induction b.
intros. simpl in *. auto.
intros.
specialize (lognat_prod_sum (expnat_2_nz b)). intros.
specialize (IHb (expnat_2_nz b)).
rewrite IHb in H.
symmetry in H.
simpl in *.
apply H.
Qed.

Lemma fold_add : forall a,
a + (a + 0) = 2*a.