Definition age1_list {A} `{ageable A} := list_forall2 age.

Lemma age1_list_join {A} {JA: Join A}{PA: Perm_alg A}{agA: ageable A}{AgeA: Age_alg A}: forall l (phi phi' phi2: A), age phi phi' -> list_join phi l phi2 -> exists l', exists phi2', age1_list l l' /\ age phi2 phi2' /\ list_join phi' l' phi2'.