induction a; intros.
assert (n = 1).
apply (lognat_0_1 pf1).
auto.
subst.
unfold lognat.
simpl.
auto.
destruct (nat_comp n).
destruct H0.
subst.
assert (x <> 0). omega.
rewrite <- (lognat_prod_sum H0) in H.
inversion H.
subst.
rewrite (lognat_odd _ pf1).
rewrite <- (lognat_prod_sum H0).
auto.
subst.
destruct (eq_nat_dec x 0).
subst.
unfold lognat in *.
simpl in *.
omega.
generalize pf2.
assert (S (S (2 * x)) = 2 * (S x)). omega.
rewrite H0.
intros.
assert (S x <> 0). omega.
assert (2 * x <> 0). omega.
rewrite (lognat_odd _ H2) in H.
rewrite <- (lognat_prod_sum n) in H.
inversion H.
rewrite <- (lognat_prod_sum H1).
specialize (IHa x n H1 H4).
destruct IHa.
subst.
rewrite H4.
auto.
subst.
right.
rewrite H3.
auto.
Qed.

Lemma lognat_succ : forall n (pf1 : n <> 0)(pf2: (S n) <> 0),
lognat pf2 = lognat pf1 \/
lognat pf2 = S (lognat pf1).