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_bits_inj :
forall x y,
(0 <= x < Zpower 2 (mw + ew + 1))%Z ->
(0 <= y < Zpower 2 (mw + ew + 1))%Z ->
split_bits x = split_bits y ->
x = y.