cut (r'a < 2 ^ p). omega. exact r'a_small. omega. apply le_0__Z. apply Zlt_le_weak. exact lt02w. cut (x_i < 2 ^ w). omega.
unfold x_i. apply lt_z__Z. rewrite Zmult_minus_distr_r. rewrite <- Zpower_exp. omega. apply Zle_ge. auto with a. apply Zle_ge.
auto with a. Qed.

(* Specifications: *)
Lemma le_x'_x'max : x' <= 2 ^ (p + w) - 1.
cut (x' < 2 ^ (p + w)). omega. exact lt_x'_x'max. Qed.