unfold ijcount.
intros.
apply count_0.
intros.
unfold UcmpCmp.
case_eq (UcmpDec (i, j) x); intros...
elimtype False.
apply H.
rewrite e0...
Qed.

Lemma hom_ijcount i j: monoidHomo monoid NatAddMonoid (ijcount i j).
Proof with auto.