auto with a. apply Zle_lt_trans with ((2 ^ (p + w) - 1) / 2 ^ p). apply Zdiv_le. apply lt_0_Zpow. auto with a.
exact le_x'_x'max. rewrite Zplus_comm. rewrite Zpower_exp. rewrite Zdiv_mult_minus. omega. apply lt_0_Zpow. auto with a.
omega. cut (0 < 2 ^ p). omega. apply lt_0_Zpow. auto with a. apply Zle_ge. auto with a. apply Zle_ge. auto with a. Qed.

(* <KE EP TOGETHER - q_i_eq_ch *)
Lemma q_i_eq_ch :
M = 1 \/ _Z q_i = x' / 2 ^ p * (2 ^ (p + w) / M mod 2 ^ w) / 2 ^ w.