(** Truth values.  Non-zero integers are treated as [True].
  The integer 0 (also used to represent the null pointer) is [False].
  Other values are neither true nor false. *)

Inductive bool_of_val: val -> bool -> Prop :=
  | bool_of_val_int:
      forall n, bool_of_val (Vint n) (negb (Int.eq n Int.zero)).(** Truth values.  Non-zero integers are treated as [True].
  The integer 0 (also used to represent the null pointer) is [False].
  Other values are neither true nor false. *)

Inductive bool_of_val: val -> bool -> Prop :=
  | bool_of_val_int:
      forall n, bool_of_val (Vint n) (negb (Int.eq n Int.zero)).(** Arithmetic operations *)

Definition neg (v: val) : val :=
  match v with
  | Vint n => Vint (Int.neg n)
  | _ => Vundef
  end.Definition negf (v: val) : val :=
  match v with
  | Vfloat f => Vfloat (Float.neg f)
  | _ => Vundef
  end.Definition absf (v: val) : val :=
  match v with
  | Vfloat f => Vfloat (Float.abs f)
  | _ => Vundef
  end.Definition negfs (v: val) : val :=
  match v with
  | Vsingle f => Vsingle (Float32.neg f)
  | _ => Vundef
  end.Definition absfs (v: val) : val :=
  match v with
  | Vsingle f => Vsingle (Float32.abs f)
  | _ => Vundef
  end.Definition maketotal (ov: option val) : val :=
  match ov with Some v => v | None => Vundef end.Definition intoffloat (v: val) : option val :=
  match v with
  | Vfloat f => option_map Vint (Float.to_int f)
  | _ => None
  end.Definition intuoffloat (v: val) : option val :=
  match v with
  | Vfloat f => option_map Vint (Float.to_intu f)
  | _ => None
  end.Definition floatofint (v: val) : option val :=
  match v with
  | Vint n => Some (Vfloat (Float.of_int n))
  | _ => None
  end.Definition floatofintu (v: val) : option val :=
  match v with
  | Vint n => Some (Vfloat (Float.of_intu n))
  | _ => None
  end.Definition intofsingle (v: val) : option val :=
  match v with
  | Vsingle f => option_map Vint (Float32.to_int f)
  | _ => None
  end.Definition intuofsingle (v: val) : option val :=
  match v with
  | Vsingle f => option_map Vint (Float32.to_intu f)
  | _ => None
  end.Definition singleofint (v: val) : option val :=
  match v with
  | Vint n => Some (Vsingle (Float32.of_int n))
  | _ => None
  end.Definition singleofintu (v: val) : option val :=
  match v with
  | Vint n => Some (Vsingle (Float32.of_intu n))
  | _ => None
  end.Definition negint (v: val) : val :=
  match v with
  | Vint n => Vint (Int.neg n)
  | _ => Vundef
  end.Definition notint (v: val) : val :=
  match v with
  | Vint n => Vint (Int.not n)
  | _ => Vundef
  end.Definition of_bool (b: bool): val := if b then Vtrue else Vfalse.Definition boolval (v: val) : val :=
  match v with
  | Vint n => of_bool (negb (Int.eq n Int.zero))
  | Vptr b ofs => Vtrue
  | _ => Vundef
  end.Definition notbool (v: val) : val :=
  match v with
  | Vint n => of_bool (Int.eq n Int.zero)
  | Vptr b ofs => Vfalse
  | _ => Vundef
  end.Definition zero_ext (nbits: Z) (v: val) : val :=
  match v with
  | Vint n => Vint(Int.zero_ext nbits n)
  | _ => Vundef
  end.Definition sign_ext (nbits: Z) (v: val) : val :=
  match v with
  | Vint n => Vint(Int.sign_ext nbits n)
  | _ => Vundef
  end.Definition singleoffloat (v: val) : val :=
  match v with
  | Vfloat f => Vsingle (Float.to_single f)
  | _ => Vundef
  end.Definition floatofsingle (v: val) : val :=
  match v with
  | Vsingle f => Vfloat (Float.of_single f)
  | _ => Vundef
  end.Definition add (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.add n1 n2)
  | Vptr b1 ofs1, Vint n2 => if Archi.ptr64 then Vundef else Vptr b1 (Ptrofs.add ofs1 (Ptrofs.of_int n2))
  | Vint n1, Vptr b2 ofs2 => if Archi.ptr64 then Vundef else Vptr b2 (Ptrofs.add ofs2 (Ptrofs.of_int n1))
  | _, _ => Vundef
  end.Definition sub (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.sub n1 n2)
  | Vptr b1 ofs1, Vint n2 => if Archi.ptr64 then Vundef else Vptr b1 (Ptrofs.sub ofs1 (Ptrofs.of_int n2))
  | Vptr b1 ofs1, Vptr b2 ofs2 =>
      if Archi.ptr64 then Vundef else
      if eq_block b1 b2 then Vint(Ptrofs.to_int (Ptrofs.sub ofs1 ofs2)) else Vundef
  | _, _ => Vundef
  end.Definition mul (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.mul n1 n2)
  | _, _ => Vundef
  end.Definition mulhs (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.mulhs n1 n2)
  | _, _ => Vundef
  end.Definition mulhu (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.mulhu n1 n2)
  | _, _ => Vundef
  end.Definition divs (v1 v2: val): option val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
      if Int.eq n2 Int.zero
      || Int.eq n1 (Int.repr Int.min_signed) && Int.eq n2 Int.mone
      then None
      else Some(Vint(Int.divs n1 n2))
  | _, _ => None
  end.Definition mods (v1 v2: val): option val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
      if Int.eq n2 Int.zero
      || Int.eq n1 (Int.repr Int.min_signed) && Int.eq n2 Int.mone
      then None
      else Some(Vint(Int.mods n1 n2))
  | _, _ => None
  end.Definition divu (v1 v2: val): option val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
      if Int.eq n2 Int.zero then None else Some(Vint(Int.divu n1 n2))
  | _, _ => None
  end.Definition modu (v1 v2: val): option val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
      if Int.eq n2 Int.zero then None else Some(Vint(Int.modu n1 n2))
  | _, _ => None
  end.Definition add_carry (v1 v2 cin: val): val :=
  match v1, v2, cin with
  | Vint n1, Vint n2, Vint c => Vint(Int.add_carry n1 n2 c)
  | _, _, _ => Vundef
  end.Definition sub_overflow (v1 v2: val) : val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.sub_overflow n1 n2 Int.zero)
  | _, _ => Vundef
  end.Definition negative (v: val) : val :=
  match v with
  | Vint n => Vint (Int.negative n)
  | _ => Vundef
  end.Definition and (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.and n1 n2)
  | _, _ => Vundef
  end.Definition or (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.or n1 n2)
  | _, _ => Vundef
  end.Definition xor (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.xor n1 n2)
  | _, _ => Vundef
  end.Definition shl (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
     if Int.ltu n2 Int.iwordsize
     then Vint(Int.shl n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shr (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
     if Int.ltu n2 Int.iwordsize
     then Vint(Int.shr n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shr_carry (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
     if Int.ltu n2 Int.iwordsize
     then Vint(Int.shr_carry n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shrx (v1 v2: val): option val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
     if Int.ltu n2 (Int.repr 31)
     then Some(Vint(Int.shrx n1 n2))
     else None
  | _, _ => None
  end.Definition shru (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
     if Int.ltu n2 Int.iwordsize
     then Vint(Int.shru n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition rol (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.rol n1 n2)
  | _, _ => Vundef
  end.Definition rolm (v: val) (amount mask: int): val :=
  match v with
  | Vint n => Vint(Int.rolm n amount mask)
  | _ => Vundef
  end.Definition ror (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vint(Int.ror n1 n2)
  | _, _ => Vundef
  end.Definition addf (v1 v2: val): val :=
  match v1, v2 with
  | Vfloat f1, Vfloat f2 => Vfloat(Float.add f1 f2)
  | _, _ => Vundef
  end.Definition subf (v1 v2: val): val :=
  match v1, v2 with
  | Vfloat f1, Vfloat f2 => Vfloat(Float.sub f1 f2)
  | _, _ => Vundef
  end.Definition mulf (v1 v2: val): val :=
  match v1, v2 with
  | Vfloat f1, Vfloat f2 => Vfloat(Float.mul f1 f2)
  | _, _ => Vundef
  end.Definition divf (v1 v2: val): val :=
  match v1, v2 with
  | Vfloat f1, Vfloat f2 => Vfloat(Float.div f1 f2)
  | _, _ => Vundef
  end.Definition floatofwords (v1 v2: val) : val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vfloat (Float.from_words n1 n2)
  | _, _ => Vundef
  end.Definition addfs (v1 v2: val): val :=
  match v1, v2 with
  | Vsingle f1, Vsingle f2 => Vsingle(Float32.add f1 f2)
  | _, _ => Vundef
  end.Definition subfs (v1 v2: val): val :=
  match v1, v2 with
  | Vsingle f1, Vsingle f2 => Vsingle(Float32.sub f1 f2)
  | _, _ => Vundef
  end.Definition mulfs (v1 v2: val): val :=
  match v1, v2 with
  | Vsingle f1, Vsingle f2 => Vsingle(Float32.mul f1 f2)
  | _, _ => Vundef
  end.Definition divfs (v1 v2: val): val :=
  match v1, v2 with
  | Vsingle f1, Vsingle f2 => Vsingle(Float32.div f1 f2)
  | _, _ => Vundef
  end.(** Operations on 64-bit integers *)

Definition longofwords (v1 v2: val) : val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vlong (Int64.ofwords n1 n2)
  | _, _ => Vundef
  end.Definition loword (v: val) : val :=
  match v with
  | Vlong n  => Vint (Int64.loword n)
  | _ => Vundef
  end.Definition hiword (v: val) : val :=
  match v with
  | Vlong n  => Vint (Int64.hiword n)
  | _ => Vundef
  end.Definition negl (v: val) : val :=
  match v with
  | Vlong n => Vlong (Int64.neg n)
  | _ => Vundef
  end.Definition notl (v: val) : val :=
  match v with
  | Vlong n => Vlong (Int64.not n)
  | _ => Vundef
  end.Definition longofint (v: val) : val :=
  match v with
  | Vint n => Vlong (Int64.repr (Int.signed n))
  | _ => Vundef
  end.Definition longofintu (v: val) : val :=
  match v with
  | Vint n => Vlong (Int64.repr (Int.unsigned n))
  | _ => Vundef
  end.Definition longoffloat (v: val) : option val :=
  match v with
  | Vfloat f => option_map Vlong (Float.to_long f)
  | _ => None
  end.Definition longuoffloat (v: val) : option val :=
  match v with
  | Vfloat f => option_map Vlong (Float.to_longu f)
  | _ => None
  end.Definition longofsingle (v: val) : option val :=
  match v with
  | Vsingle f => option_map Vlong (Float32.to_long f)
  | _ => None
  end.Definition longuofsingle (v: val) : option val :=
  match v with
  | Vsingle f => option_map Vlong (Float32.to_longu f)
  | _ => None
  end.Definition floatoflong (v: val) : option val :=
  match v with
  | Vlong n => Some (Vfloat (Float.of_long n))
  | _ => None
  end.Definition floatoflongu (v: val) : option val :=
  match v with
  | Vlong n => Some (Vfloat (Float.of_longu n))
  | _ => None
  end.Definition singleoflong (v: val) : option val :=
  match v with
  | Vlong n => Some (Vsingle (Float32.of_long n))
  | _ => None
  end.Definition singleoflongu (v: val) : option val :=
  match v with
  | Vlong n => Some (Vsingle (Float32.of_longu n))
  | _ => None
  end.Definition addl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.add n1 n2)
  | Vptr b1 ofs1, Vlong n2 => if Archi.ptr64 then Vptr b1 (Ptrofs.add ofs1 (Ptrofs.of_int64 n2)) else Vundef
  | Vlong n1, Vptr b2 ofs2 => if Archi.ptr64 then Vptr b2 (Ptrofs.add ofs2 (Ptrofs.of_int64 n1)) else Vundef
  | _, _ => Vundef
  end.Definition subl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.sub n1 n2)
  | Vptr b1 ofs1, Vlong n2 =>
      if Archi.ptr64 then Vptr b1 (Ptrofs.sub ofs1 (Ptrofs.of_int64 n2)) else Vundef
  | Vptr b1 ofs1, Vptr b2 ofs2 =>
      if negb Archi.ptr64 then Vundef else
      if eq_block b1 b2 then Vlong(Ptrofs.to_int64 (Ptrofs.sub ofs1 ofs2)) else Vundef
  | _, _ => Vundef
  end.Definition mull (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.mul n1 n2)
  | _, _ => Vundef
  end.Definition mull' (v1 v2: val): val :=
  match v1, v2 with
  | Vint n1, Vint n2 => Vlong(Int64.mul' n1 n2)
  | _, _ => Vundef
  end.Definition mullhs (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.mulhs n1 n2)
  | _, _ => Vundef
  end.Definition mullhu (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.mulhu n1 n2)
  | _, _ => Vundef
  end.Definition divls (v1 v2: val): option val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 =>
      if Int64.eq n2 Int64.zero
      || Int64.eq n1 (Int64.repr Int64.min_signed) && Int64.eq n2 Int64.mone
      then None
      else Some(Vlong(Int64.divs n1 n2))
  | _, _ => None
  end.Definition modls (v1 v2: val): option val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 =>
      if Int64.eq n2 Int64.zero
      || Int64.eq n1 (Int64.repr Int64.min_signed) && Int64.eq n2 Int64.mone
      then None
      else Some(Vlong(Int64.mods n1 n2))
  | _, _ => None
  end.Definition divlu (v1 v2: val): option val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 =>
      if Int64.eq n2 Int64.zero then None else Some(Vlong(Int64.divu n1 n2))
  | _, _ => None
  end.Definition modlu (v1 v2: val): option val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 =>
      if Int64.eq n2 Int64.zero then None else Some(Vlong(Int64.modu n1 n2))
  | _, _ => None
  end.Definition addl_carry (v1 v2 cin: val): val :=
  match v1, v2, cin with
  | Vlong n1, Vlong n2, Vlong c => Vlong(Int64.add_carry n1 n2 c)
  | _, _, _ => Vundef
  end.Definition subl_overflow (v1 v2: val) : val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vint (Int.repr (Int64.unsigned (Int64.sub_overflow n1 n2 Int64.zero)))
  | _, _ => Vundef
  end.Definition negativel (v: val) : val :=
  match v with
  | Vlong n => Vint (Int.repr (Int64.unsigned (Int64.negative n)))
  | _ => Vundef
  end.Definition andl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.and n1 n2)
  | _, _ => Vundef
  end.Definition orl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.or n1 n2)
  | _, _ => Vundef
  end.Definition xorl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Vlong(Int64.xor n1 n2)
  | _, _ => Vundef
  end.Definition shll (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 =>
     if Int.ltu n2 Int64.iwordsize'
     then Vlong(Int64.shl' n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shrl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 =>
     if Int.ltu n2 Int64.iwordsize'
     then Vlong(Int64.shr' n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shrlu (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 =>
     if Int.ltu n2 Int64.iwordsize'
     then Vlong(Int64.shru' n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition shrxl (v1 v2: val): option val :=
  match v1, v2 with
  | Vlong n1, Vint n2 =>
     if Int.ltu n2 (Int.repr 63)
     then Some(Vlong(Int64.shrx' n1 n2))
     else None
  | _, _ => None
  end.Definition shrl_carry (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 =>
     if Int.ltu n2 Int64.iwordsize'
     then Vlong(Int64.shr_carry' n1 n2)
     else Vundef
  | _, _ => Vundef
  end.Definition roll (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 => Vlong(Int64.rol n1 (Int64.repr (Int.unsigned n2)))
  | _, _ => Vundef
  end.Definition rorl (v1 v2: val): val :=
  match v1, v2 with
  | Vlong n1, Vint n2 => Vlong(Int64.ror n1 (Int64.repr (Int.unsigned n2)))
  | _, _ => Vundef
  end.Definition rolml (v: val) (amount: int) (mask: int64): val :=
  match v with
  | Vlong n => Vlong(Int64.rolm n (Int64.repr (Int.unsigned amount)) mask)
  | _ => Vundef
  end.Definition cmp_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vint n1, Vint n2 => Some (Int.cmp c n1 n2)
  | _, _ => None
  end.Definition cmp_different_blocks (c: comparison): option bool :=
  match c with
  | Ceq => Some false
  | Cne => Some true
  | _   => None
  end.Definition cmpu_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vint n1, Vint n2 =>
      Some (Int.cmpu c n1 n2)
  | Vint n1, Vptr b2 ofs2 =>
      if Archi.ptr64 then None else
      if Int.eq n1 Int.zero && weak_valid_ptr b2 (Ptrofs.unsigned ofs2)
      then cmp_different_blocks c
      else None
  | Vptr b1 ofs1, Vptr b2 ofs2 =>
      if Archi.ptr64 then None else
      if eq_block b1 b2 then
        if weak_valid_ptr b1 (Ptrofs.unsigned ofs1)
           && weak_valid_ptr b2 (Ptrofs.unsigned ofs2)
        then Some (Ptrofs.cmpu c ofs1 ofs2)
        else None
      else
        if valid_ptr b1 (Ptrofs.unsigned ofs1)
           && valid_ptr b2 (Ptrofs.unsigned ofs2)
        then cmp_different_blocks c
        else None
  | Vptr b1 ofs1, Vint n2 =>
      if Archi.ptr64 then None else
      if Int.eq n2 Int.zero && weak_valid_ptr b1 (Ptrofs.unsigned ofs1)
      then cmp_different_blocks c
      else None
  | _, _ => None
  end.Definition cmpf_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vfloat f1, Vfloat f2 => Some (Float.cmp c f1 f2)
  | _, _ => None
  end.Definition cmpfs_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vsingle f1, Vsingle f2 => Some (Float32.cmp c f1 f2)
  | _, _ => None
  end.Definition cmpl_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Some (Int64.cmp c n1 n2)
  | _, _ => None
  end.Definition cmplu_bool (c: comparison) (v1 v2: val): option bool :=
  match v1, v2 with
  | Vlong n1, Vlong n2 => Some (Int64.cmpu c n1 n2)
  | Vlong n1, Vptr b2 ofs2 =>
      if negb Archi.ptr64 then None else
      if Int64.eq n1 Int64.zero && weak_valid_ptr b2 (Ptrofs.unsigned ofs2)
      then cmp_different_blocks c
      else None
  | Vptr b1 ofs1, Vptr b2 ofs2 =>
      if negb Archi.ptr64 then None else
      if eq_block b1 b2 then
        if weak_valid_ptr b1 (Ptrofs.unsigned ofs1)
           && weak_valid_ptr b2 (Ptrofs.unsigned ofs2)
        then Some (Ptrofs.cmpu c ofs1 ofs2)
        else None
      else
        if valid_ptr b1 (Ptrofs.unsigned ofs1)
           && valid_ptr b2 (Ptrofs.unsigned ofs2)
        then cmp_different_blocks c
        else None
  | Vptr b1 ofs1, Vlong n2 =>
      if negb Archi.ptr64 then None else
      if Int64.eq n2 Int64.zero && weak_valid_ptr b1 (Ptrofs.unsigned ofs1)
      then cmp_different_blocks c
      else None
  | _, _ => None
  end.Definition of_optbool (ob: option bool): val :=
  match ob with Some true => Vtrue | Some false => Vfalse | None => Vundef end.Definition cmp (c: comparison) (v1 v2: val): val :=
  of_optbool (cmp_bool c v1 v2).Definition cmpu (c: comparison) (v1 v2: val): val :=
  of_optbool (cmpu_bool c v1 v2).Definition cmpf (c: comparison) (v1 v2: val): val :=
  of_optbool (cmpf_bool c v1 v2).Definition cmpfs (c: comparison) (v1 v2: val): val :=
  of_optbool (cmpfs_bool c v1 v2).Definition cmpl (c: comparison) (v1 v2: val): option val :=
  option_map of_bool (cmpl_bool c v1 v2).Definition cmplu (c: comparison) (v1 v2: val): option val :=
  option_map of_bool (cmplu_bool c v1 v2).Definition maskzero_bool (v: val) (mask: int): option bool :=
  match v with
  | Vint n => Some (Int.eq (Int.and n mask) Int.zero)
  | _ => None
  end.(** Add the given offset to the given pointer. *)

Definition offset_ptr (v: val) (delta: ptrofs) : val :=
  match v with
  | Vptr b ofs => Vptr b (Ptrofs.add ofs delta)
  | _ => Vundef
  end.(** [load_result] reflects the effect of storing a value with a given
  memory chunk, then reading it back with the same chunk.  Depending
  on the chunk and the type of the value, some normalization occurs.
  For instance, consider storing the integer value [0xFFF] on 1 byte
  at a given address, and reading it back.  If it is read back with
  chunk [Mint8unsigned], zero-extension must be performed, resulting
  in [0xFF].  If it is read back as a [Mint8signed], sign-extension is
  performed and [0xFFFFFFFF] is returned. *)

Definition load_result (chunk: memory_chunk) (v: val) :=
  match chunk, v with
  | Mint8signed, Vint n => Vint (Int.sign_ext 8 n)
  | Mint8unsigned, Vint n => Vint (Int.zero_ext 8 n)
  | Mint16signed, Vint n => Vint (Int.sign_ext 16 n)
  | Mint16unsigned, Vint n => Vint (Int.zero_ext 16 n)
  | Mint32, Vint n => Vint n
  | Mint32, Vptr b ofs => if Archi.ptr64 then Vundef else Vptr b ofs
  | Mint64, Vlong n => Vlong n
  | Mint64, Vptr b ofs => if Archi.ptr64 then Vptr b ofs else Vundef
  | Mfloat32, Vsingle f => Vsingle f
  | Mfloat64, Vfloat f => Vfloat f
  | Many32, (Vint _ | Vsingle _) => v
  | Many32, Vptr _ _ => if Archi.ptr64 then Vundef else v
  | Many64, _ => v
  | _, _ => Vundef
  end.

Theorem modu_divu: forall x y z, modu x y = Some z -> exists v, divu x y = Some v /\ z = sub x (mul v y).