Section NESTED_PRED.
Context {cs: compspecs}.
Definition nested_pred (atom_pred: type -> bool): type -> bool :=
type_func
(fun _ => bool)
(fun t => atom_pred t)
(fun t n a b => (atom_pred (Tarray t n a) && b)%bool)
(fun id a bl => (atom_pred (Tstruct id a) && fold_right andb true (decay bl))%bool)
(fun id a bl => (atom_pred (Tunion id a) && fold_right andb true (decay bl))%bool).
Definition nested_fields_pred (atom_pred: type -> bool) (m: members) : bool :=
fold_right (fun it b => (nested_pred atom_pred (field_type (fst it) m) && b)%bool) true m.

Lemma nested_pred_eq: forall atom_pred t,
nested_pred atom_pred t =
match t with
| Tarray t0 _ _ => (atom_pred t && nested_pred atom_pred t0)%bool
| Tstruct id _
| Tunion id _ => (atom_pred t && nested_fields_pred atom_pred (co_members (get_co id)))%bool
| _ => atom_pred t
end.