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

(** Identities are exactly units for themselves (are idempotent).*)
Lemma identity_unit_equiv {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}:
forall a,  identity a <-> unit_for a a.