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 constructive_join_sub_smash {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}: (forall x:A, {identity x}+{~identity x}) -> forall a c : lifted JA, constructive_join_sub (proj1_sig a) (proj1_sig c) -> @constructive_join_sub (option (lifted JA)) _ (Some a) (Some c).