Definition nonunit {A} `{Join A}  (a: A) := forall x, ~ unit_for a x.

(* Specifications: *)
Lemma nonidentity_nonunit {A} {JA: Join A} {PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}: forall {a}, nonidentity a -> nonunit a.
Lemma nonunit_nonidentity {A}{JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall x, nonunit x -> ~identity x.