simple induction p; auto.
intros; simpl (nat_log_inf (xI p0)) in |- *; simpl (log_inf (xI p0)) in |- *.
rewrite <- H; apply Znat.inj_S.
intros; simpl (nat_log_inf (xO p0)) in |- *; simpl (log_inf (xO p0)) in |- *.
rewrite <- H; apply Znat.inj_S.
Qed.

Lemma log_sup_log_sup :
forall p : positive, Z_of_nat (nat_log_sup p) = log_sup p.