clear x_i_in_Z_ x_i s_i s'_i s''_i h'_i q_i q'_i y_i h'_i' x'.
replace (h mod M * 2 ^ w) with (0 + h mod M * 2 ^ (w - p + p)). rewrite Zpower_exp. rewrite Zmult_assoc.
rewrite Zmod_plus_distr_l with (m := 2 ^ p). rewrite Z_mod_plus. rewrite modred with (x := 0).
replace (0 + l) with l. exact r_Si_eq_lit. ring. omega. apply lt_0_Zpow. auto with a.
apply Zlt_gt. apply Zlt_le_trans with M. omega. auto with a. apply lt_0_Zpow. auto with a. cut (p < w). omega. auto with a.
apply Zle_ge. auto with a. replace (w - p + p) with w. ring. ring. Qed.

Lemma r_Si_eq_b :
_Z r_Si = ( d_Si + ((h mod M + M) * 2 ^ w + l) mod 2 ^ p) mod 2 ^ w.