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

(** A unit for an element is a unit for itself (is idempotent). *)
Lemma unit_self_unit {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A} :
forall a ea,   unit_for ea a ->  unit_for ea ea.