Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Ctypes.
Require Archi.

Syntax of operators.

Inductive unary_operation : Type :=
  | Onotbool : unary_operation (* boolean negation (! in C) *)
  | Onotint : unary_operation (* integer complement (~ in C) *)
  | Oneg : unary_operation (* opposite (unary -) *)
  | Oabsfloat : unary_operation. (* floating-point absolute value *)

Inductive binary_operation : Type :=
  | Oadd : binary_operation (* addition (binary +) *)
  | Osub : binary_operation (* subtraction (binary -) *)
  | Omul : binary_operation (* multiplication (binary *) *)
  | Odiv : binary_operation (* division (/) *)
  | Omod : binary_operation (* remainder (%) *)
  | Oand : binary_operation (* bitwise and (&) *)
  | Oor : binary_operation (* bitwise or (|) *)
  | Oxor : binary_operation (* bitwise xor (^) *)
  | Oshl : binary_operation (* left shift (<<) *)
  | Oshr : binary_operation (* right shift (>>) *)
  | Oeq: binary_operation (* comparison (==) *)
  | One: binary_operation (* comparison (!=) *)
  | Olt: binary_operation (* comparison (<) *)
  | Ogt: binary_operation (* comparison (>) *)
  | Ole: binary_operation (* comparison (<=) *)
  | Oge: binary_operation. (* comparison (>=) *)

Inductive incr_or_decr : Type := Incr | Decr.

Type classification and semantics of operators.

Casts and truth values

Inductive classify_cast_cases : Type :=
  | cast_case_pointer (* between pointer types or intptr_t types *)
  | cast_case_i2i (sz2:intsize) (si2:signedness) (* int -> int *)
  | cast_case_f2f (* double -> double *)
  | cast_case_s2s (* single -> single *)
  | cast_case_f2s (* double -> single *)
  | cast_case_s2f (* single -> double *)
  | cast_case_i2f (si1: signedness) (* int -> double *)
  | cast_case_i2s (si1: signedness) (* int -> single *)
  | cast_case_f2i (sz2:intsize) (si2:signedness) (* double -> int *)
  | cast_case_s2i (sz2:intsize) (si2:signedness) (* single -> int *)
  | cast_case_l2l (* long -> long *)
  | cast_case_i2l (si1: signedness) (* int -> long *)
  | cast_case_l2i (sz2: intsize) (si2: signedness) (* long -> int *)
  | cast_case_l2f (si1: signedness) (* long -> double *)
  | cast_case_l2s (si1: signedness) (* long -> single *)
  | cast_case_f2l (si2:signedness) (* double -> long *)
  | cast_case_s2l (si2:signedness) (* single -> long *)
  | cast_case_i2bool (* int -> bool *)
  | cast_case_l2bool (* long -> bool *)
  | cast_case_f2bool (* double -> bool *)
  | cast_case_s2bool (* single -> bool *)
  | cast_case_struct (id1 id2: ident) (* struct -> struct *)
  | cast_case_union (id1 id2: ident) (* union -> union *)
  | cast_case_void (* any -> void *)
  | cast_case_default.

Definition classify_cast (tfrom tto: type) : classify_cast_cases :=
  match tto, tfrom with
  | Tvoid, _ => cast_case_void
  | Tint IBool _ _, Tint _ _ _ => cast_case_i2bool
  | Tint IBool _ _, Tlong _ _ => cast_case_l2bool
  | Tint IBool _ _, Tfloat F64 _ => cast_case_f2bool
  | Tint IBool _ _, Tfloat F32 _ => cast_case_s2bool
  | Tint IBool _ _, (Tpointer _ _ | Tarray _ _ _ | Tfunction _ _ _) =>
      if Archi.ptr64 then cast_case_l2bool else cast_case_i2bool
  | Tint sz2 si2 _, Tint _ _ _ =>
      if Archi.ptr64 then cast_case_i2i sz2 si2
      else if intsize_eq sz2 I32 then cast_case_pointer
      else cast_case_i2i sz2 si2
  | Tint sz2 si2 _, Tlong _ _ => cast_case_l2i sz2 si2
  | Tint sz2 si2 _, Tfloat F64 _ => cast_case_f2i sz2 si2
  | Tint sz2 si2 _, Tfloat F32 _ => cast_case_s2i sz2 si2
  | Tint sz2 si2 _, (Tpointer _ _ | Tarray _ _ _ | Tfunction _ _ _) =>
      if Archi.ptr64 then cast_case_l2i sz2 si2
      else if intsize_eq sz2 I32 then cast_case_pointer
      else cast_case_i2i sz2 si2
  | Tlong _ _, Tlong _ _ =>
      if Archi.ptr64 then cast_case_pointer else cast_case_l2l
  | Tlong _ _, Tint sz1 si1 _ => cast_case_i2l si1
  | Tlong si2 _, Tfloat F64 _ => cast_case_f2l si2
  | Tlong si2 _, Tfloat F32 _ => cast_case_s2l si2
  | Tlong si2 _, (Tpointer _ _ | Tarray _ _ _ | Tfunction _ _ _) =>
      if Archi.ptr64 then cast_case_pointer else cast_case_i2l si2
  | Tfloat F64 _, Tint sz1 si1 _ => cast_case_i2f si1
  | Tfloat F32 _, Tint sz1 si1 _ => cast_case_i2s si1
  | Tfloat F64 _, Tlong si1 _ => cast_case_l2f si1
  | Tfloat F32 _, Tlong si1 _ => cast_case_l2s si1
  | Tfloat F64 _, Tfloat F64 _ => cast_case_f2f
  | Tfloat F32 _, Tfloat F32 _ => cast_case_s2s
  | Tfloat F64 _, Tfloat F32 _ => cast_case_s2f
  | Tfloat F32 _, Tfloat F64 _ => cast_case_f2s
  | Tpointer _ _, Tint _ _ _ =>
      if Archi.ptr64 then cast_case_i2l Unsigned else cast_case_pointer
  | Tpointer _ _, Tlong _ _ =>
      if Archi.ptr64 then cast_case_pointer else cast_case_l2i I32 Unsigned
  | Tpointer _ _, (Tpointer _ _ | Tarray _ _ _ | Tfunction _ _ _) => cast_case_pointer
  | Tstruct id2 _, Tstruct id1 _ => cast_case_struct id1 id2
  | Tunion id2 _, Tunion id1 _ => cast_case_union id1 id2
  | _, _ => cast_case_default
  end.


Definition cast_int_int (sz: intsize) (sg: signedness) (i: int) : int :=
  match sz, sg with
  | I8, Signed => Int.sign_ext 8 i
  | I8, Unsigned => Int.zero_ext 8 i
  | I16, Signed => Int.sign_ext 16 i
  | I16, Unsigned => Int.zero_ext 16 i
  | I32, _ => i
  | IBool, _ => if Int.eq i Int.zero then Int.zero else Int.one
  end.

Definition cast_int_float (si: signedness) (i: int) : float :=
  match si with
  | Signed => Float.of_int i
  | Unsigned => Float.of_intu i
  end.

Definition cast_float_int (si : signedness) (f: float) : option int :=
  match si with
  | Signed => Float.to_int f
  | Unsigned => Float.to_intu f
  end.

Definition cast_int_single (si: signedness) (i: int) : float32 :=
  match si with
  | Signed => Float32.of_int i
  | Unsigned => Float32.of_intu i
  end.

Definition cast_single_int (si : signedness) (f: float32) : option int :=
  match si with
  | Signed => Float32.to_int f
  | Unsigned => Float32.to_intu f
  end.

Definition cast_int_long (si: signedness) (i: int) : int64 :=
  match si with
  | Signed => Int64.repr (Int.signed i)
  | Unsigned => Int64.repr (Int.unsigned i)
  end.

Definition cast_long_float (si: signedness) (i: int64) : float :=
  match si with
  | Signed => Float.of_long i
  | Unsigned => Float.of_longu i
  end.

Definition cast_long_single (si: signedness) (i: int64) : float32 :=
  match si with
  | Signed => Float32.of_long i
  | Unsigned => Float32.of_longu i
  end.

Definition cast_float_long (si : signedness) (f: float) : option int64 :=
  match si with
  | Signed => Float.to_long f
  | Unsigned => Float.to_longu f
  end.

Definition cast_single_long (si : signedness) (f: float32) : option int64 :=
  match si with
  | Signed => Float32.to_long f
  | Unsigned => Float32.to_longu f
  end.


Class SemCast {T: Type} (valid_pointer: T -> block -> Z -> bool): Prop :=
  {
    weak_valid_pointer m b o :=
      valid_pointer m b o || valid_pointer m b (o - 1)
  }.

Section WITHSEMCAST.
Context `{sem_cast_prf: SemCast}.

Definition sem_cast (v: val) (t1 t2: type) (m: T): option val :=
  match classify_cast t1 t2 with
  | cast_case_pointer =>
      match v with
      | Vptr _ _ => Some v
      | Vint _ => if Archi.ptr64 then None else Some v
      | Vlong _ => if Archi.ptr64 then Some v else None
      | _ => None
      end
  | cast_case_i2i sz2 si2 =>
      match v with
      | Vint i => Some (Vint (cast_int_int sz2 si2 i))
      | _ => None
      end
  | cast_case_f2f =>
      match v with
      | Vfloat f => Some (Vfloat f)
      | _ => None
      end
  | cast_case_s2s =>
      match v with
      | Vsingle f => Some (Vsingle f)
      | _ => None
      end
  | cast_case_s2f =>
      match v with
      | Vsingle f => Some (Vfloat (Float.of_single f))
      | _ => None
      end
  | cast_case_f2s =>
      match v with
      | Vfloat f => Some (Vsingle (Float.to_single f))
      | _ => None
      end
  | cast_case_i2f si1 =>
      match v with
      | Vint i => Some (Vfloat (cast_int_float si1 i))
      | _ => None
      end
  | cast_case_i2s si1 =>
      match v with
      | Vint i => Some (Vsingle (cast_int_single si1 i))
      | _ => None
      end
  | cast_case_f2i sz2 si2 =>
      match v with
      | Vfloat f =>
          match cast_float_int si2 f with
          | Some i => Some (Vint (cast_int_int sz2 si2 i))
          | None => None
          end
      | _ => None
      end
  | cast_case_s2i sz2 si2 =>
      match v with
      | Vsingle f =>
          match cast_single_int si2 f with
          | Some i => Some (Vint (cast_int_int sz2 si2 i))
          | None => None
          end
      | _ => None
      end
  | cast_case_i2bool =>
      match v with
      | Vint n =>
          Some(Vint(if Int.eq n Int.zero then Int.zero else Int.one))
      | Vptr b ofs =>
          if Archi.ptr64 then None else
          if weak_valid_pointer m b (Ptrofs.unsigned ofs) then Some Vone else None
      | _ => None
      end
  | cast_case_l2bool =>
      match v with
      | Vlong n =>
          Some(Vint(if Int64.eq n Int64.zero then Int.zero else Int.one))
      | Vptr b ofs =>
          if negb Archi.ptr64 then None else
            if weak_valid_pointer m b (Ptrofs.unsigned ofs) then Some Vone else None
      | _ => None
      end
  | cast_case_f2bool =>
      match v with
      | Vfloat f =>
          Some(Vint(if Float.cmp Ceq f Float.zero then Int.zero else Int.one))
      | _ => None
      end
  | cast_case_s2bool =>
      match v with
      | Vsingle f =>
          Some(Vint(if Float32.cmp Ceq f Float32.zero then Int.zero else Int.one))
      | _ => None
      end
  | cast_case_l2l =>
      match v with
      | Vlong n => Some (Vlong n)
      | _ => None
      end
  | cast_case_i2l si =>
      match v with
      | Vint n => Some(Vlong (cast_int_long si n))
      | _ => None
      end
  | cast_case_l2i sz si =>
      match v with
      | Vlong n => Some(Vint (cast_int_int sz si (Int.repr (Int64.unsigned n))))
      | _ => None
      end
  | cast_case_l2f si1 =>
      match v with
      | Vlong i => Some (Vfloat (cast_long_float si1 i))
      | _ => None
      end
  | cast_case_l2s si1 =>
      match v with
      | Vlong i => Some (Vsingle (cast_long_single si1 i))
      | _ => None
      end
  | cast_case_f2l si2 =>
      match v with
      | Vfloat f =>
          match cast_float_long si2 f with
          | Some i => Some (Vlong i)
          | None => None
          end
      | _ => None
      end
  | cast_case_s2l si2 =>
      match v with
      | Vsingle f =>
          match cast_single_long si2 f with
          | Some i => Some (Vlong i)
          | None => None
          end
      | _ => None
      end
  | cast_case_struct id1 id2 =>
      match v with
      | Vptr b ofs =>
          if ident_eq id1 id2 then Some v else None
      | _ => None
      end
  | cast_case_union id1 id2 =>
      match v with
      | Vptr b ofs =>
          if ident_eq id1 id2 then Some v else None
      | _ => None
      end
  | cast_case_void =>
      Some v
  | cast_case_default =>
      None
  end.


Inductive classify_bool_cases : Type :=
  | bool_case_i (* integer *)
  | bool_case_l (* long *)
  | bool_case_f (* double float *)
  | bool_case_s (* single float *)
  | bool_default.

Definition classify_bool (ty: type) : classify_bool_cases :=
  match typeconv ty with
  | Tint _ _ _ => bool_case_i
  | Tpointer _ _ => if Archi.ptr64 then bool_case_l else bool_case_i
  | Tfloat F64 _ => bool_case_f
  | Tfloat F32 _ => bool_case_s
  | Tlong _ _ => bool_case_l
  | _ => bool_default
  end.


Definition bool_val (v: val) (t: type) (m: T) : option bool :=
  match classify_bool t with
  | bool_case_i =>
      match v with
      | Vint n => Some (negb (Int.eq n Int.zero))
      | Vptr b ofs =>
          if Archi.ptr64 then None else
          if weak_valid_pointer m b (Ptrofs.unsigned ofs) then Some true else None
      | _ => None
      end
  | bool_case_l =>
      match v with
      | Vlong n => Some (negb (Int64.eq n Int64.zero))
      | Vptr b ofs =>
          if negb Archi.ptr64 then None else
          if weak_valid_pointer m b (Ptrofs.unsigned ofs) then Some true else None
      | _ => None
      end
  | bool_case_f =>
      match v with
      | Vfloat f => Some (negb (Float.cmp Ceq f Float.zero))
      | _ => None
      end
  | bool_case_s =>
      match v with
      | Vsingle f => Some (negb (Float32.cmp Ceq f Float32.zero))
      | _ => None
      end
  | bool_default => None
  end.

Unary operators

Boolean negation

Definition sem_notbool (v: val) (ty: type) (m: T): option val :=
  option_map (fun b => Val.of_bool (negb b)) (bool_val v ty m).

Opposite and absolute value

Inductive classify_neg_cases : Type :=
  | neg_case_i(s: signedness) (* int *)
  | neg_case_f (* double float *)
  | neg_case_s (* single float *)
  | neg_case_l(s: signedness) (* long *)
  | neg_default.

Definition classify_neg (ty: type) : classify_neg_cases :=
  match ty with
  | Tint I32 Unsigned _ => neg_case_i Unsigned
  | Tint _ _ _ => neg_case_i Signed
  | Tfloat F64 _ => neg_case_f
  | Tfloat F32 _ => neg_case_s
  | Tlong si _ => neg_case_l si
  | _ => neg_default
  end.

Definition sem_neg (v: val) (ty: type) : option val :=
  match classify_neg ty with
  | neg_case_i sg =>
      match v with
      | Vint n => Some (Vint (Int.neg n))
      | _ => None
      end
  | neg_case_f =>
      match v with
      | Vfloat f => Some (Vfloat (Float.neg f))
      | _ => None
      end
  | neg_case_s =>
      match v with
      | Vsingle f => Some (Vsingle (Float32.neg f))
      | _ => None
      end
  | neg_case_l sg =>
      match v with
      | Vlong n => Some (Vlong (Int64.neg n))
      | _ => None
      end
  | neg_default => None
  end.

Definition sem_absfloat (v: val) (ty: type) : option val :=
  match classify_neg ty with
  | neg_case_i sg =>
      match v with
      | Vint n => Some (Vfloat (Float.abs (cast_int_float sg n)))
      | _ => None
      end
  | neg_case_f =>
      match v with
      | Vfloat f => Some (Vfloat (Float.abs f))
      | _ => None
      end
  | neg_case_s =>
      match v with
      | Vsingle f => Some (Vfloat (Float.abs (Float.of_single f)))
      | _ => None
      end
  | neg_case_l sg =>
      match v with
      | Vlong n => Some (Vfloat (Float.abs (cast_long_float sg n)))
      | _ => None
      end
  | neg_default => None
  end.

Bitwise complement

Inductive classify_notint_cases : Type :=
  | notint_case_i(s: signedness) (* int *)
  | notint_case_l(s: signedness) (* long *)
  | notint_default.

Definition classify_notint (ty: type) : classify_notint_cases :=
  match ty with
  | Tint I32 Unsigned _ => notint_case_i Unsigned
  | Tint _ _ _ => notint_case_i Signed
  | Tlong si _ => notint_case_l si
  | _ => notint_default
  end.

Definition sem_notint (v: val) (ty: type): option val :=
  match classify_notint ty with
  | notint_case_i sg =>
      match v with
      | Vint n => Some (Vint (Int.not n))
      | _ => None
      end
  | notint_case_l sg =>
      match v with
      | Vlong n => Some (Vlong (Int64.not n))
      | _ => None
      end
  | notint_default => None
  end.

Binary operators

Inductive binarith_cases: Type :=
  | bin_case_i (s: signedness) (* at int type *)
  | bin_case_l (s: signedness) (* at long int type *)
  | bin_case_f (* at double float type *)
  | bin_case_s (* at single float type *)
  | bin_default. (* error *)

Definition classify_binarith (ty1: type) (ty2: type) : binarith_cases :=
  match ty1, ty2 with
  | Tint I32 Unsigned _, Tint _ _ _ => bin_case_i Unsigned
  | Tint _ _ _, Tint I32 Unsigned _ => bin_case_i Unsigned
  | Tint _ _ _, Tint _ _ _ => bin_case_i Signed
  | Tlong Signed _, Tlong Signed _ => bin_case_l Signed
  | Tlong _ _, Tlong _ _ => bin_case_l Unsigned
  | Tlong sg _, Tint _ _ _ => bin_case_l sg
  | Tint _ _ _, Tlong sg _ => bin_case_l sg
  | Tfloat F32 _, Tfloat F32 _ => bin_case_s
  | Tfloat _ _, Tfloat _ _ => bin_case_f
  | Tfloat F64 _, (Tint _ _ _ | Tlong _ _) => bin_case_f
  | (Tint _ _ _ | Tlong _ _), Tfloat F64 _ => bin_case_f
  | Tfloat F32 _, (Tint _ _ _ | Tlong _ _) => bin_case_s
  | (Tint _ _ _ | Tlong _ _), Tfloat F32 _ => bin_case_s
  | _, _ => bin_default
  end.


Definition binarith_type (c: binarith_cases) : type :=
  match c with
  | bin_case_i sg => Tint I32 sg noattr
  | bin_case_l sg => Tlong sg noattr
  | bin_case_f => Tfloat F64 noattr
  | bin_case_s => Tfloat F32 noattr
  | bin_default => Tvoid
  end.

Definition sem_binarith
    (sem_int: signedness -> int -> int -> option val)
    (sem_long: signedness -> int64 -> int64 -> option val)
    (sem_float: float -> float -> option val)
    (sem_single: float32 -> float32 -> option val)
    (v1: val) (t1: type) (v2: val) (t2: type) (m: T): option val :=
  let c := classify_binarith t1 t2 in
  let t := binarith_type c in
  match sem_cast v1 t1 t m with
  | None => None
  | Some v1' =>
  match sem_cast v2 t2 t m with
  | None => None
  | Some v2' =>
  match c with
  | bin_case_i sg =>
      match v1', v2' with
      | Vint n1, Vint n2 => sem_int sg n1 n2
      | _, _ => None
      end
  | bin_case_f =>
      match v1', v2' with
      | Vfloat n1, Vfloat n2 => sem_float n1 n2
      | _, _ => None
      end
  | bin_case_s =>
      match v1', v2' with
      | Vsingle n1, Vsingle n2 => sem_single n1 n2
      | _, _ => None
      end
  | bin_case_l sg =>
      match v1', v2' with
      | Vlong n1, Vlong n2 => sem_long sg n1 n2
      | _, _ => None
      end
  | bin_default => None
  end end end.

Addition

Inductive classify_add_cases : Type :=
  | add_case_pi (ty: type) (si: signedness) (* pointer, int *)
  | add_case_pl (ty: type) (* pointer, long *)
  | add_case_ip (si: signedness) (ty: type) (* int, pointer *)
  | add_case_lp (ty: type) (* long, pointer *)
  | add_default. (* numerical type, numerical type *)

Definition classify_add (ty1: type) (ty2: type) :=
  match typeconv ty1, typeconv ty2 with
  | Tpointer ty _, Tint _ si _ => add_case_pi ty si
  | Tpointer ty _, Tlong _ _ => add_case_pl ty
  | Tint _ si _, Tpointer ty _ => add_case_ip si ty
  | Tlong _ _, Tpointer ty _ => add_case_lp ty
  | _, _ => add_default
  end.

Definition ptrofs_of_int (si: signedness) (n: int) : ptrofs :=
  match si with
  | Signed => Ptrofs.of_ints n
  | Unsigned => Ptrofs.of_intu n
  end.

Definition sem_add_ptr_int (cenv: composite_env) (ty: type) (si: signedness) (v1 v2: val): option val :=
  match v1, v2 with
  | Vptr b1 ofs1, Vint n2 =>
      let n2 := ptrofs_of_int si n2 in
      Some (Vptr b1 (Ptrofs.add ofs1 (Ptrofs.mul (Ptrofs.repr (sizeof cenv ty)) n2)))
  | Vint n1, Vint n2 =>
      if Archi.ptr64 then None else Some (Vint (Int.add n1 (Int.mul (Int.repr (sizeof cenv ty)) n2)))
  | Vlong n1, Vint n2 =>
      let n2 := cast_int_long si n2 in
      if Archi.ptr64 then Some (Vlong (Int64.add n1 (Int64.mul (Int64.repr (sizeof cenv ty)) n2))) else None
  | _, _ => None
  end.

Definition sem_add_ptr_long (cenv: composite_env) (ty: type) (v1 v2: val): option val :=
  match v1, v2 with
  | Vptr b1 ofs1, Vlong n2 =>
      let n2 := Ptrofs.of_int64 n2 in
      Some (Vptr b1 (Ptrofs.add ofs1 (Ptrofs.mul (Ptrofs.repr (sizeof cenv ty)) n2)))
  | Vint n1, Vlong n2 =>
      let n2 := Int.repr (Int64.unsigned n2) in
      if Archi.ptr64 then None else Some (Vint (Int.add n1 (Int.mul (Int.repr (sizeof cenv ty)) n2)))
  | Vlong n1, Vlong n2 =>
      if Archi.ptr64 then Some (Vlong (Int64.add n1 (Int64.mul (Int64.repr (sizeof cenv ty)) n2))) else None
  | _, _ => None
  end.

Definition sem_add (cenv: composite_env) (v1:val) (t1:type) (v2: val) (t2:type) (m: T): option val :=
  match classify_add t1 t2 with
  | add_case_pi ty si => (* pointer plus integer *)
      sem_add_ptr_int cenv ty si v1 v2
  | add_case_pl ty => (* pointer plus long *)
      sem_add_ptr_long cenv ty v1 v2
  | add_case_ip si ty => (* integer plus pointer *)
      sem_add_ptr_int cenv ty si v2 v1
  | add_case_lp ty => (* long plus pointer *)
      sem_add_ptr_long cenv ty v2 v1
  | add_default =>
      sem_binarith
        (fun sg n1 n2 => Some(Vint(Int.add n1 n2)))
        (fun sg n1 n2 => Some(Vlong(Int64.add n1 n2)))
        (fun n1 n2 => Some(Vfloat(Float.add n1 n2)))
        (fun n1 n2 => Some(Vsingle(Float32.add n1 n2)))
        v1 t1 v2 t2 m
  end.

Subtraction

Inductive classify_sub_cases : Type :=
  | sub_case_pi (ty: type) (si: signedness) (* pointer, int *)
  | sub_case_pp (ty: type) (* pointer, pointer *)
  | sub_case_pl (ty: type) (* pointer, long *)
  | sub_default. (* numerical type, numerical type *)

Definition classify_sub (ty1: type) (ty2: type) :=
  match typeconv ty1, typeconv ty2 with
  | Tpointer ty _, Tint _ si _ => sub_case_pi ty si
  | Tpointer ty _ , Tpointer _ _ => sub_case_pp ty
  | Tpointer ty _, Tlong _ _ => sub_case_pl ty
  | _, _ => sub_default
  end.

Definition sem_sub (cenv: composite_env) (v1:val) (t1:type) (v2: val) (t2:type) (m:T): option val :=
  match classify_sub t1 t2 with
  | sub_case_pi ty si => (* pointer minus integer *)
      match v1, v2 with
      | Vptr b1 ofs1, Vint n2 =>
          let n2 := ptrofs_of_int si n2 in
          Some (Vptr b1 (Ptrofs.sub ofs1 (Ptrofs.mul (Ptrofs.repr (sizeof cenv ty)) n2)))
      | Vint n1, Vint n2 =>
          if Archi.ptr64 then None else Some (Vint (Int.sub n1 (Int.mul (Int.repr (sizeof cenv ty)) n2)))
      | Vlong n1, Vint n2 =>
          let n2 := cast_int_long si n2 in
          if Archi.ptr64 then Some (Vlong (Int64.sub n1 (Int64.mul (Int64.repr (sizeof cenv ty)) n2))) else None
      | _, _ => None
      end
  | sub_case_pl ty => (* pointer minus long *)
      match v1, v2 with
      | Vptr b1 ofs1, Vlong n2 =>
          let n2 := Ptrofs.of_int64 n2 in
          Some (Vptr b1 (Ptrofs.sub ofs1 (Ptrofs.mul (Ptrofs.repr (sizeof cenv ty)) n2)))
      | Vint n1, Vlong n2 =>
          let n2 := Int.repr (Int64.unsigned n2) in
          if Archi.ptr64 then None else Some (Vint (Int.sub n1 (Int.mul (Int.repr (sizeof cenv ty)) n2)))
      | Vlong n1, Vlong n2 =>
          if Archi.ptr64 then Some (Vlong (Int64.sub n1 (Int64.mul (Int64.repr (sizeof cenv ty)) n2))) else None
      | _, _ => None
      end
  | sub_case_pp ty => (* pointer minus pointer *)
      match v1,v2 with
      | Vptr b1 ofs1, Vptr b2 ofs2 =>
          if eq_block b1 b2 then
            let sz := sizeof cenv ty in
            if zlt 0 sz && zle sz Ptrofs.max_signed
            then Some (Vptrofs (Ptrofs.divs (Ptrofs.sub ofs1 ofs2) (Ptrofs.repr sz)))
            else None
          else None
      | _, _ => None
      end
  | sub_default =>
      sem_binarith
        (fun sg n1 n2 => Some(Vint(Int.sub n1 n2)))
        (fun sg n1 n2 => Some(Vlong(Int64.sub n1 n2)))
        (fun n1 n2 => Some(Vfloat(Float.sub n1 n2)))
        (fun n1 n2 => Some(Vsingle(Float32.sub n1 n2)))
        v1 t1 v2 t2 m
  end.

Multiplication, division, modulus

Definition sem_mul (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 => Some(Vint(Int.mul n1 n2)))
    (fun sg n1 n2 => Some(Vlong(Int64.mul n1 n2)))
    (fun n1 n2 => Some(Vfloat(Float.mul n1 n2)))
    (fun n1 n2 => Some(Vsingle(Float32.mul n1 n2)))
    v1 t1 v2 t2 m.

Definition sem_div (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 =>
      match sg with
      | Signed =>
          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))
      | Unsigned =>
          if Int.eq n2 Int.zero
          then None else Some(Vint(Int.divu n1 n2))
      end)
    (fun sg n1 n2 =>
      match sg with
      | Signed =>
          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))
      | Unsigned =>
          if Int64.eq n2 Int64.zero
          then None else Some(Vlong(Int64.divu n1 n2))
      end)
    (fun n1 n2 => Some(Vfloat(Float.div n1 n2)))
    (fun n1 n2 => Some(Vsingle(Float32.div n1 n2)))
    v1 t1 v2 t2 m.

Definition sem_mod (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 =>
      match sg with
      | Signed =>
          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))
      | Unsigned =>
          if Int.eq n2 Int.zero
          then None else Some(Vint(Int.modu n1 n2))
      end)
    (fun sg n1 n2 =>
      match sg with
      | Signed =>
          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))
      | Unsigned =>
          if Int64.eq n2 Int64.zero
          then None else Some(Vlong(Int64.modu n1 n2))
      end)
    (fun n1 n2 => None)
    (fun n1 n2 => None)
    v1 t1 v2 t2 m.

Definition sem_and (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 => Some(Vint(Int.and n1 n2)))
    (fun sg n1 n2 => Some(Vlong(Int64.and n1 n2)))
    (fun n1 n2 => None)
    (fun n1 n2 => None)
    v1 t1 v2 t2 m.

Definition sem_or (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 => Some(Vint(Int.or n1 n2)))
    (fun sg n1 n2 => Some(Vlong(Int64.or n1 n2)))
    (fun n1 n2 => None)
    (fun n1 n2 => None)
    v1 t1 v2 t2 m.

Definition sem_xor (v1:val) (t1:type) (v2: val) (t2:type) (m:T) : option val :=
  sem_binarith
    (fun sg n1 n2 => Some(Vint(Int.xor n1 n2)))
    (fun sg n1 n2 => Some(Vlong(Int64.xor n1 n2)))
    (fun n1 n2 => None)
    (fun n1 n2 => None)
    v1 t1 v2 t2 m.

Shifts

Inductive classify_shift_cases : Type:=
  | shift_case_ii(s: signedness) (* int , int *)
  | shift_case_ll(s: signedness) (* long, long *)
  | shift_case_il(s: signedness) (* int, long *)
  | shift_case_li(s: signedness) (* long, int *)
  | shift_default.

Definition classify_shift (ty1: type) (ty2: type) :=
  match typeconv ty1, typeconv ty2 with
  | Tint I32 Unsigned _, Tint _ _ _ => shift_case_ii Unsigned
  | Tint _ _ _, Tint _ _ _ => shift_case_ii Signed
  | Tint I32 Unsigned _, Tlong _ _ => shift_case_il Unsigned
  | Tint _ _ _, Tlong _ _ => shift_case_il Signed
  | Tlong s _, Tint _ _ _ => shift_case_li s
  | Tlong s _, Tlong _ _ => shift_case_ll s
  | _,_ => shift_default
  end.

Definition sem_shift
    (sem_int: signedness -> int -> int -> int)
    (sem_long: signedness -> int64 -> int64 -> int64)
    (v1: val) (t1: type) (v2: val) (t2: type) : option val :=
  match classify_shift t1 t2 with
  | shift_case_ii sg =>
      match v1, v2 with
      | Vint n1, Vint n2 =>
          if Int.ltu n2 Int.iwordsize
          then Some(Vint(sem_int sg n1 n2)) else None
      | _, _ => None
      end
  | shift_case_il sg =>
      match v1, v2 with
      | Vint n1, Vlong n2 =>
          if Int64.ltu n2 (Int64.repr 32)
          then Some(Vint(sem_int sg n1 (Int64.loword n2))) else None
      | _, _ => None
      end
  | shift_case_li sg =>
      match v1, v2 with
      | Vlong n1, Vint n2 =>
          if Int.ltu n2 Int64.iwordsize'
          then Some(Vlong(sem_long sg n1 (Int64.repr (Int.unsigned n2)))) else None
      | _, _ => None
      end
  | shift_case_ll sg =>
      match v1, v2 with
      | Vlong n1, Vlong n2 =>
          if Int64.ltu n2 Int64.iwordsize
          then Some(Vlong(sem_long sg n1 n2)) else None
      | _, _ => None
      end
  | shift_default => None
  end.

Definition sem_shl (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
  sem_shift
    (fun sg n1 n2 => Int.shl n1 n2)
    (fun sg n1 n2 => Int64.shl n1 n2)
    v1 t1 v2 t2.

Definition sem_shr (v1:val) (t1:type) (v2: val) (t2:type) : option val :=
  sem_shift
    (fun sg n1 n2 => match sg with Signed => Int.shr n1 n2 | Unsigned => Int.shru n1 n2 end)
    (fun sg n1 n2 => match sg with Signed => Int64.shr n1 n2 | Unsigned => Int64.shru n1 n2 end)
    v1 t1 v2 t2.

Comparisons

Inductive classify_cmp_cases : Type :=
  | cmp_case_pp (* pointer, pointer *)
  | cmp_case_pi (si: signedness) (* pointer, int *)
  | cmp_case_ip (si: signedness) (* int, pointer *)
  | cmp_case_pl (* pointer, long *)
  | cmp_case_lp (* long, pointer *)
  | cmp_default. (* numerical, numerical *)

Definition classify_cmp (ty1: type) (ty2: type) :=
  match typeconv ty1, typeconv ty2 with
  | Tpointer _ _ , Tpointer _ _ => cmp_case_pp
  | Tpointer _ _ , Tint _ si _ => cmp_case_pi si
  | Tint _ si _, Tpointer _ _ => cmp_case_ip si
  | Tpointer _ _ , Tlong _ _ => cmp_case_pl
  | Tlong _ _ , Tpointer _ _ => cmp_case_lp
  | _, _ => cmp_default
  end.

Definition cmp_ptr (m: T) (c: comparison) (v1 v2: val): option val :=
  option_map Val.of_bool
   (if Archi.ptr64
    then Val.cmplu_bool (valid_pointer m) c v1 v2
    else Val.cmpu_bool (valid_pointer m) c v1 v2).

Definition sem_cmp (c:comparison)
                  (v1: val) (t1: type) (v2: val) (t2: type)
                  (m: T): option val :=
  match classify_cmp t1 t2 with
  | cmp_case_pp =>
      cmp_ptr m c v1 v2
  | cmp_case_pi si =>
      match v2 with
      | Vint n2 =>
          let v2' := Vptrofs (ptrofs_of_int si n2) in
          cmp_ptr m c v1 v2'
      | Vptr b ofs =>
          if Archi.ptr64 then None else cmp_ptr m c v1 v2
      | _ =>
          None
      end
  | cmp_case_ip si =>
      match v1 with
      | Vint n1 =>
          let v1' := Vptrofs (ptrofs_of_int si n1) in
          cmp_ptr m c v1' v2
      | Vptr b ofs =>
          if Archi.ptr64 then None else cmp_ptr m c v1 v2
      | _ =>
          None
      end
  | cmp_case_pl =>
      match v2 with
      | Vlong n2 =>
          let v2' := Vptrofs (Ptrofs.of_int64 n2) in
          cmp_ptr m c v1 v2'
      | Vptr b ofs =>
          if Archi.ptr64 then cmp_ptr m c v1 v2 else None
      | _ =>
          None
      end
  | cmp_case_lp =>
      match v1 with
      | Vlong n1 =>
          let v1' := Vptrofs (Ptrofs.of_int64 n1) in
          cmp_ptr m c v1' v2
      | Vptr b ofs =>
          if Archi.ptr64 then cmp_ptr m c v1 v2 else None
      | _ =>
          None
      end
  | cmp_default =>
      sem_binarith
        (fun sg n1 n2 =>
            Some(Val.of_bool(match sg with Signed => Int.cmp c n1 n2 | Unsigned => Int.cmpu c n1 n2 end)))
        (fun sg n1 n2 =>
            Some(Val.of_bool(match sg with Signed => Int64.cmp c n1 n2 | Unsigned => Int64.cmpu c n1 n2 end)))
        (fun n1 n2 =>
            Some(Val.of_bool(Float.cmp c n1 n2)))
        (fun n1 n2 =>
            Some(Val.of_bool(Float32.cmp c n1 n2)))
        v1 t1 v2 t2 m
  end.

Function applications

Inductive classify_fun_cases : Type :=
  | fun_case_f (targs: typelist) (tres: type) (cc: calling_convention) (* (pointer to) function *)
  | fun_default.

Definition classify_fun (ty: type) :=
  match ty with
  | Tfunction args res cc => fun_case_f args res cc
  | Tpointer (Tfunction args res cc) _ => fun_case_f args res cc
  | _ => fun_default
  end.

Argument of a switch statement

Inductive classify_switch_cases : Type :=
  | switch_case_i
  | switch_case_l
  | switch_default.

Definition classify_switch (ty: type) :=
  match ty with
  | Tint _ _ _ => switch_case_i
  | Tlong _ _ => switch_case_l
  | _ => switch_default
  end.

Definition sem_switch_arg (v: val) (ty: type): option Z :=
  match classify_switch ty with
  | switch_case_i =>
      match v with Vint n => Some(Int.unsigned n) | _ => None end
  | switch_case_l =>
      match v with Vlong n => Some(Int64.unsigned n) | _ => None end
  | switch_default =>
      None
  end.

Combined semantics of unary and binary operators

Definition sem_unary_operation
            (op: unary_operation) (v: val) (ty: type) (m: T): option val :=
  match op with
  | Onotbool => sem_notbool v ty m
  | Onotint => sem_notint v ty
  | Oneg => sem_neg v ty
  | Oabsfloat => sem_absfloat v ty
  end.

Definition sem_binary_operation
    (cenv: composite_env)
    (op: binary_operation)
    (v1: val) (t1: type) (v2: val) (t2:type)
    (m: T): option val :=
  match op with
  | Oadd => sem_add cenv v1 t1 v2 t2 m
  | Osub => sem_sub cenv v1 t1 v2 t2 m
  | Omul => sem_mul v1 t1 v2 t2 m
  | Omod => sem_mod v1 t1 v2 t2 m
  | Odiv => sem_div v1 t1 v2 t2 m
  | Oand => sem_and v1 t1 v2 t2 m
  | Oor => sem_or v1 t1 v2 t2 m
  | Oxor => sem_xor v1 t1 v2 t2 m
  | Oshl => sem_shl v1 t1 v2 t2
  | Oshr => sem_shr v1 t1 v2 t2
  | Oeq => sem_cmp Ceq v1 t1 v2 t2 m
  | One => sem_cmp Cne v1 t1 v2 t2 m
  | Olt => sem_cmp Clt v1 t1 v2 t2 m
  | Ogt => sem_cmp Cgt v1 t1 v2 t2 m
  | Ole => sem_cmp Cle v1 t1 v2 t2 m
  | Oge => sem_cmp Cge v1 t1 v2 t2 m
  end.

Definition sem_incrdecr (cenv: composite_env) (id: incr_or_decr) (v: val) (ty: type) (m: T) :=
  match id with
  | Incr => sem_add cenv v ty (Vint Int.one) type_int32s m
  | Decr => sem_sub cenv v ty (Vint Int.one) type_int32s m
  end.

Definition incrdecr_type (ty: type) :=
  match typeconv ty with
  | Tpointer ty a => Tpointer ty a
  | Tint sz sg a => Tint sz sg noattr
  | Tlong sg a => Tlong sg noattr
  | Tfloat sz a => Tfloat sz noattr
  | _ => Tvoid
  end.

End WITHSEMCAST.

Global Instance sem_cast_unit: SemCast (fun (_: unit) _ _ => false) := {}.

Global Instance sem_cast_mem `{memory_model_ops: Mem.MemoryModelOps}:
  SemCast Mem.valid_pointer
  := {}.

Section WITHMEMORYMODEL.
Context `{memory_model_prf: Mem.MemoryModel}.

Lemma weak_valid_pointer_eq:
  weak_valid_pointer = Mem.weak_valid_pointer.

Lemma sem_cast_unit_to_mem (m: mem) (u: unit) v t1 t2 v':
  sem_cast v t1 t2 u = Some v' ->
  sem_cast v t1 t2 m = Some v'.

Lemma bool_val_unit_to_mem (m: mem) (u: unit) v t v':
  bool_val v t u = Some v' ->
  bool_val v t m = Some v'.

Lemma sem_notbool_unit_to_mem (m: mem) (u: unit) v ty v':
  sem_notbool v ty u = Some v' ->
  sem_notbool v ty m = Some v'.

Lemma sem_binarith_unit_to_mem
      (m: mem) (u: unit)
      sem_int sem_long sem_float sem_single
      v1 t1 v2 t2
      v':
  sem_binarith sem_int sem_long sem_float sem_single v1 t1 v2 t2 u = Some v' ->
  sem_binarith sem_int sem_long sem_float sem_single v1 t1 v2 t2 m = Some v'.

Lemma sem_add_unit_to_mem
      (m: mem) (u: unit)
      cenv v1 t1 v2 t2 v':
  sem_add cenv v1 t1 v2 t2 u = Some v' ->
  sem_add cenv v1 t1 v2 t2 m = Some v'.

Lemma sem_sub_unit_to_mem
      (m: mem) (u: unit)
      cenv v1 t1 v2 t2 v':
  sem_sub cenv v1 t1 v2 t2 u = Some v' ->
  sem_sub cenv v1 t1 v2 t2 m = Some v'.

Lemma sem_mul_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_mul v1 t1 v2 t2 u = Some v' ->
  sem_mul v1 t1 v2 t2 m = Some v'.

Lemma sem_div_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_div v1 t1 v2 t2 u = Some v' ->
  sem_div v1 t1 v2 t2 m = Some v'.

Lemma sem_mod_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_mod v1 t1 v2 t2 u = Some v' ->
  sem_mod v1 t1 v2 t2 m = Some v'.

Lemma sem_and_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_and v1 t1 v2 t2 u = Some v' ->
  sem_and v1 t1 v2 t2 m = Some v'.

Lemma sem_or_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_or v1 t1 v2 t2 u = Some v' ->
  sem_or v1 t1 v2 t2 m = Some v'.

Lemma sem_xor_unit_to_mem
      (m: mem) (u: unit)
      v1 t1 v2 t2 v':
  sem_xor v1 t1 v2 t2 u = Some v' ->
  sem_xor v1 t1 v2 t2 m = Some v'.

Lemma option_of_bool_cmpu_unit_to_mem
      (m: mem)
      c v1 v2 v':
  option_map Val.of_bool
             (Val.cmpu_bool (fun (_ : block) (_ : Z) => false) c v1 v2) =
  Some v' ->
  option_map Val.of_bool (Val.cmpu_bool (Mem.valid_pointer m) c v1 v2) =
  Some v'.


Lemma option_of_bool_cmplu_unit_to_mem
      (m: mem)
      c v1 v2 v':
  option_map Val.of_bool
             (Val.cmplu_bool (fun (_ : block) (_ : Z) => false) c v1 v2) =
  Some v' ->
  option_map Val.of_bool (Val.cmplu_bool (Mem.valid_pointer m) c v1 v2) =
  Some v'.

Lemma cmp_ptr_unit_to_mem:
  forall u m c v1 v2 v',
    cmp_ptr (T := unit) (valid_pointer := fun _ _ _ => false) u c v1 v2 = Some v' ->
    cmp_ptr (T := mem) (valid_pointer := Mem.valid_pointer) m c v1 v2 = Some v'.

Lemma sem_cmp_unit_to_mem
      (m: mem) (u: unit)
      c
      v1 t1 v2 t2
      v':
  sem_cmp c v1 t1 v2 t2 u = Some v' ->
  sem_cmp c v1 t1 v2 t2 m = Some v'.

Lemma sem_unary_operation_unit_to_mem
      (m: mem) (u: unit)
      op v ty v':
  sem_unary_operation op v ty u = Some v' ->
  sem_unary_operation op v ty m = Some v'.

Lemma sem_binary_operation_unit_to_mem
      (m: mem) (u: unit)
      cenv op v1 t1 v2 t2 v':
  sem_binary_operation cenv op v1 t1 v2 t2 u = Some v' ->
  sem_binary_operation cenv op v1 t1 v2 t2 m = Some v'.

Lemma sem_incrdecr_unit_to_mem
      (m: mem) (u: unit)
      cenv id v ty v':
  sem_incrdecr cenv id v ty u = Some v' ->
  sem_incrdecr cenv id v ty m = Some v'.

Compatibility with extensions and injections

Section GENERIC_INJECTION.

Variable f: meminj.
Variables m m': mem.

Hypothesis valid_pointer_inj:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.valid_pointer m b1 (Ptrofs.unsigned ofs) = true ->
  Mem.valid_pointer m' b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.

Hypothesis weak_valid_pointer_inj:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.weak_valid_pointer m b1 (Ptrofs.unsigned ofs) = true ->
  Mem.weak_valid_pointer m' b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.

Hypothesis weak_valid_pointer_no_overflow:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.weak_valid_pointer m b1 (Ptrofs.unsigned ofs) = true ->
  0 <= Ptrofs.unsigned ofs + Ptrofs.unsigned (Ptrofs.repr delta) <= Ptrofs.max_unsigned.

Hypothesis valid_different_pointers_inj:
  forall b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m b1 (Ptrofs.unsigned ofs1) = true ->
  Mem.valid_pointer m b2 (Ptrofs.unsigned ofs2) = true ->
  f b1 = Some (b1', delta1) ->
  f b2 = Some (b2', delta2) ->
  b1' <> b2' \/
  Ptrofs.unsigned (Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned (Ptrofs.add ofs2 (Ptrofs.repr delta2)).

Remark val_inject_vtrue: forall f, Val.inject f Vtrue Vtrue.

Remark val_inject_vfalse: forall f, Val.inject f Vfalse Vfalse.

Remark val_inject_of_bool: forall f b, Val.inject f (Val.of_bool b) (Val.of_bool b).

Remark val_inject_vptrofs: forall n, Val.inject f (Vptrofs n) (Vptrofs n).

Hint Resolve val_inject_vtrue val_inject_vfalse val_inject_of_bool val_inject_vptrofs.

Ltac TrivialInject :=
  match goal with
  | [ H: None = Some _ |- _ ] => discriminate
  | [ H: Some _ = Some _ |- _ ] => inv H; TrivialInject
  | [ H: match ?x with Some _ => _ | None => _ end = Some _ |- _ ] => destruct x; TrivialInject
  | [ H: match ?x with true => _ | false => _ end = Some _ |- _ ] => destruct x eqn:?; TrivialInject
  | [ |- exists v', Some ?v = Some v' /\ _ ] => exists v; split; auto
  | _ => idtac
  end.

Lemma sem_cast_inj:
  forall v1 ty1 ty v tv1,
  sem_cast v1 ty1 ty m = Some v ->
  Val.inject f v1 tv1 ->
  exists tv, sem_cast tv1 ty1 ty m'= Some tv /\ Val.inject f v tv.

Lemma bool_val_inj:
  forall v ty b tv,
  bool_val v ty m = Some b ->
  Val.inject f v tv ->
  bool_val tv ty m' = Some b.

Lemma sem_unary_operation_inj:
  forall op v1 ty v tv1,
  sem_unary_operation op v1 ty m = Some v ->
  Val.inject f v1 tv1 ->
  exists tv, sem_unary_operation op tv1 ty m' = Some tv /\ Val.inject f v tv.

Definition optval_self_injects (ov: option val) : Prop :=
  match ov with
  | Some (Vptr b ofs) => False
  | _ => True
  end.

Remark sem_binarith_inject:
  forall sem_int sem_long sem_float sem_single v1 t1 v2 t2 v v1' v2',
  sem_binarith sem_int sem_long sem_float sem_single v1 t1 v2 t2 m = Some v ->
  Val.inject f v1 v1' -> Val.inject f v2 v2' ->
  (forall sg n1 n2, optval_self_injects (sem_int sg n1 n2)) ->
  (forall sg n1 n2, optval_self_injects (sem_long sg n1 n2)) ->
  (forall n1 n2, optval_self_injects (sem_float n1 n2)) ->
  (forall n1 n2, optval_self_injects (sem_single n1 n2)) ->
  exists v', sem_binarith sem_int sem_long sem_float sem_single v1' t1 v2' t2 m' = Some v' /\ Val.inject f v v'.

Remark sem_shift_inject:
  forall sem_int sem_long v1 t1 v2 t2 v v1' v2',
  sem_shift sem_int sem_long v1 t1 v2 t2 = Some v ->
  Val.inject f v1 v1' -> Val.inject f v2 v2' ->
  exists v', sem_shift sem_int sem_long v1' t1 v2' t2 = Some v' /\ Val.inject f v v'.

Remark sem_cmp_ptr_inj:
  forall c v1 v2 v tv1 tv2,
  cmp_ptr (valid_pointer := Mem.valid_pointer) m c v1 v2 = Some v ->
  Val.inject f v1 tv1 ->
  Val.inject f v2 tv2 ->
  exists tv, cmp_ptr (valid_pointer := Mem.valid_pointer) m' c tv1 tv2 = Some tv /\ Val.inject f v tv.

Remark sem_cmp_inj:
  forall cmp v1 tv1 ty1 v2 tv2 ty2 v,
  sem_cmp cmp v1 ty1 v2 ty2 m = Some v ->
  Val.inject f v1 tv1 ->
  Val.inject f v2 tv2 ->
  exists tv, sem_cmp cmp tv1 ty1 tv2 ty2 m' = Some tv /\ Val.inject f v tv.

Lemma sem_binary_operation_inj:
  forall cenv op v1 ty1 v2 ty2 v tv1 tv2,
  sem_binary_operation cenv op v1 ty1 v2 ty2 m = Some v ->
  Val.inject f v1 tv1 -> Val.inject f v2 tv2 ->
  exists tv, sem_binary_operation cenv op tv1 ty1 tv2 ty2 m' = Some tv /\ Val.inject f v tv.

End GENERIC_INJECTION.

Context {injperm: InjectPerm}.

Lemma sem_cast_inject:
  forall f g v1 ty1 ty m v tv1 tm,
  sem_cast v1 ty1 ty m = Some v ->
  Val.inject f v1 tv1 ->
  Mem.inject f g m tm ->
  exists tv, sem_cast tv1 ty1 ty tm = Some tv /\ Val.inject f v tv.

Lemma sem_unary_operation_inject:
  forall f g m m' op v1 ty1 v tv1,
  sem_unary_operation op v1 ty1 m = Some v ->
  Val.inject f v1 tv1 ->
  Mem.inject f g m m' ->
  exists tv, sem_unary_operation op tv1 ty1 m' = Some tv /\ Val.inject f v tv.

Lemma sem_binary_operation_inject:
  forall f g m m' cenv op v1 ty1 v2 ty2 v tv1 tv2,
  sem_binary_operation cenv op v1 ty1 v2 ty2 m = Some v ->
  Val.inject f v1 tv1 -> Val.inject f v2 tv2 ->
  Mem.inject f g m m' ->
  exists tv, sem_binary_operation cenv op tv1 ty1 tv2 ty2 m' = Some tv /\ Val.inject f v tv.

Lemma bool_val_inject:
  forall f g m m' v ty b tv,
  bool_val v ty m = Some b ->
  Val.inject f v tv ->
  Mem.inject f g m m' ->
  bool_val tv ty m' = Some b.

Some properties of operator semantics

Lemma cast_bool_bool_val:
  forall v t m,
  sem_cast v t (Tint IBool Signed noattr) m =
  match bool_val v t m with None => None | Some b => Some(Val.of_bool b) end.
  intros.
  assert (A: classify_bool t =
    match t with
    | Tint _ _ _ => bool_case_i
    | Tpointer _ _ | Tarray _ _ _ | Tfunction _ _ _ => if Archi.ptr64 then bool_case_l else bool_case_i
    | Tfloat F64 _ => bool_case_f
    | Tfloat F32 _ => bool_case_s
    | Tlong _ _ => bool_case_l
    | _ => bool_default
    end).
  {
    unfold classify_bool; destruct t; simpl; auto. destruct i; auto.
  }
  unfold bool_val. rewrite A.
  unfold sem_cast, classify_cast; remember Archi.ptr64 as ptr64; destruct t; simpl; auto; destruct v; auto;
    try rewrite weak_valid_pointer_eq in * |- *.
  destruct (Int.eq i0 Int.zero); auto.
  destruct ptr64; auto. destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i0)); auto.
  destruct (Int64.eq i Int64.zero); auto.
  destruct (negb ptr64); auto. destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct f; auto.
  destruct f; auto.
  destruct f; auto.
  destruct f; auto.
  destruct (Float.cmp Ceq f0 Float.zero); auto.
  destruct f; auto.
  destruct (Float32.cmp Ceq f0 Float32.zero); auto.
  destruct f; auto.
  destruct ptr64; auto.
  destruct (Int.eq i Int.zero); auto.
  destruct ptr64; auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Int64.eq i Int64.zero); auto.
  destruct ptr64; auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Int.eq i Int.zero); auto.
  destruct ptr64; auto. destruct (Int64.eq i Int64.zero); auto.
  destruct ptr64; auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Int.eq i Int.zero); auto.
  destruct ptr64; auto. destruct (Int64.eq i Int64.zero); auto.
  destruct ptr64; auto.
  destruct ptr64; auto.
  destruct ptr64; auto. destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
  destruct (Mem.weak_valid_pointer m b (Ptrofs.unsigned i)); auto.
Qed.