(** Lemmas common to both parts of the progress/preservation simulation results *)

Lemma join_sub_join {A} {J : Join A}
{PA : Perm_alg A} {SA : Sep_alg A} {_ : Canc_alg A} {DA : Disj_alg A} {CA : Cross_alg A}
(a b c x : A) :
join a b c ->
join_sub a x ->
join_sub b x ->
join_sub c x.