intros n H H0. apply P'foldlz. rewrite Zdiv_small_0. simpl. unfold P'. split. left.
replace (_Z (x_ (S n))) with 0. replace (0 mod M) with 0. replace (0 * 2 ^ w + 0) with 0. rewrite Zdiv_small_0.
replace (0 * 2 ^ p) with 0. rewrite modred. trivial. omega. omega. ring. omega. apply lt_0_Zpow. unfold p. unfold p_in_Z_w.
apply le_0_p. trivial. symmetry. apply modred. omega. omega. transitivity (x / 2 ^ (Z_of_nat (S n) * w) mod 2 ^ w).
rewrite Zdiv_small_0. symmetry. apply modred. omega. exact lt02w. assumption. assumption. trivial.
replace (_Z (x_ (S n))) with 0. replace (0 mod 2 ^ p) with 0. simpl. symmetry. apply modred. omega. exact lt02w. symmetry.
apply modred. omega. apply lt_0_Zpow. unfold p. unfold p_in_Z_w. apply le_0_p.
transitivity (x / 2 ^ (Z_of_nat (S n) * w) mod 2 ^ w). rewrite Zdiv_small_0. symmetry. apply modred. omega. exact lt02w.
assumption. assumption. trivial. assumption. apply Zlt_trans with (2 ^ (Z_of_nat (S n) * w)). assumption. apply Zlt_pow_lt.
apply Zmult_le_0_compat. omega. exact le_0_w. apply Zmult_gt_0_lt_compat_r. apply Zlt_gt. exact lt_0_w. rewrite inj_S. omega.
Qed.

(* Specifications: *)
Lemma P_O :
0 <= x -> x < 2 ^ (Z_of_nat O * w) ->
P' M p (fst (foldlz f' (zero_in_Z_, zero_in_Z_) O))
(snd (foldlz f' (zero_in_Z_, zero_in_Z_) O)) (x / 2 ^ w) (x_ 0).