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_split_fpm : forall A B (JB: Join B) (PB: Perm_alg B)(SB : Sep_alg B)(CB: Disj_alg B) (Bdec: forall x:B, {identity x}+{~identity x}) , Cross_alg B -> Cross_alg (fpm A (lifted JB)) .