Definition array_pred: forall {A: Type}{d: Inhabitant A} (lo hi: Z) (P: Z -> A -> val -> mpred) (v: list A),
    val -> mpred := @array_pred.Definition struct_pred: forall (m: members) {A: ident * type -> Type} (P: forall it, A it -> val -> mpred) (v: compact_prod (map A m)) (p: val), mpred := @struct_pred.Definition union_pred: forall (m: members) {A: ident * type -> Type} (P: forall it, A it -> val -> mpred) (v: compact_sum (map A m)) (p: val), mpred := @union_pred.Definition array_Prop: forall {A: Type} (d:A) (lo hi: Z) (P: Z -> A -> Prop) (v: list A), Prop := @array_Prop.Definition struct_Prop: forall (m: members) {A: ident * type -> Type} (P: forall it, A it -> Prop) (v: compact_prod (map A m)), Prop := @struct_Prop.Definition union_Prop: forall (m: members) {A: ident * type -> Type} (P: forall it, A it -> Prop) (v: compact_sum (map A m)), Prop := union_Prop.Definition array_pred_len_0: forall {A}{d: Inhabitant A} lo hi P p,
  hi = lo ->
  array_pred lo hi P nil p = emp
:= @array_pred_len_0.Definition array_pred_len_1: forall {A}{d: Inhabitant A} i P v p,
  array_pred i (i + 1) P (v :: nil) p =  P i v p
:= @array_pred_len_1.Definition split_array_pred: forall  {A}{d: Inhabitant A} lo mid hi P v p,
  lo <= mid <= hi ->
  Zlength v = (hi-lo) ->
  array_pred lo hi P v p =
  array_pred lo mid P (sublist 0 (mid-lo) v) p *
  array_pred mid hi P (sublist (mid-lo) (hi-lo) v) p
:= @split_array_pred.Definition array_pred_shift: forall {A} {d: Inhabitant A} lo hi lo' hi' mv 
              P' P v p,
  lo - lo' = mv ->
  hi - hi' = mv ->
  (forall i i', lo <= i < hi -> i - i' = mv ->
           P' i' (Znth (i - lo) v) p = P i (Znth (i - lo) v) p) ->
  array_pred lo' hi' P' v p = array_pred lo hi P v p
:= @array_pred_shift.Definition array_pred_ext:
  forall {A B} {dA: Inhabitant A} {dB: Inhabitant B} lo hi P0 P1 (v0: list A) (v1: list B)  p,
  Zlength v0 = Zlength v1 ->
  (forall i, lo <= i < hi ->
     P0 i (Znth (i - lo) v0) p = P1 i (Znth (i - lo) v1) p) ->
  array_pred lo hi P0 v0 p = array_pred lo hi P1 v1 p
:= @array_pred_ext.Definition at_offset_array_pred: forall {A} {d: Inhabitant A} lo hi P v ofs p,
  at_offset (array_pred lo hi P v) ofs p = array_pred lo hi (fun i v => at_offset (P i v) ofs) v p
:= @at_offset_array_pred.Definition array_pred_sepcon: forall  {A} {d: Inhabitant A} lo hi P Q v p,
  array_pred lo hi P v p * array_pred lo hi Q v p = array_pred lo hi (P * Q) v p
:= @array_pred_sepcon.Definition struct_pred_ext:
  forall m {A0 A1} (P0: forall it, A0 it -> val -> mpred) (P1: forall it, A1 it -> val -> mpred) v0 v1 p,
  members_no_replicate m = true ->
  (forall i d0 d1, in_members i m ->
     P0 _ (proj_struct i m v0 d0) p = P1 _ (proj_struct i m v1 d1) p) ->
  struct_pred m P0 v0 p = struct_pred m P1 v1 p
:= @struct_pred_ext.Definition at_offset_struct_pred: forall m {A} (P: forall it, A it -> val -> mpred) v p ofs,
  at_offset (struct_pred m P v) ofs p = struct_pred m (fun it v => at_offset (P it v) ofs) v p
:= @at_offset_struct_pred.Definition andp_struct_pred: forall m {A} (P: forall it, A it -> val -> mpred) v p Q,
  corable Q ->
  Q && struct_pred m P v p =
  match m with
  | nil => Q && emp
  | _ => struct_pred m (fun it v p => Q && P it v p) v p
  end
:= @corable_andp_struct_pred.Definition struct_pred_sepcon: forall m {A} (P Q: forall it, A it -> val -> mpred) v p,
  struct_pred m P v p * struct_pred m Q v p = struct_pred m (fun it => P it * Q it) v p
:= @struct_pred_sepcon.Definition union_pred_ext:
  forall m {A0 A1} (P0: forall it, A0 it -> val -> mpred) (P1: forall it, A1 it -> val -> mpred) v0 v1 p,
  members_no_replicate m = true ->
  (forall it, members_union_inj v0 it <-> members_union_inj v1 it) ->
  (forall i d0 d1, members_union_inj v0 (i, field_type i m) -> members_union_inj v1 (i, field_type i m) ->
     P0 _ (proj_union i m v0 d0) p = P1 _ (proj_union i m v1 d1) p) ->
  union_pred m P0 v0 p = union_pred m P1 v1 p
:= @union_pred_ext.Definition at_offset_union_pred: forall m {A} (P: forall it, A it -> val -> mpred) v p ofs,
  at_offset (union_pred m P v) ofs p = union_pred m (fun it v => at_offset (P it v) ofs) v p
:= at_offset_union_pred.Definition andp_union_pred: forall m {A} (P: forall it, A it -> val -> mpred) v p Q,
  Q && union_pred m P v p =
  match m with
  | nil => Q && emp
  | _ => union_pred m (fun it v p => Q && P it v p) v p
  end
:= @andp_union_pred.Definition union_pred_sepcon: forall m {A} (P Q: forall it, A it -> val -> mpred) v p,
  union_pred m P v p * union_pred m Q v p = union_pred m (fun it => P it * Q it) v p
:= @union_pred_sepcon.Definition struct_Prop_compact_prod_gen: forall m (F: ident * type -> Type) (P: forall it, F it -> Prop) (f: forall it, F it),
  members_no_replicate m = true ->
  (forall i, in_members i m -> P (i, field_type i m) (f (i, field_type i m))) ->
  struct_Prop m P (compact_prod_gen f m)
:= @struct_Prop_compact_prod_gen.Definition struct_Prop_proj: forall m (F: ident * type -> Type) (P: forall it, F it -> Prop) v i d,
  in_members i m ->
  struct_Prop m P v ->
  P (i, field_type i m) (proj_struct i m v d)
:= @struct_Prop_proj.Definition union_Prop_compact_sum_gen: forall m (F: ident * type -> Type) (P: forall it, F it -> Prop) (f: forall it, F it),
  members_no_replicate m = true ->
  (forall i, in_members i m -> P (i, field_type i m) (f (i, field_type i m))) ->
  union_Prop m P (compact_sum_gen (fun _ => true) f m)
:= @union_Prop_compact_sum_gen.Definition union_Prop_proj: forall m (F: ident * type -> Type) (P: forall it, F it -> Prop) v i d,
  members_union_inj v (i, field_type i m) ->
  union_Prop m P v ->
  P (i, field_type i m) (proj_union i m v d)
:= @union_Prop_proj.Definition struct_data_at_rec_aux (m m0: members) (sz: Z) (P: ListType (map (fun it => reptype (field_type (fst it) m0) -> (val -> mpred)) m)) (v: compact_prod (map (fun it => reptype (field_type (fst it) m0)) m)) : (val -> mpred).Definition union_data_at_rec_aux (m m0: members) (sz: Z) (P: ListType (map (fun it => reptype (field_type (fst it) m0) -> (val -> mpred)) m)) (v: compact_sum (map (fun it => reptype (field_type (fst it) m0)) m)) : (val -> mpred).

(* Specifications: *)
Lemma struct_data_at_rec_aux_spec: forall m m0 sz v P, struct_data_at_rec_aux m m0 sz (ListTypeGen (fun it => reptype (field_type (fst it) m0) -> val -> mpred) P m) v = struct_pred m (fun it v => withspacer sh (field_offset cenv_cs (fst it) m0 + sizeof (field_type (fst it) m0)) (field_offset_next cenv_cs (fst it) m0 sz) (at_offset (P it v) (field_offset cenv_cs (fst it) m0))) v.
Lemma union_data_at_rec_aux_spec: forall m m0 sz v P, union_data_at_rec_aux m m0 sz (ListTypeGen (fun it => reptype (field_type (fst it) m0) -> val -> mpred) P m) v = union_pred m (fun it v => withspacer sh (sizeof (field_type (fst it) m0)) sz (P it v)) v.