Inductive efield : Type :=
  | eArraySubsc: forall i: expr, efield
  | eStructField: forall i: ident, efield
  | eUnionField: forall i: ident, efield.Inductive efield : Type :=
  | eArraySubsc: forall i: expr, efield
  | eStructField: forall i: ident, efield
  | eUnionField: forall i: ident, efield.Inductive efield : Type :=
  | eArraySubsc: forall i: expr, efield
  | eStructField: forall i: ident, efield
  | eUnionField: forall i: ident, efield.Inductive efield : Type :=
  | eArraySubsc: forall i: expr, efield
  | eStructField: forall i: ident, efield
  | eUnionField: forall i: ident, efield.Fixpoint nested_efield (e: expr) (efs: list efield) (tts: list type) : expr :=
  match efs, tts with
  | nil, _ => e
  | _, nil => e
  | cons ef efs', cons t0 tts' =>
    match ef with
    | eArraySubsc ei => Ederef (Ebinop Cop.Oadd (nested_efield e efs' tts') ei (tptr t0)) t0
    | eStructField i => Efield (nested_efield e efs' tts') i t0
    | eUnionField i => Efield (nested_efield e efs' tts') i t0
    end
  end.Inductive array_subsc_denote {cs: compspecs}: expr -> Z -> environ -> Prop :=
  | array_subsc_denote_intro_int:
      forall e i rho, Vint (Int.repr i) = eval_expr e rho -> array_subsc_denote e i rho
  | array_subsc_denote_intro_long:
      forall e i rho, Vlong (Int64.repr i) = eval_expr e rho -> array_subsc_denote e i rho.Inductive array_subsc_denote {cs: compspecs}: expr -> Z -> environ -> Prop :=
  | array_subsc_denote_intro_int:
      forall e i rho, Vint (Int.repr i) = eval_expr e rho -> array_subsc_denote e i rho
  | array_subsc_denote_intro_long:
      forall e i rho, Vlong (Int64.repr i) = eval_expr e rho -> array_subsc_denote e i rho.Inductive array_subsc_denote {cs: compspecs}: expr -> Z -> environ -> Prop :=
  | array_subsc_denote_intro_int:
      forall e i rho, Vint (Int.repr i) = eval_expr e rho -> array_subsc_denote e i rho
  | array_subsc_denote_intro_long:
      forall e i rho, Vlong (Int64.repr i) = eval_expr e rho -> array_subsc_denote e i rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Inductive efield_denote {cs: compspecs}: list efield -> list gfield -> environ -> Prop :=
  | efield_denote_nil: forall rho, efield_denote nil nil rho
  | efield_denote_ArraySubsc_int: forall ei efs i gfs rho,
       match typeconv (typeof ei) with
       | Tint _ Signed _ => Int.min_signed <= i <= Int.max_signed
       | Tint _ Unsigned _ => 0 <= i <= Int.max_unsigned
       | _ => False
       end ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_ArraySubsc: forall ei efs i gfs rho,
      is_ptrofs_type (typeof ei) = true ->
      array_subsc_denote ei i rho ->
      efield_denote efs gfs rho ->
      efield_denote (eArraySubsc ei :: efs) (ArraySubsc i :: gfs) rho
  | efield_denote_StructField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eStructField i :: efs) (StructField i :: gfs) rho
  | efield_denote_UnionField: forall i efs gfs rho,
      efield_denote efs gfs rho ->
      efield_denote (eUnionField i :: efs) (UnionField i :: gfs) rho.Fixpoint typecheck_efield {cs: compspecs} (Delta: tycontext) (efs: list efield) : tc_assert :=
  match efs with
  | nil => tc_TT
  | eArraySubsc ei :: efs' =>
    tc_andp (typecheck_expr Delta ei) (typecheck_efield Delta efs')
  | eStructField i :: efs' =>
    typecheck_efield Delta efs'
  | eUnionField i :: efs' =>
    typecheck_efield Delta efs'
  end.Definition tc_efield {cs: compspecs} (Delta: tycontext) (efs: list efield) : environ -> mpred := denote_tc_assert (typecheck_efield Delta efs).Definition typeconv' (ty: type): type :=
match ty with
| Tvoid => remove_attributes ty
| Tint I8 _ _ => Tint I32 Signed noattr
| Tint I16 _ _ => Tint I32 Signed noattr
| Tint I32 _ _ => remove_attributes ty
| Tint IBool _ _ => Tint I32 Signed noattr
| Tlong _ _ => remove_attributes ty
| Tfloat _ _ => remove_attributes ty
| Tpointer _ _ => if eqb_type ty int_or_ptr_type then ty else remove_attributes ty
| Tarray t _ _ => Tpointer t noattr
| Tfunction _ _ _ => Tpointer ty noattr
| Tstruct _ _ => remove_attributes ty
| Tunion _ _ => remove_attributes ty
end.(* Null Empty Path situation *)
Definition type_almost_match e t lr:=
  match typeof e, t, lr with
  | _, Tarray t1 _ a1, RRRR => eqb_type (typeconv' (typeof e)) (Tpointer t1 noattr)
  | _, _, LLLL => eqb_type (typeof e) t
  | _, _, _ => false
  end.(* TODO: remove almost_match' and use "type_is_by_value" in proof for assistent. *)
(* Empty Path situation *)
Definition type_almost_match' e t lr:=
  match typeof e, t, lr with
  | _, _, LLLL => eqb_type (typeof e) t
  | _, _, _ => false
  end.Fixpoint legal_nested_efield_rec t_root (gfs: list gfield) (tts: list type): bool :=
  match gfs, tts with
  | nil, nil => true
  | nil, _ => false
  | _ , nil => false
  | gf :: gfs0, t0 :: tts0 => (legal_nested_efield_rec t_root gfs0 tts0 && eqb_type (nested_field_type t_root gfs) t0)%bool
  end.Definition legal_nested_efield t_root e gfs tts lr :=
 (match gfs with
  | nil => type_almost_match' e t_root lr
  | _ => type_almost_match e t_root lr
  end &&
  legal_nested_efield_rec t_root gfs tts)%bool.

(* Specifications: *)
Lemma legal_nested_efield_rec_cons: forall t_root gf gfs t tts, legal_nested_efield_rec t_root (gf :: gfs) (t :: tts) = true -> legal_nested_efield_rec t_root gfs tts = true.
Lemma typeconv_typeconv'_eq: forall t1 t2, typeconv' t1 = typeconv' t2 -> typeconv t1 = typeconv t2.
Lemma tc_efield_ind: forall {cs: compspecs} (Delta: tycontext) (efs: list efield), tc_efield Delta efs = match efs with | nil => TT | eArraySubsc ei :: efs' => tc_expr Delta ei && tc_efield Delta efs' | eStructField i :: efs' => tc_efield Delta efs' | eUnionField i :: efs' => tc_efield Delta efs' end.
Lemma typeof_nested_efield': forall rho t_root e ef efs gf gfs t tts, legal_nested_efield_rec t_root (gf :: gfs) (t :: tts) = true -> efield_denote (ef :: efs) (gf :: gfs) rho -> nested_field_type t_root (gf :: gfs) = typeof (nested_efield e (ef :: efs) (t :: tts)).
Lemma typeof_nested_efield: forall rho t_root e efs gfs tts lr, legal_nested_efield t_root e gfs tts lr = true -> efield_denote efs gfs rho -> nested_field_type t_root gfs = typeof (nested_efield e efs tts).
Lemma offset_val_sem_add_pi: forall ofs t0 si e rho i, match si with | Signed => Int.min_signed <= i <= Int.max_signed | Unsigned => 0 <= i <= Int.max_unsigned end -> offset_val ofs (force_val (Cop.sem_add_ptr_int _ t0 si (eval_expr e rho) (Vint (Int.repr i)))) = offset_val ofs (offset_val (sizeof t0 * i) (eval_expr e rho)).