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

(** This lemma uses the full power of self_join and eliminates both N and Z. *)
Lemma join_sub_joins_identity {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{DS: Disj_alg A}: forall a b,
join_sub a b -> joins a b -> identity a.