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)).