(* Lemma union_default_filter_legal: forall m, m <> nil -> legal_compact_sum_filter (union_default_filter m) m = true. *)
Definition union_default_val (m : members) := compact_sum_gen (fun it => true) (fun it => default_val (field_type (fst it) m)) m.

Lemma compact_sum_sigT_compact_sum_gen:
forall {A B} {P: A -> Type} (genT: B -> A) (genV: forall b: B, P (genT b)) (filter: B -> bool) (gen: B -> sigT P) (l: list B),
(forall b, gen b = existT P (genT b) (genV b)) ->
is_default_filter filter l ->
JMeq (compact_sum_sigT_value (map gen l)) (compact_sum_gen filter genV l).