Definition UcmpCmp (x y: nat * nat): bool := unsum_bool (UcmpDec x y).Definition ijcount (i j: nat): monoid -> nat := count (UcmpCmp (i, j)).

(* Specifications: *)
Lemma ijcount_0 i j l: ~ In (i, j) l -> ijcount i j l = 0.
Lemma hom_ijcount i j: monoidHomo monoid NatAddMonoid (ijcount i j).
Lemma ijcount_eq_count i j: ijcount i j = eq_count UcmpDec (i, j).