auto with a. unfold x_i. apply le_0__Z. Qed.
Local Hint Resolve le_0_x' : a.

Lemma lt_x'_x'max : x' < 2 ^ (p + w).
unfold x'. apply Zle_lt_trans with ((2 ^ p - 1) * 2 ^ w + (2 ^ w - 1)). apply Zplus_le_compat. apply Zmult_le_compat.