rewrite p_M. elim (Zle_lt_or_eq 1 M). intro H. left. apply Zle_trans with (Zlog_sup 2). simpl. omega.
apply Zlog_sup_seq. omega. intro H. right. symmetry. trivial. assumption. Qed.

(* Specifications: *)
Lemma lt_0_p : 0 < p \/ M = 1.
clear x_i_in_Z_ x_i.