Hint Resolve @join_joins @join_joins' @join_join_sub @join_join_sub'.
Definition nonidentity {A} `{Perm_alg A} (a: A) := ~(identity a).

(** Sometimes it is useful to use a negative form of the previous lemma *)
Lemma join_overlap {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{DS: Disj_alg A}: forall a b,
join_sub a b -> nonidentity a -> overlap a b.