intros a b c'. replace w with (w - p + p). rewrite Zpower_exp. do 2 rewrite Zmult_assoc. do 2 rewrite Zplus_comm with (m := b).
rewrite Z_mod_plus. rewrite Z_mod_plus. trivial. apply Zlt_gt. apply lt_0_Zpow. auto with a. apply Zlt_gt. apply lt_0_Zpow.
auto with a. cut (p < w). omega. auto with a. apply Zle_ge. auto with a. ring. Qed.

(* <KE EP TOGETHER - r'_i_eq *)
Lemma r'_i_eq : _Z r'_i = (h mod M * 2 ^ w + l) / 2 ^ p * 2 ^ p mod M + (h mod M * 2 ^ w + l) mod 2 ^ p.
clear x_i_in_Z_ x_i s_i s'_i s''_i h'_i q_i q'_i y_i h'_i h'_i' x'.