Hint Resolve @None_identity.

Lemma lower_inv: forall {A}{JA: Join A} {PA: Perm_alg A} {psa_A: Pos_alg A} (a b c : option A),
join a b c ->
(a = None /\ b = c) + (a = c /\ b = None) +
({a' : A & {b' : A & {c' : A | a = Some a' /\ b = Some b' /\ c = Some c' /\
join a' b' c'}}}).