Definition zero_in_Z_ := exist (in_Z_ (2 ^ w)) 0 zero_is_in_Z_.

Lemma Zpow_2_w_is_2Zpow_2_wm1 : 2 ^ w = 2 ^ (w - 1) + 2 ^ (w - 1).
transitivity (2 ^ (1 + (w - 1))). rewrite Zplus_minus. trivial. rewrite Zpower_exp.