(* KEEP TOGETHER> *)
fold (0 * 0). apply Zmult_le_compat. auto with a. apply Zlt_le_weak. apply lt_0_Zpow. auto with a. omega.
omega. apply Zlt_le_trans with (2 ^ p * 2 ^ (w - p)). apply Zmult_lt_compat_r. auto with a.
apply lt_0_Zpow. auto with a. assumption. rewrite <- Zpower_exp. replace (p + (w - p)) with w. omega.
ring. apply Zle_ge. auto with a. apply Zle_ge. auto with a. rewrite H in p_M. simpl in p_M.
rewrite p_M in v_M. replace (2 ^ 0) with 1 in v_M. cut (v = 0). intro H0. unfold x. rewrite H0.
replace (2 ^ t * 0) with 0. simpl. rewrite Zdiv_small_0. simpl. apply modred. omega. auto with a.
auto with a. auto with a. ring. cut (0 <= v). omega. unfold v. apply le_0__Z. ring. Qed.

(* <KEEP TOGETHER - h'_eq *)
Lemma h'_eq : _Z h' = x / 2 ^ p.
unfold h'. rewrite Z_from_plusw. rewrite h_eq. simpl. fold u p. rewrite l_eq. rewrite <- Zdiv_times_plus.