s m e = x.