elim (Zle_lt_or_eq 1 M). intro H. replace (2 ^ p) with (1 * M + (2 ^ p - M)). apply Zdiv_mult_plus. omega.
cut (M <= 2 ^ p). omega. auto with a. cut (2 ^ p < M + M). omega.
apply Zlt_le_trans with (2 ^ (p - 1) + 1 + (2 ^ (p - 1) + 1)).
replace (2 ^ (p - 1) + 1 + (2 ^ (p - 1) + 1)) with (2 ^ (p - 1) + 2 ^ (p - 1) + 2). rewrite Z_pow_plus.
replace (p - 1 + 1) with p. omega. ring. elim le_0_pm1. omega. intro H0. absurd (1 < M). omega.
assumption. ring. cut (2 ^ (p - 1) < M). omega. elim lt_2pm1_M. omega. intro H0. absurd (1 < M). omega.
assumption. ring. intro H0. rewrite p_M. rewrite <- H0. trivial. assumption. Qed.

Lemma large_M'_eq : 2 ^ w + (2 ^ (p + w) / M) mod 2 ^ w = 2 ^ (p + w) / M.
transitivity (2 ^ (p + w) / M / 2 ^ w * 2 ^ w + (2 ^ (p + w) / M) mod 2 ^ w).