Module Op


Operators and addressing modes. The abstract syntax and dynamic semantics for the CminorSel, RTL, LTL and Mach languages depend on the following types, defined in this library: These types are PowerPC-specific and correspond roughly to what the processor can compute in one instruction. In other terms, these types reflect the state of the program after instruction selection. For a processor-independent set of operations, see the abstract syntax and dynamic semantics of the Cminor language.

Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Globalenvs.
Require Import Events.

Set Implicit Arguments.
Local Transparent Archi.ptr64.

Conditions (boolean-valued operators).

Inductive condition : Type :=
  | Ccomp: comparison -> condition (* signed integer comparison *)
  | Ccompu: comparison -> condition (* unsigned integer comparison *)
  | Ccompimm: comparison -> int -> condition (* signed integer comparison with a constant *)
  | Ccompuimm: comparison -> int -> condition (* unsigned integer comparison with a constant *)
  | Ccompf: comparison -> condition (* floating-point comparison *)
  | Cnotcompf: comparison -> condition (* negation of a floating-point comparison *)
  | Cmaskzero: int -> condition (* test (arg & constant) == 0 *)
  | Cmasknotzero: int -> condition. (* test (arg & constant) != 0 *)

Arithmetic and logical operations. In the descriptions, rd is the result of the operation and r1, r2, etc, are the arguments.

Inductive operation : Type :=
  | Omove: operation (* rd = r1 *)
  | Ointconst: int -> operation (* rd is set to the given integer constant *)
  | Ofloatconst: float -> operation (* rd is set to the given float constant *)
  | Osingleconst: float32 -> operation (* rd is set to the given float constant *)
  | Oaddrsymbol: ident -> ptrofs -> operation (* rd is set to the the address of the symbol plus the offset *)
  | Oaddrstack: ptrofs -> operation (* rd is set to the stack pointer plus the given offset *)
  | Ocast8signed: operation (* rd is 8-bit sign extension of r1 *)
  | Ocast16signed: operation (* rd is 16-bit sign extension of r1 *)
  | Oadd: operation (* rd = r1 + r2 *)
  | Oaddimm: int -> operation (* rd = r1 + n *)
  | Oaddsymbol: ident -> ptrofs -> operation (* rd = addr(id + ofs) + r1 *)
  | Osub: operation (* rd = r1 - r2 *)
  | Osubimm: int -> operation (* rd = n - r1 *)
  | Omul: operation (* rd = r1 * r2 *)
  | Omulimm: int -> operation (* rd = r1 * n *)
  | Omulhs: operation (* rd = high part of r1 * r2, signed *)
  | Omulhu: operation (* rd = high part of r1 * r2, unsigned *)
  | Odiv: operation (* rd = r1 / r2 (signed) *)
  | Odivu: operation (* rd = r1 / r2 (unsigned) *)
  | Oand: operation (* rd = r1 & r2 *)
  | Oandimm: int -> operation (* rd = r1 & n *)
  | Oor: operation (* rd = r1 | r2 *)
  | Oorimm: int -> operation (* rd = r1 | n *)
  | Oxor: operation (* rd = r1 ^ r2 *)
  | Oxorimm: int -> operation (* rd = r1 ^ n *)
  | Onot: operation (* rd = ~r1 *)
  | Onand: operation (* rd = ~(r1 & r2) *)
  | Onor: operation (* rd = ~(r1 | r2) *)
  | Onxor: operation (* rd = ~(r1 ^ r2) *)
  | Oandc: operation (* rd = r1 & ~r2 *)
  | Oorc: operation (* rd = r1 | ~r2 *)
  | Oshl: operation (* rd = r1 << r2 *)
  | Oshr: operation (* rd = r1 >> r2 (signed) *)
  | Oshrimm: int -> operation (* rd = r1 >> n (signed) *)
  | Oshrximm: int -> operation (* rd = r1 / 2^n (signed) *)
  | Oshru: operation (* rd = r1 >> r2 (unsigned) *)
  | Orolm: int -> int -> operation (* rotate left and mask *)
  | Oroli: int -> int -> operation (* rotate left and insert *)
  | Onegf: operation (* rd = - r1 *)
  | Oabsf: operation (* rd = abs(r1) *)
  | Oaddf: operation (* rd = r1 + r2 *)
  | Osubf: operation (* rd = r1 - r2 *)
  | Omulf: operation (* rd = r1 * r2 *)
  | Odivf: operation (* rd = r1 / r2 *)
  | Onegfs: operation (* rd = - r1 *)
  | Oabsfs: operation (* rd = abs(r1) *)
  | Oaddfs: operation (* rd = r1 + r2 *)
  | Osubfs: operation (* rd = r1 - r2 *)
  | Omulfs: operation (* rd = r1 * r2 *)
  | Odivfs: operation (* rd = r1 / r2 *)
  | Osingleoffloat: operation (* rd is r1 truncated to single-precision float *)
  | Ofloatofsingle: operation (* rd is r1 extended to double-precision float *)
  | Ointoffloat: operation (* rd = signed_int_of_float(r1) *)
  | Ointuoffloat: operation (* rd = unsigned_int_of_float(r1) (PPC64 only) *)
  | Ofloatofint: operation (* rd = float_of_signed_int(r1) (PPC64 only) *)
  | Ofloatofintu: operation (* rd = float_of_unsigned_int(r1) (PPC64 only *)
  | Ofloatofwords: operation (* rd = float_of_words(r1,r2) *)
  | Omakelong: operation (* rd = r1 << 32 | r2 *)
  | Olowlong: operation (* rd = low-word(r1) *)
  | Ohighlong: operation (* rd = high-word(r1) *)
  | Ocmp: condition -> operation. (* rd = 1 if condition holds, rd = 0 otherwise. *)

Addressing modes. r1, r2, etc, are the arguments to the addressing.

Inductive addressing: Type :=
  | Aindexed: int -> addressing (* Address is r1 + offset *)
  | Aindexed2: addressing (* Address is r1 + r2 *)
  | Aglobal: ident -> ptrofs -> addressing (* Address is symbol + offset *)
  | Abased: ident -> ptrofs -> addressing (* Address is symbol + offset + r1 *)
  | Ainstack: ptrofs -> addressing. (* Address is stack_pointer + offset *)

Comparison functions (used in module CSE).

Definition eq_condition (x y: condition) : {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec; intro.
  assert (forall (x y: comparison), {x=y}+{x<>y}). decide equality.
  decide equality.
Defined.


Definition eq_operation (x y: operation): {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec Ptrofs.eq_dec ident_eq; intro.
  generalize Float.eq_dec Float32.eq_dec; intros.
  generalize eq_condition; intro.
  decide equality.
Defined.

Definition eq_addressing (x y: addressing) : {x=y} + {x<>y}.
Proof.
  generalize Int.eq_dec Ptrofs.eq_dec ident_eq; intro.
  decide equality.
Defined.

Global Opaque eq_condition eq_addressing eq_operation.

Evaluation functions


Evaluation of conditions, operators and addressing modes applied to lists of values. Return None when the computation can trigger an error, e.g. integer division by zero. eval_condition returns a boolean, eval_operation and eval_addressing return a value.

Definition eval_condition (cond: condition) (vl: list val) (m: mem): option bool :=
  match cond, vl with
  | Ccomp c, v1 :: v2 :: nil => Val.cmp_bool c v1 v2
  | Ccompu c, v1 :: v2 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 v2
  | Ccompimm c n, v1 :: nil => Val.cmp_bool c v1 (Vint n)
  | Ccompuimm c n, v1 :: nil => Val.cmpu_bool (Mem.valid_pointer m) c v1 (Vint n)
  | Ccompf c, v1 :: v2 :: nil => Val.cmpf_bool c v1 v2
  | Cnotcompf c, v1 :: v2 :: nil => option_map negb (Val.cmpf_bool c v1 v2)
  | Cmaskzero n, v1 :: nil => Val.maskzero_bool v1 n
  | Cmasknotzero n, v1 :: nil => option_map negb (Val.maskzero_bool v1 n)
  | _, _ => None
  end.

Definition eval_operation
             (F V: Type) (genv: Genv.t F V) (sp: val)
             (op: operation) (vl: list val) (m: mem): option val :=
  match op, vl with
  | Omove, v1::nil => Some v1
  | Ointconst n, nil => Some (Vint n)
  | Ofloatconst n, nil => Some (Vfloat n)
  | Osingleconst n, nil => Some (Vsingle n)
  | Oaddrsymbol s ofs, nil => Some (Genv.symbol_address genv s ofs)
  | Oaddrstack ofs, nil => Some (Val.offset_ptr sp ofs)
  | Ocast8signed, v1::nil => Some (Val.sign_ext 8 v1)
  | Ocast16signed, v1::nil => Some (Val.sign_ext 16 v1)
  | Oadd, v1::v2::nil => Some (Val.add v1 v2)
  | Oaddimm n, v1::nil => Some (Val.add v1 (Vint n))
  | Oaddsymbol s ofs, v1::nil => Some (Val.add (Genv.symbol_address genv s ofs) v1)
  | Osub, v1::v2::nil => Some (Val.sub v1 v2)
  | Osubimm n, v1::nil => Some (Val.sub (Vint n) v1)
  | Omul, v1::v2::nil => Some (Val.mul v1 v2)
  | Omulimm n, v1::nil => Some (Val.mul v1 (Vint n))
  | Omulhs, v1::v2::nil => Some (Val.mulhs v1 v2)
  | Omulhu, v1::v2::nil => Some (Val.mulhu v1 v2)
  | Odiv, v1::v2::nil => Val.divs v1 v2
  | Odivu, v1::v2::nil => Val.divu v1 v2
  | Oand, v1::v2::nil => Some(Val.and v1 v2)
  | Oandimm n, v1::nil => Some (Val.and v1 (Vint n))
  | Oor, v1::v2::nil => Some(Val.or v1 v2)
  | Oorimm n, v1::nil => Some (Val.or v1 (Vint n))
  | Oxor, v1::v2::nil => Some(Val.xor v1 v2)
  | Oxorimm n, v1::nil => Some (Val.xor v1 (Vint n))
  | Onot, v1::nil => Some(Val.notint v1)
  | Onand, v1::v2::nil => Some (Val.notint (Val.and v1 v2))
  | Onor, v1::v2::nil => Some (Val.notint (Val.or v1 v2))
  | Onxor, v1::v2::nil => Some (Val.notint (Val.xor v1 v2))
  | Oandc, v1::v2::nil => Some (Val.and v1 (Val.notint v2))
  | Oorc, v1::v2::nil => Some (Val.or v1 (Val.notint v2))
  | Oshl, v1::v2::nil => Some (Val.shl v1 v2)
  | Oshr, v1::v2::nil => Some (Val.shr v1 v2)
  | Oshrimm n, v1::nil => Some (Val.shr v1 (Vint n))
  | Oshrximm n, v1::nil => Val.shrx v1 (Vint n)
  | Oshru, v1::v2::nil => Some (Val.shru v1 v2)
  | Orolm amount mask, v1::nil => Some (Val.rolm v1 amount mask)
  | Oroli amount mask, v1::v2::nil =>
      Some(Val.or (Val.and v1 (Vint (Int.not mask))) (Val.rolm v2 amount mask))
  | Onegf, v1::nil => Some(Val.negf v1)
  | Oabsf, v1::nil => Some(Val.absf v1)
  | Oaddf, v1::v2::nil => Some(Val.addf v1 v2)
  | Osubf, v1::v2::nil => Some(Val.subf v1 v2)
  | Omulf, v1::v2::nil => Some(Val.mulf v1 v2)
  | Odivf, v1::v2::nil => Some(Val.divf v1 v2)
  | Onegfs, v1::nil => Some(Val.negfs v1)
  | Oabsfs, v1::nil => Some(Val.absfs v1)
  | Oaddfs, v1::v2::nil => Some(Val.addfs v1 v2)
  | Osubfs, v1::v2::nil => Some(Val.subfs v1 v2)
  | Omulfs, v1::v2::nil => Some(Val.mulfs v1 v2)
  | Odivfs, v1::v2::nil => Some(Val.divfs v1 v2)
  | Osingleoffloat, v1::nil => Some(Val.singleoffloat v1)
  | Ofloatofsingle, v1::nil => Some(Val.floatofsingle v1)
  | Ointoffloat, v1::nil => Val.intoffloat v1
  | Ointuoffloat, v1::nil => Val.intuoffloat v1
  | Ofloatofint, v1::nil => Val.floatofint v1
  | Ofloatofintu, v1::nil => Val.floatofintu v1
  | Ofloatofwords, v1::v2::nil => Some(Val.floatofwords v1 v2)
  | Omakelong, v1::v2::nil => Some(Val.longofwords v1 v2)
  | Olowlong, v1::nil => Some(Val.loword v1)
  | Ohighlong, v1::nil => Some(Val.hiword v1)
  | Ocmp c, _ => Some(Val.of_optbool (eval_condition c vl m))
  | _, _ => None
  end.

Definition eval_addressing
    (F V: Type) (genv: Genv.t F V) (sp: val)
    (addr: addressing) (vl: list val) : option val :=
  match addr, vl with
  | Aindexed n, v1::nil => Some (Val.add v1 (Vint n))
  | Aindexed2, v1::v2::nil => Some (Val.add v1 v2)
  | Aglobal s ofs, nil => Some (Genv.symbol_address genv s ofs)
  | Abased s ofs, v1::nil => Some (Val.add (Genv.symbol_address genv s ofs) v1)
  | Ainstack ofs, nil => Some(Val.offset_ptr sp ofs)
  | _, _ => None
  end.

Remark eval_addressing_Ainstack:
  forall (F V: Type) (genv: Genv.t F V) sp ofs,
  eval_addressing genv sp (Ainstack ofs) nil = Some (Val.offset_ptr sp ofs).
Proof.
  intros. reflexivity.
Qed.

Remark eval_addressing_Ainstack_inv:
  forall (F V: Type) (genv: Genv.t F V) sp ofs vl v,
  eval_addressing genv sp (Ainstack ofs) vl = Some v -> vl = nil /\ v = Val.offset_ptr sp ofs.
Proof.
  unfold eval_addressing; intros; destruct vl; inv H; auto.
Qed.

Ltac FuncInv :=
  match goal with
  | H: (match ?x with nil => _ | _ :: _ => _ end = Some _) |- _ =>
      destruct x; simpl in H; try discriminate; FuncInv
  | H: (match ?v with Vundef => _ | Vint _ => _ | Vlong _ => _ | Vfloat _ => _ | Vptr _ _ => _ end = Some _) |- _ =>
      destruct v; simpl in H; try discriminate; FuncInv
  | H: (Some _ = Some _) |- _ =>
      injection H; intros; clear H; FuncInv
  | _ =>
      idtac
  end.

Static typing of conditions, operators and addressing modes.


Definition type_of_condition (c: condition) : list typ :=
  match c with
  | Ccomp _ => Tint :: Tint :: nil
  | Ccompu _ => Tint :: Tint :: nil
  | Ccompimm _ _ => Tint :: nil
  | Ccompuimm _ _ => Tint :: nil
  | Ccompf _ => Tfloat :: Tfloat :: nil
  | Cnotcompf _ => Tfloat :: Tfloat :: nil
  | Cmaskzero _ => Tint :: nil
  | Cmasknotzero _ => Tint :: nil
  end.

Definition type_of_operation (op: operation) : list typ * typ :=
  match op with
  | Omove => (nil, Tint)
  | Ointconst _ => (nil, Tint)
  | Ofloatconst f => (nil, Tfloat)
  | Osingleconst f => (nil, Tsingle)
  | Oaddrsymbol _ _ => (nil, Tint)
  | Oaddrstack _ => (nil, Tint)
  | Ocast8signed => (Tint :: nil, Tint)
  | Ocast16signed => (Tint :: nil, Tint)
  | Oadd => (Tint :: Tint :: nil, Tint)
  | Oaddimm _ => (Tint :: nil, Tint)
  | Oaddsymbol _ _ => (Tint :: nil, Tint)
  | Osub => (Tint :: Tint :: nil, Tint)
  | Osubimm _ => (Tint :: nil, Tint)
  | Omul => (Tint :: Tint :: nil, Tint)
  | Omulimm _ => (Tint :: nil, Tint)
  | Omulhs => (Tint :: Tint :: nil, Tint)
  | Omulhu => (Tint :: Tint :: nil, Tint)
  | Odiv => (Tint :: Tint :: nil, Tint)
  | Odivu => (Tint :: Tint :: nil, Tint)
  | Oand => (Tint :: Tint :: nil, Tint)
  | Oandimm _ => (Tint :: nil, Tint)
  | Oor => (Tint :: Tint :: nil, Tint)
  | Oorimm _ => (Tint :: nil, Tint)
  | Oxor => (Tint :: Tint :: nil, Tint)
  | Oxorimm _ => (Tint :: nil, Tint)
  | Onot => (Tint :: nil, Tint)
  | Onand => (Tint :: Tint :: nil, Tint)
  | Onor => (Tint :: Tint :: nil, Tint)
  | Onxor => (Tint :: Tint :: nil, Tint)
  | Oandc => (Tint :: Tint :: nil, Tint)
  | Oorc => (Tint :: Tint :: nil, Tint)
  | Oshl => (Tint :: Tint :: nil, Tint)
  | Oshr => (Tint :: Tint :: nil, Tint)
  | Oshrimm _ => (Tint :: nil, Tint)
  | Oshrximm _ => (Tint :: nil, Tint)
  | Oshru => (Tint :: Tint :: nil, Tint)
  | Orolm _ _ => (Tint :: nil, Tint)
  | Oroli _ _ => (Tint :: Tint :: nil, Tint)
  | Onegf => (Tfloat :: nil, Tfloat)
  | Oabsf => (Tfloat :: nil, Tfloat)
  | Oaddf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Osubf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Omulf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Odivf => (Tfloat :: Tfloat :: nil, Tfloat)
  | Onegfs => (Tsingle :: nil, Tsingle)
  | Oabsfs => (Tsingle :: nil, Tsingle)
  | Oaddfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Osubfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Omulfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Odivfs => (Tsingle :: Tsingle :: nil, Tsingle)
  | Osingleoffloat => (Tfloat :: nil, Tsingle)
  | Ofloatofsingle => (Tsingle :: nil, Tfloat)
  | Ointoffloat => (Tfloat :: nil, Tint)
  | Ointuoffloat => (Tfloat :: nil, Tint)
  | Ofloatofint => (Tint :: nil, Tfloat)
  | Ofloatofintu => (Tint :: nil, Tfloat)
  | Ofloatofwords => (Tint :: Tint :: nil, Tfloat)
  | Omakelong => (Tint :: Tint :: nil, Tlong)
  | Olowlong => (Tlong :: nil, Tint)
  | Ohighlong => (Tlong :: nil, Tint)
  | Ocmp c => (type_of_condition c, Tint)
  end.

Definition type_of_addressing (addr: addressing) : list typ :=
  match addr with
  | Aindexed _ => Tint :: nil
  | Aindexed2 => Tint :: Tint :: nil
  | Aglobal _ _ => nil
  | Abased _ _ => Tint :: nil
  | Ainstack _ => nil
  end.

Weak type soundness results for eval_operation: the result values, when defined, are always of the type predicted by type_of_operation.

Section SOUNDNESS.

Variable A V: Type.
Variable genv: Genv.t A V.

Lemma type_of_operation_sound:
  forall op vl sp v m,
  op <> Omove ->
  eval_operation genv sp op vl m = Some v ->
  Val.has_type v (snd (type_of_operation op)).
Proof with
(try exact I; try reflexivity).
  intros.
  destruct op; simpl in H0; FuncInv; subst; simpl.
  congruence.
  exact I.
  auto.
  auto.
  unfold Genv.symbol_address. destruct (Genv.find_symbol genv i)...
  destruct sp...
  destruct v0...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  unfold Genv.symbol_address. destruct (Genv.find_symbol genv i)... destruct v0...
  destruct v0; destruct v1... simpl. destruct (eq_block b b0)...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1; simpl in *; inv H0.
    destruct (Int.eq i0 Int.zero
         || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2...
  destruct v0; destruct v1; simpl in *; inv H0. destruct (Int.eq i0 Int.zero); inv H2...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0; simpl... destruct (Int.ltu i Int.iwordsize)...
  destruct v0; simpl in *; inv H0. destruct (Int.ltu i (Int.repr 31)); inv H2...
  destruct v0; destruct v1; simpl... destruct (Int.ltu i0 Int.iwordsize)...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct v0; simpl in H0; inv H0. destruct (Float.to_int f); inv H2...
  destruct v0; simpl in H0; inv H0. destruct (Float.to_intu f); inv H2...
  destruct v0; simpl in H0; inv H0...
  destruct v0; simpl in H0; inv H0...
  destruct v0; destruct v1...
  destruct v0; destruct v1...
  destruct v0...
  destruct v0...
  destruct (eval_condition c vl m); simpl... destruct b...
Qed.

End SOUNDNESS.

Manipulating and transforming operations


Recognition of move operations.

Definition is_move_operation
    (A: Type) (op: operation) (args: list A) : option A :=
  match op, args with
  | Omove, arg :: nil => Some arg
  | _, _ => None
  end.

Lemma is_move_operation_correct:
  forall (A: Type) (op: operation) (args: list A) (a: A),
  is_move_operation op args = Some a ->
  op = Omove /\ args = a :: nil.
Proof.
  intros until a. unfold is_move_operation; destruct op;
  try (intros; discriminate).
  destruct args. intros; discriminate.
  destruct args. intros. intuition congruence.
  intros; discriminate.
Qed.

negate_condition cond returns a condition that is logically equivalent to the negation of cond.

Definition negate_condition (cond: condition): condition :=
  match cond with
  | Ccomp c => Ccomp(negate_comparison c)
  | Ccompu c => Ccompu(negate_comparison c)
  | Ccompimm c n => Ccompimm (negate_comparison c) n
  | Ccompuimm c n => Ccompuimm (negate_comparison c) n
  | Ccompf c => Cnotcompf c
  | Cnotcompf c => Ccompf c
  | Cmaskzero n => Cmasknotzero n
  | Cmasknotzero n => Cmaskzero n
  end.

Lemma eval_negate_condition:
  forall cond vl m,
  eval_condition (negate_condition cond) vl m = option_map negb (eval_condition cond vl m).
Proof.
  intros. destruct cond; simpl.
  repeat (destruct vl; auto). apply Val.negate_cmp_bool.
  repeat (destruct vl; auto). apply Val.negate_cmpu_bool.
  repeat (destruct vl; auto). apply Val.negate_cmp_bool.
  repeat (destruct vl; auto). apply Val.negate_cmpu_bool.
  repeat (destruct vl; auto).
  repeat (destruct vl; auto). destruct (Val.cmpf_bool c v v0); auto. destruct b; auto.
  repeat (destruct vl; auto).
  repeat (destruct vl; auto). destruct (Val.maskzero_bool v i) as [[]|]; auto.
Qed.

Shifting stack-relative references. This is used in Stacking.

Definition shift_stack_addressing (delta: Z) (addr: addressing) :=
  match addr with
  | Ainstack ofs => Ainstack (Ptrofs.add (Ptrofs.repr delta) ofs)
  | _ => addr
  end.

Definition shift_stack_operation (delta: Z) (op: operation) :=
  match op with
  | Oaddrstack ofs => Oaddrstack (Ptrofs.add (Ptrofs.repr delta) ofs)
  | _ => op
  end.

Lemma type_shift_stack_addressing:
  forall delta addr, type_of_addressing (shift_stack_addressing delta addr) = type_of_addressing addr.
Proof.
  intros. destruct addr; auto.
Qed.

Lemma type_shift_stack_operation:
  forall delta op, type_of_operation (shift_stack_operation delta op) = type_of_operation op.
Proof.
  intros. destruct op; auto.
Qed.

Lemma eval_shift_stack_addressing:
  forall F V (ge: Genv.t F V) sp addr vl delta,
  eval_addressing ge (Vptr sp Ptrofs.zero) (shift_stack_addressing delta addr) vl =
  eval_addressing ge (Vptr sp (Ptrofs.repr delta)) addr vl.
Proof.
  intros. destruct addr; simpl; auto.
  rewrite Ptrofs.add_zero_l; auto.
Qed.

Lemma eval_shift_stack_operation:
  forall F V (ge: Genv.t F V) sp op vl m delta,
  eval_operation ge (Vptr sp Ptrofs.zero) (shift_stack_operation delta op) vl m =
  eval_operation ge (Vptr sp (Ptrofs.repr delta)) op vl m.
Proof.
  intros. destruct op; simpl; auto.
  rewrite Ptrofs.add_zero_l; auto.
Qed.

Offset an addressing mode addr by a quantity delta, so that it designates the pointer delta bytes past the pointer designated by addr. May be undefined, in which case None is returned.

Definition offset_addressing (addr: addressing) (delta: Z) : option addressing :=
  match addr with
  | Aindexed n => Some(Aindexed (Int.add n (Int.repr delta)))
  | Aindexed2 => None
  | Aglobal s n => Some(Aglobal s (Ptrofs.add n (Ptrofs.repr delta)))
  | Abased s n => Some(Abased s (Ptrofs.add n (Ptrofs.repr delta)))
  | Ainstack n => Some(Ainstack (Ptrofs.add n (Ptrofs.repr delta)))
  end.

Lemma eval_offset_addressing:
  forall (F V: Type) (ge: Genv.t F V) sp addr args delta addr' v,
  offset_addressing addr delta = Some addr' ->
  eval_addressing ge sp addr args = Some v ->
  eval_addressing ge sp addr' args = Some(Val.add v (Vint (Int.repr delta))).
Proof.
  intros.
  assert (D: Ptrofs.repr delta = Ptrofs.of_int (Int.repr delta)) by (symmetry; auto with ptrofs).
  destruct addr; simpl in H; inv H; simpl in *; FuncInv; subst.
- rewrite Val.add_assoc; auto.
- unfold Genv.symbol_address. destruct (Genv.find_symbol ge i); auto. rewrite D; auto.
- unfold Genv.symbol_address. destruct (Genv.find_symbol ge i); auto.
  rewrite Val.add_assoc. rewrite Val.add_permut. rewrite Val.add_commut.
  simpl. rewrite D. auto.
- destruct sp; simpl; auto. rewrite Ptrofs.add_assoc, D. auto.
Qed.

Operations that are so cheap to recompute that CSE should not factor them out.

Definition is_trivial_op (op: operation) : bool :=
  match op with
  | Omove => true
  | Ointconst _ => true
  | Oaddrsymbol _ _ => true
  | Oaddrstack _ => true
  | _ => false
  end.

Operations that depend on the memory state.

Definition op_depends_on_memory (op: operation) : bool :=
  match op with
  | Ocmp (Ccompu _) => true
  | Ocmp (Ccompuimm _ _) => true
  | _ => false
  end.

Lemma op_depends_on_memory_correct:
  forall (F V: Type) (ge: Genv.t F V) sp op args m1 m2,
  op_depends_on_memory op = false ->
  eval_operation ge sp op args m1 = eval_operation ge sp op args m2.
Proof.
  intros until m2. destruct op; simpl; try congruence. unfold eval_condition.
  destruct c; simpl; auto; try discriminate.
Qed.

Global variables mentioned in an operation or addressing mode

Definition globals_operation (op: operation) : list ident :=
  match op with
  | Oaddrsymbol s ofs => s :: nil
  | Oaddsymbol s ofs => s :: nil
  | _ => nil
  end.

Definition globals_addressing (addr: addressing) : list ident :=
  match addr with
  | Aglobal s n => s :: nil
  | Abased s n => s :: nil
  | _ => nil
  end.

Invariance and compatibility properties.


eval_operation and eval_addressing depend on a global environment for resolving references to global symbols. We show that they give the same results if a global environment is replaced by another that assigns the same addresses to the same symbols.

Section GENV_TRANSF.

Variable F1 F2 V1 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.
Hypothesis agree_on_symbols:
  forall (s: ident), Genv.find_symbol ge2 s = Genv.find_symbol ge1 s.

Remark symbol_address_preserved:
  forall s ofs, Genv.symbol_address ge2 s ofs = Genv.symbol_address ge1 s ofs.
Proof.
  unfold Genv.symbol_address; intros. rewrite agree_on_symbols; auto.
Qed.

Lemma eval_operation_preserved:
  forall sp op vl m,
  eval_operation ge2 sp op vl m = eval_operation ge1 sp op vl m.
Proof.
  intros. destruct op; simpl; auto; rewrite symbol_address_preserved; auto.
Qed.

Lemma eval_addressing_preserved:
  forall sp addr vl,
  eval_addressing ge2 sp addr vl = eval_addressing ge1 sp addr vl.
Proof.
  intros. destruct addr; simpl; auto; rewrite symbol_address_preserved; auto.
Qed.

End GENV_TRANSF.

Compatibility of the evaluation functions with value injections.

Section EVAL_COMPAT.

Variable F1 F2 V1 V2: Type.
Variable ge1: Genv.t F1 V1.
Variable ge2: Genv.t F2 V2.
Variable f: meminj.

Variable m1: mem.
Variable m2: mem.

Hypothesis valid_pointer_inj:
  forall b1 ofs b2 delta,
  f b1 = Some(b2, delta) ->
  Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true ->
  Mem.valid_pointer m2 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 m1 b1 (Ptrofs.unsigned ofs) = true ->
  Mem.weak_valid_pointer m2 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 m1 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 m1 b1 (Ptrofs.unsigned ofs1) = true ->
  Mem.valid_pointer m1 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)).

Ltac InvInject :=
  match goal with
  | [ H: Val.inject _ (Vint _) _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject _ (Vfloat _) _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject _ (Vsingle _) _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject _ (Vptr _ _) _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject_list _ nil _ |- _ ] =>
      inv H; InvInject
  | [ H: Val.inject_list _ (_ :: _) _ |- _ ] =>
      inv H; InvInject
  | _ => idtac
  end.

Lemma eval_condition_inj:
  forall cond vl1 vl2 b,
  Val.inject_list f vl1 vl2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. destruct cond; simpl in H0; FuncInv; InvInject; simpl; auto.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  eauto 3 using Val.cmpu_bool_inject, Mem.valid_pointer_implies.
  inv H3; simpl in H0; inv H0; auto.
  eauto 3 using Val.cmpu_bool_inject, Mem.valid_pointer_implies.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  inv H3; inv H2; simpl in H0; inv H0; auto.
  inv H3; try discriminate; auto.
  inv H3; try discriminate; auto.
Qed.

Ltac TrivialExists :=
  match goal with
  | [ |- exists v2, Some ?v1 = Some v2 /\ Val.inject _ _ v2 ] =>
      exists v1; split; auto
  | _ => idtac
  end.

Lemma eval_operation_inj:
  forall op sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_operation op) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_operation ge1 sp1 op vl1 m1 = Some v1 ->
  exists v2, eval_operation ge2 sp2 op vl2 m2 = Some v2 /\ Val.inject f v1 v2.
Proof.
  intros until v1; intros GL; intros. destruct op; simpl in H1; simpl; FuncInv; InvInject; TrivialExists.
  apply GL; simpl; auto.
  apply Val.offset_ptr_inject; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  apply Val.add_inject; auto.
  apply Val.add_inject; auto.
  apply Val.add_inject; auto. apply GL; simpl; auto.
  apply Val.sub_inject; auto.
  inv H4; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H3; simpl in H1; inv H1. simpl.
    destruct (Int.eq i0 Int.zero
         || Int.eq i (Int.repr Int.min_signed) && Int.eq i0 Int.mone); inv H2. TrivialExists.
  inv H4; inv H3; simpl in H1; inv H1. simpl.
    destruct (Int.eq i0 Int.zero); inv H2. TrivialExists.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; simpl; auto. destruct (Int.ltu i Int.iwordsize); auto.
  inv H4; simpl in *; inv H1. destruct (Int.ltu i (Int.repr 31)); inv H2. econstructor; eauto.
  inv H4; inv H2; simpl; auto. destruct (Int.ltu i0 Int.iwordsize); auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl in H1; inv H1. simpl. destruct (Float.to_int f0); simpl in H2; inv H2.
  exists (Vint i); auto.
  inv H4; simpl in H1; inv H1. simpl. destruct (Float.to_intu f0); simpl in H2; inv H2.
  exists (Vint i); auto.
  inv H4; simpl in H1; inv H1; simpl. TrivialExists.
  inv H4; simpl in H1; inv H1; simpl. TrivialExists.
  inv H4; inv H2; simpl; auto.
  inv H4; inv H2; simpl; auto.
  inv H4; simpl; auto.
  inv H4; simpl; auto.
  subst. destruct (eval_condition c vl1 m1) eqn:?.
  exploit eval_condition_inj; eauto. intros EQ; rewrite EQ.
  destruct b; simpl; constructor.
  simpl; constructor.
Qed.

Lemma eval_addressing_inj:
  forall addr sp1 vl1 sp2 vl2 v1,
  (forall id ofs,
      In id (globals_addressing addr) ->
      Val.inject f (Genv.symbol_address ge1 id ofs) (Genv.symbol_address ge2 id ofs)) ->
  Val.inject f sp1 sp2 ->
  Val.inject_list f vl1 vl2 ->
  eval_addressing ge1 sp1 addr vl1 = Some v1 ->
  exists v2, eval_addressing ge2 sp2 addr vl2 = Some v2 /\ Val.inject f v1 v2.
Proof.
  intros. destruct addr; simpl in H2; simpl; FuncInv; InvInject; TrivialExists;
    auto using Val.add_inject, Val.offset_ptr_inject.
  apply H; simpl; auto.
  apply Val.add_inject; auto. apply H; simpl; auto.
Qed.

End EVAL_COMPAT.

Compatibility of the evaluation functions with the ``is less defined'' relation over values.

Section EVAL_LESSDEF.

Variable F V: Type.
Variable genv: Genv.t F V.

Remark valid_pointer_extends:
  forall m1 m2, Mem.extends m1 m2 ->
  forall b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true ->
  Mem.valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.
Proof.
  intros. inv H0. rewrite Ptrofs.add_zero. eapply Mem.valid_pointer_extends; eauto.
Qed.

Remark weak_valid_pointer_extends:
  forall m1 m2, Mem.extends m1 m2 ->
  forall b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true ->
  Mem.weak_valid_pointer m2 b2 (Ptrofs.unsigned (Ptrofs.add ofs (Ptrofs.repr delta))) = true.
Proof.
  intros. inv H0. rewrite Ptrofs.add_zero. eapply Mem.weak_valid_pointer_extends; eauto.
Qed.

Remark weak_valid_pointer_no_overflow_extends:
  forall m1 b1 ofs b2 delta,
  Some(b1, 0) = Some(b2, delta) ->
  Mem.weak_valid_pointer m1 b1 (Ptrofs.unsigned ofs) = true ->
  0 <= Ptrofs.unsigned ofs + Ptrofs.unsigned (Ptrofs.repr delta) <= Ptrofs.max_unsigned.
Proof.
  intros. inv H. rewrite Zplus_0_r. apply Ptrofs.unsigned_range_2.
Qed.

Remark valid_different_pointers_extends:
  forall m1 b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,
  b1 <> b2 ->
  Mem.valid_pointer m1 b1 (Ptrofs.unsigned ofs1) = true ->
  Mem.valid_pointer m1 b2 (Ptrofs.unsigned ofs2) = true ->
  Some(b1, 0) = Some (b1', delta1) ->
  Some(b2, 0) = Some (b2', delta2) ->
  b1' <> b2' \/
  Ptrofs.unsigned(Ptrofs.add ofs1 (Ptrofs.repr delta1)) <> Ptrofs.unsigned(Ptrofs.add ofs2 (Ptrofs.repr delta2)).
Proof.
  intros. inv H2; inv H3. auto.
Qed.

Lemma eval_condition_lessdef:
  forall cond vl1 vl2 b m1 m2,
  Val.lessdef_list vl1 vl2 ->
  Mem.extends m1 m2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. eapply eval_condition_inj with (f := fun b => Some(b, 0)) (m1 := m1).
  apply valid_pointer_extends; auto.
  apply weak_valid_pointer_extends; auto.
  apply weak_valid_pointer_no_overflow_extends; auto.
  apply valid_different_pointers_extends; auto.
  rewrite <- val_inject_list_lessdef. eauto. auto.
Qed.

Lemma eval_operation_lessdef:
  forall sp op vl1 vl2 v1 m1 m2,
  Val.lessdef_list vl1 vl2 ->
  Mem.extends m1 m2 ->
  eval_operation genv sp op vl1 m1 = Some v1 ->
  exists v2, eval_operation genv sp op vl2 m2 = Some v2 /\ Val.lessdef v1 v2.
Proof.
  intros. rewrite val_inject_list_lessdef in H.
  assert (exists v2 : val,
          eval_operation genv sp op vl2 m2 = Some v2
          /\ Val.inject (fun b => Some(b, 0)) v1 v2).
  eapply eval_operation_inj with (m1 := m1) (sp1 := sp).
  apply valid_pointer_extends; auto.
  apply weak_valid_pointer_extends; auto.
  apply weak_valid_pointer_no_overflow_extends; auto.
  apply valid_different_pointers_extends; auto.
  intros. rewrite <- val_inject_lessdef; auto.
  rewrite <- val_inject_lessdef; auto.
  eauto. auto.
  destruct H2 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto.
Qed.

Lemma eval_addressing_lessdef:
  forall sp addr vl1 vl2 v1,
  Val.lessdef_list vl1 vl2 ->
  eval_addressing genv sp addr vl1 = Some v1 ->
  exists v2, eval_addressing genv sp addr vl2 = Some v2 /\ Val.lessdef v1 v2.
Proof.
  intros. rewrite val_inject_list_lessdef in H.
  assert (exists v2 : val,
          eval_addressing genv sp addr vl2 = Some v2
          /\ Val.inject (fun b => Some(b, 0)) v1 v2).
  eapply eval_addressing_inj with (sp1 := sp).
  intros. rewrite <- val_inject_lessdef; auto.
  rewrite <- val_inject_lessdef; auto.
  eauto. auto.
  destruct H1 as [v2 [A B]]. exists v2; split; auto. rewrite val_inject_lessdef; auto.
Qed.

End EVAL_LESSDEF.

Compatibility of the evaluation functions with memory injections.

Section EVAL_INJECT.

Variable F V: Type.
Variable genv: Genv.t F V.
Variable f: meminj.
Hypothesis globals: meminj_preserves_globals genv f.
Variable sp1: block.
Variable sp2: block.
Variable delta: Z.
Hypothesis sp_inj: f sp1 = Some(sp2, delta).

Remark symbol_address_inject:
  forall id ofs, Val.inject f (Genv.symbol_address genv id ofs) (Genv.symbol_address genv id ofs).
Proof.
  intros. unfold Genv.symbol_address. destruct (Genv.find_symbol genv id) eqn:?; auto.
  exploit (proj1 globals); eauto. intros.
  econstructor; eauto. rewrite Ptrofs.add_zero; auto.
Qed.

Lemma eval_condition_inject:
  forall cond vl1 vl2 b m1 m2,
  Val.inject_list f vl1 vl2 ->
  Mem.inject f m1 m2 ->
  eval_condition cond vl1 m1 = Some b ->
  eval_condition cond vl2 m2 = Some b.
Proof.
  intros. eapply eval_condition_inj with (f := f) (m1 := m1); eauto.
  intros; eapply Mem.valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto.
  intros; eapply Mem.different_pointers_inject; eauto.
Qed.

Lemma eval_addressing_inject:
  forall addr vl1 vl2 v1,
  Val.inject_list f vl1 vl2 ->
  eval_addressing genv (Vptr sp1 Ptrofs.zero) addr vl1 = Some v1 ->
  exists v2,
     eval_addressing genv (Vptr sp2 Ptrofs.zero) (shift_stack_addressing delta addr) vl2 = Some v2
  /\ Val.inject f v1 v2.
Proof.
  intros.
  rewrite eval_shift_stack_addressing.
  eapply eval_addressing_inj with (sp1 := Vptr sp1 Ptrofs.zero); eauto.
  intros. apply symbol_address_inject.
  econstructor; eauto. rewrite Ptrofs.add_zero_l; auto.
Qed.

Lemma eval_operation_inject:
  forall op vl1 vl2 v1 m1 m2,
  Val.inject_list f vl1 vl2 ->
  Mem.inject f m1 m2 ->
  eval_operation genv (Vptr sp1 Ptrofs.zero) op vl1 m1 = Some v1 ->
  exists v2,
     eval_operation genv (Vptr sp2 Ptrofs.zero) (shift_stack_operation delta op) vl2 m2 = Some v2
  /\ Val.inject f v1 v2.
Proof.
  intros.
  rewrite eval_shift_stack_operation. simpl.
  eapply eval_operation_inj with (sp1 := Vptr sp1 Ptrofs.zero) (m1 := m1); eauto.
  intros; eapply Mem.valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_val; eauto.
  intros; eapply Mem.weak_valid_pointer_inject_no_overflow; eauto.
  intros; eapply Mem.different_pointers_inject; eauto.
  intros. apply symbol_address_inject.
  econstructor; eauto. rewrite Ptrofs.add_zero_l; auto.
Qed.

End EVAL_INJECT.

Masks for rotate and mask instructions


Recognition of integers that are acceptable as immediate operands to the rlwim PowerPC instruction. These integers are of the form 000011110000 or 111100001111, that is, a run of one bits surrounded by zero bits, or conversely. We recognize these integers by running the following automaton on the bits. The accepting states are 2, 3, 4, 5, and 6.
               0          1          0
              / \        / \        / \
              \ /        \ /        \ /
        -0--> [1] --1--> [2] --0--> [3]
       /
     [0]
       \
        -1--> [4] --0--> [5] --1--> [6]
              / \        / \        / \
              \ /        \ /        \ /
               1          0          1

Inductive rlw_state: Type :=
  | RLW_S0 : rlw_state
  | RLW_S1 : rlw_state
  | RLW_S2 : rlw_state
  | RLW_S3 : rlw_state
  | RLW_S4 : rlw_state
  | RLW_S5 : rlw_state
  | RLW_S6 : rlw_state
  | RLW_Sbad : rlw_state.

Definition rlw_transition (s: rlw_state) (b: bool) : rlw_state :=
  match s, b with
  | RLW_S0, false => RLW_S1
  | RLW_S0, true => RLW_S4
  | RLW_S1, false => RLW_S1
  | RLW_S1, true => RLW_S2
  | RLW_S2, false => RLW_S3
  | RLW_S2, true => RLW_S2
  | RLW_S3, false => RLW_S3
  | RLW_S3, true => RLW_Sbad
  | RLW_S4, false => RLW_S5
  | RLW_S4, true => RLW_S4
  | RLW_S5, false => RLW_S5
  | RLW_S5, true => RLW_S6
  | RLW_S6, false => RLW_Sbad
  | RLW_S6, true => RLW_S6
  | RLW_Sbad, _ => RLW_Sbad
  end.

Definition rlw_accepting (s: rlw_state) : bool :=
  match s with
  | RLW_S0 => false
  | RLW_S1 => false
  | RLW_S2 => true
  | RLW_S3 => true
  | RLW_S4 => true
  | RLW_S5 => true
  | RLW_S6 => true
  | RLW_Sbad => false
  end.

Fixpoint is_rlw_mask_rec (n: nat) (s: rlw_state) (x: Z) {struct n} : bool :=
  match n with
  | O =>
      rlw_accepting s
  | S m =>
      is_rlw_mask_rec m (rlw_transition s (Z.odd x)) (Z.div2 x)
  end.

Definition is_rlw_mask (x: int) : bool :=
  is_rlw_mask_rec Int.wordsize RLW_S0 (Int.unsigned x).