Hint Resolve @join_sub_refl.

(* Specifications: *)
Lemma join_sub_trans {A} {J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall a b c,
join_sub a b ->
join_sub b c ->
join_sub a c.
Lemma join_sub_unit_for {A} {J: Join A}{PA: Perm_alg A}:
forall e a b, unit_for e a -> join_sub a b -> unit_for e b.
Lemma join_sub_same_identity {A} {J: Join A}{PA: Perm_alg A}:
forall e e' a c,  identity e -> unit_for e a -> identity e' -> unit_for e' c ->
join_sub a c -> e = e'.
Lemma join_sub_joins {A} `{HA: Perm_alg A}: forall {a b},
join_sub a b -> joins a b -> joins a a.