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

Lemma join_sub_to_joining {A} {J : Join A}
{_ : Perm_alg A} {_ : Sep_alg A} {_ : Canc_alg A} {_ : Disj_alg A}
(a b e : A) :
join_sub e a ->
join_sub e b ->
joins a b ->
identity e.