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

(** If something is an identity and it joins with an element then it is a unit for that element. *)
Lemma identity_unit {A} `{HA: Perm_alg A}: forall e a,
identity e ->
joins e a ->
unit_for e a.