Definition valid_2 a b := exists c, join a b c /\ valid c.Definition fp_update_ND a B := forall c, valid_2 a c -> exists b, B b /\ valid_2 b c.Definition fp_update a b := forall c, valid_2 a c -> valid_2 b c.

Lemma fp_update_equiv: forall a b, fp_update a b <-> fp_update_ND a (eq b).