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 split_join_bits :
forall s m e,
(0 <= m < Zpower 2 mw)%Z ->
(0 <= e < Zpower 2 ew)%Z ->
split_bits (join_bits s m e) = (s, m, e).