intros.
unfold lognat in *.
rewrite (factor_s_conv pf1 pf2).
assert (a <> 0). omega.
rewrite (factor_double_conv H pf1).
simpl.
auto.
Qed.

Lemma double_s : forall a, (S (S (2 * a))) = 2 * (S a).
intros. omega.