Instance Cross_prod : forall A B saA saB,
@Cross_alg A saA ->
@Cross_alg B saB ->
@Cross_alg (A * B) (Join_prod _ saA _ saB).

Lemma Cross_smash : forall A (JA: Join A) {PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A},
(forall x:A, {identity x}+{~identity x}) ->
Cross_alg A ->
Cross_alg (option (lifted JA)).