Fixpoint reptype (t: type): Type :=
match t with
| Tint => val
| Tstruct _ fs => reptype_fieldlist fs
end
with reptype_fieldlist (fs: fieldlist) : Type :=
match fs with
| Fnil => unit
| Fcons (_, t) fs_tl => prod (reptype t) (reptype_fieldlist fs_tl)
end.
Fixpoint default_val (t: type): reptype t :=
match t as t_PAT return reptype t_PAT with
| Tint => Vundef
| Tstruct _ fs => default_val_fieldlist fs
end
with default_val_fieldlist (fs: fieldlist) : reptype_fieldlist fs :=
match fs as fs_PAT return reptype_fieldlist fs_PAT with
| Fnil => tt
| Fcons (_, t) fs_tl => (default_val t, default_val_fieldlist fs_tl)
end.
Open Scope logic.
Fixpoint data_at_rec (t: type)
(v: reptype t) : val -> Pred :=
match t as t_PAT
return reptype t_PAT -> val -> Pred with
| Tint => fun v p =>
mapsto p v
| Tstruct _ fs => fun v p =>
data_at_rec_fieldlist fs v fs p
end v
with data_at_rec_fieldlist (fs: fieldlist)
(v: reptype_fieldlist fs) (full_fs: fieldlist): val -> Pred :=
match fs as fs_PAT
return reptype_fieldlist fs_PAT -> val -> Pred with
| Fnil => fun v p =>
emp
| Fcons (id, t) fs_tl => fun v p=>
data_at_rec t (fst v)
(offset_val (field_offset id full_fs) p)*
data_at_rec_fieldlist fs_tl (snd v) full_fs p
end v.
Fixpoint nested_pred (atom_pred: type -> Prop) (t: type): Prop :=
match t with
| Tint => atom_pred t
| Tstruct _ fs => atom_pred t /\ nested_pred_fieldlist atom_pred fs
end
with nested_pred_fieldlist (atom_pred: type -> Prop) (fs: fieldlist): Prop :=
match fs with
| Fnil => True
| Fcons (i0, t0) fs0 => nested_pred atom_pred t0 /\ nested_pred_fieldlist atom_pred fs0
end.

Lemma nested_pred_atom_pred: forall atom_pred t, nested_pred atom_pred t -> atom_pred t.