Definition match_reif_option {B: Type} (e: expr typ func) (somef : typ -> expr typ func -> B)
           (nonef : typ -> B) (d : B) :=
match e with
| (App (Inj (inr (Other (fsome t)))) e) => somef t e
| (Inj (inr (Other (fnone t)))) => nonef t
| _ => d
end.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Inductive val_e :=
    Vundef : val_e
  | Vint : int -> val_e
  | Vlong : int64 -> val_e
  | Vfloat : float -> val_e
  | Vsingle : float32 -> val_e
  | Vexpr : expr typ func -> val_e
  | Vunop : Cop.unary_operation -> type ->  val_e -> val_e
  | Vbinop : Cop.binary_operation -> type -> type -> val_e -> val_e -> val_e
  | Veval_cast : type -> type -> val_e -> val_e
  | Vforce_ptr : val_e -> val_e
  | Veval_field : type -> ident -> val_e -> val_e.Definition Vderef_noload (t: type) (e: val_e) : val_e :=
  match access_mode t with
  | By_reference => e
  | _ => Vundef
  end.Definition val_e_binarith op ty1 ty2 e1 e2 :=
  match op, ty1, ty2, e1, e2 with
  | Oand, Tint _ _ _, Tint _ _ _,
    App (Inj (inr (Value fVint))) e1',
    App (Inj (inr (Value fVint))) e2' =>
      match e1', e2' with
      | App (Inj (inr (Intop fint_repr))) e1'',
        App (Inj (inr (Intop fint_repr))) e2'' =>
                appR (Value fVint) (appR (Intop fint_repr) (App (appR (Zop fZ_land) e1'') e2''))
      | _, _ => appR (Value fVint) (App (appR (Intop fint_and) e1') e2')
      end
  | Oadd, Tint _ _ _, Tint _ _ _,
    App (Inj (inr (Value fVint))) e1',
    App (Inj (inr (Value fVint))) e2' =>
      match e1', e2' with
      | App (Inj (inr (Intop fint_repr))) e1'',
        App (Inj (inr (Intop fint_repr))) e2'' =>
                appR (Value fVint) (appR (Intop fint_repr) (App (appR (Zop fZ_add) e1'') e2''))
      | _, _ => appR (Value fVint) (App (appR (Intop fint_add) e1') e2')
      end
  | _, _, _, _, _ => App (appR (Eval_f (feval_binop op
                                         ty1 ty2)) e1) e2
  end.Fixpoint val_e_to_expr (v : val_e) : (expr typ func) :=
match v with
  | Vundef => injR (Value fVundef)
  | Vlong l => (appR (Value fVlong) (injR (Const (fint64 l))))
  | Vint i => (appR (Value fVint) (appR (Intop fint_repr) (injR (Const (fZ (Int.unsigned i))))))
  | Vfloat f => (appR (Value fVfloat) (injR (Const (ffloat f))))
  | Vsingle f => (appR (Value fVsingle) (injR (Const (ffloat32 f))))
  | Vexpr e => e
  | Vunop op ty e => appR (Eval_f (feval_unop op ty)) (val_e_to_expr e)
  | Vbinop op ty1 ty2 e1 e2 => val_e_binarith op ty1 ty2 (val_e_to_expr e1) (val_e_to_expr e2)
  | Veval_cast ty1 ty2 v => (appR (Eval_f (feval_cast ty1 ty2))) (val_e_to_expr v)
  | Vforce_ptr v => (appR (Other (fforce_ptr))) (val_e_to_expr v)
  | Veval_field t id v => (appR (Eval_f (feval_field t id))) (val_e_to_expr v)
end.Definition msubst_var id T2 ty :=
match get_reif id T2 (typrod tyc_type tyval) with
  | App (Inj (inr (Other (fsome t))))
        (App (App (Inj (inr (Data (fpair t1 t2))))
                  (Inj (inr (Const (fCtype ty')))))
             v) =>
    if eqb_type ty ty'
    then Some (Vexpr v)
    else None
  | _ => None
end.Fixpoint msubst_eval_expr_reif (T1: ExprCore.expr typ func) (T2: ExprCore.expr typ func) (e: Clight.expr) : option (val_e) :=
  match e with
  | Econst_int i ty => Some (Vint i)
  | Econst_long i ty => Some (Vlong i)
  | Econst_float f ty => Some (Vfloat f)
  | Econst_single f ty => Some (Vsingle f)
  | Etempvar id ty => match get_reif id T1 tyval with
                        | (App (Inj (inr (Other (fsome t)))) v) => Some (Vexpr v)
                        | _ => None
                      end
  | Eaddrof a ty => msubst_eval_lvalue_reif T1 T2 a
  | Eunop op a ty =>  option_map (Vunop op (typeof a)) (msubst_eval_expr_reif T1 T2 a)
  | Ebinop op a1 a2 ty => match (msubst_eval_expr_reif T1 T2 a1), (msubst_eval_expr_reif T1 T2 a2) with
                            | Some v1, Some v2 => Some (Vbinop op (typeof a1) (typeof a2) v1 v2)
                            | _, _ => None
                          end
  | Ecast a ty => option_map (Veval_cast (typeof a) ty) (msubst_eval_expr_reif T1 T2 a)
  | Evar id ty => option_map (Vderef_noload ty) (msubst_var id T2 ty)
  | Ederef a ty => option_map (Vderef_noload ty) (option_map Vforce_ptr (msubst_eval_expr_reif T1 T2 a))
  | Efield a i ty => option_map (Vderef_noload ty) (option_map (Veval_field (typeof a) i) (msubst_eval_lvalue_reif T1 T2 a))
  end
with
msubst_eval_lvalue_reif (T1: ExprCore.expr typ func) (T2: ExprCore.expr typ func) (e: Clight.expr) : option val_e :=
  match e with
  | Evar id ty => (msubst_var id T2 ty)
  | Ederef a ty => option_map Vforce_ptr (msubst_eval_expr_reif T1 T2 a)
  | Efield a i ty => option_map (Veval_field (typeof a) i) (msubst_eval_lvalue_reif T1 T2 a)
  | _  => Some Vundef
  end.Definition rmsubst_eval_expr (T1: (ExprCore.expr typ func)) (T2: ExprCore.expr typ func) (e: Clight.expr) :=
match msubst_eval_expr_reif T1 T2 e with
| Some e => some_reif (val_e_to_expr e) tyval
| None => none_reif tyval
end.Definition rmsubst_eval_lvalue (T1: (ExprCore.expr typ func)) (T2: ExprCore.expr typ func) (e: Clight.expr) :=
match msubst_eval_lvalue_reif T1 T2 e with
| Some e => some_reif (val_e_to_expr e) tyval
| None => none_reif tyval
end.Definition rmsubst_eval_LR (T1: (ExprCore.expr typ func)) (T2: ExprCore.expr typ func) (e: Clight.expr) (lr : LLRR) :=
match lr with
| LLLL => rmsubst_eval_lvalue T1 T2 e
| RRRR => rmsubst_eval_expr T1 T2 e
end.Fixpoint msubst_efield_denote_reif (T1: ExprCore.expr typ func) (T2: ExprCore.expr typ func) (efs : list efield) :=
  match efs with
  | nil => Some (injR (Data (fnil tygfield)))
  | cons (eStructField i) efs0 => option_map (App
                                (appR (Data (fcons tygfield))
                                      (appR (Smx fstruct_field) (injR (Const (fident i))))))
                                 (msubst_efield_denote_reif T1 T2 efs0)
  | cons (eUnionField i) efs0 => option_map (App
                                (appR (Data (fcons tygfield))
                                      (appR (Smx funion_field) (injR (Const (fident i))))))
                                 (msubst_efield_denote_reif T1 T2 efs0)
  | cons (eArraySubsc ei) efs0 =>
      match typeof ei, rmsubst_eval_expr T1 T2 ei with
      | Tint _ _ _,
        App (Inj (inr (Other (fsome _))))
         (App (Inj (inr (Value fVint))) i) =>
          match i with
          | App (Inj (inr (Intop fint_repr))) i' =>
                             option_map (App
                                (appR (Data (fcons tygfield))
                                      (appR (Smx farray_subsc) i')))
                                 (msubst_efield_denote_reif T1 T2 efs0)
          | _ =>
                             option_map (App
                                (appR (Data (fcons tygfield))
                                      (appR (Smx farray_subsc) (appR (Intop fint_unsigned) i))))
                                 (msubst_efield_denote_reif T1 T2 efs0)
          end
      | _, _ => None
      end
  end.Definition rmsubst_efield_denote (T1: ExprCore.expr typ func) (T2: ExprCore.expr typ func) (efs : list efield) :=
match msubst_efield_denote_reif T1 T2 efs with
| Some e => some_reif e (tylist tygfield)
| None => none_reif (tylist tygfield)
end.

(* Specifications: *)
Lemma Forall_reverse : forall A P (l: list A), Forall P l <-> Forall P (rev l).
Lemma in_fst : forall T T2 (p : T) (v: T2) l, In (p, v) l -> In p (map fst l).
Lemma elt_eq : forall T l p (v:T) v0 ls, (p, v0) :: l = rev (PTree.elements (PTree.set p v ls)) -> v = v0.