(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** Static analysis and strength reduction for operators 
  and conditions.  This is the machine-dependent part of [Constprop]. *)

Require Import Coqlib.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Op.
Require Import Registers.

(** * Static analysis *)

(** To each pseudo-register at each program point, the static analysis 
  associates a compile-time approximation taken from the following set. *)

Inductive approx : Type :=
  | Novalue: approx      (** No value possible, code is unreachable. *)
  | Unknown: approx      (** All values are possible,
                             no compile-time information is available. *)
  | I: int -> approx     (** A known integer value. *)
  | F: float -> approx   (** A known floating-point value. *)
  | G: ident -> int -> approx
                         (** The value is the address of the given global
                             symbol plus the given integer offset. *)
  | S: int -> approx.    (** The value is the stack pointer plus the offset. *)

(** We now define the abstract interpretations of conditions and operators
  over this set of approximations.  For instance, the abstract interpretation
  of the operator [Oaddf] applied to two expressions [a] and [b] is
  [F(Float.add f g)] if [a] and [b] have static approximations [Vfloat f]
  and [Vfloat g] respectively, and [Unknown] otherwise.

  The static approximations are defined by large pattern-matchings over
  the approximations of the results.  We write these matchings in the
  indirect style described in file [SelectOp] to avoid excessive
  duplication of cases in proofs. *)

(** Original definition:
<<
Nondetfunction eval_static_condition (cond: condition) (vl: list approx) :=
  match cond, vl with
  | Ccomp c, I n1 :: I n2 :: nil => Some(Int.cmp c n1 n2)
  | Ccompu c, I n1 :: I n2 :: nil => Some(Int.cmpu c n1 n2)
  | Ccompimm c n, I n1 :: nil => Some(Int.cmp c n1 n)
  | Ccompuimm c n, I n1 :: nil => Some(Int.cmpu c n1 n)
  | Ccompf c, F n1 :: F n2 :: nil => Some(Float.cmp c n1 n2)
  | Cnotcompf c, F n1 :: F n2 :: nil => Some(negb(Float.cmp c n1 n2))
  | Cmaskzero n, I n1 :: nil => Some(Int.eq (Int.and n1 n) Int.zero)
  | Cmasknotzero n, I n1::nil => Some(negb(Int.eq (Int.and n1 n) Int.zero))
  | _, _ => None
  end.
>>
*)

Inductive eval_static_condition_cases: forall (cond: condition) (vl: list approx), Type :=
  | eval_static_condition_case1: forall c n1 n2, eval_static_condition_cases (Ccomp c) (I n1 :: I n2 :: nil)
  | eval_static_condition_case2: forall c n1 n2, eval_static_condition_cases (Ccompu c) (I n1 :: I n2 :: nil)
  | eval_static_condition_case3: forall c n n1, eval_static_condition_cases (Ccompimm c n) (I n1 :: nil)
  | eval_static_condition_case4: forall c n n1, eval_static_condition_cases (Ccompuimm c n) (I n1 :: nil)
  | eval_static_condition_case5: forall c n1 n2, eval_static_condition_cases (Ccompf c) (F n1 :: F n2 :: nil)
  | eval_static_condition_case6: forall c n1 n2, eval_static_condition_cases (Cnotcompf c) (F n1 :: F n2 :: nil)
  | eval_static_condition_case7: forall n n1, eval_static_condition_cases (Cmaskzero n) (I n1 :: nil)
  | eval_static_condition_case8: forall n n1, eval_static_condition_cases (Cmasknotzero n) (I n1::nil)
  | eval_static_condition_default: forall (cond: condition) (vl: list approx), eval_static_condition_cases cond vl.

Definition eval_static_condition_match (cond: condition) (vl: list approx) :=
  match cond as zz1, vl as zz2 return eval_static_condition_cases zz1 zz2 with
  | Ccomp c, I n1 :: I n2 :: nil => eval_static_condition_case1 c n1 n2
  | Ccompu c, I n1 :: I n2 :: nil => eval_static_condition_case2 c n1 n2
  | Ccompimm c n, I n1 :: nil => eval_static_condition_case3 c n n1
  | Ccompuimm c n, I n1 :: nil => eval_static_condition_case4 c n n1
  | Ccompf c, F n1 :: F n2 :: nil => eval_static_condition_case5 c n1 n2
  | Cnotcompf c, F n1 :: F n2 :: nil => eval_static_condition_case6 c n1 n2
  | Cmaskzero n, I n1 :: nil => eval_static_condition_case7 n n1
  | Cmasknotzero n, I n1::nil => eval_static_condition_case8 n n1
  | cond, vl => eval_static_condition_default cond vl
  end.

Definition eval_static_condition (cond: condition) (vl: list approx) :=
  match eval_static_condition_match cond vl with
  | eval_static_condition_case1 c n1 n2 => (* Ccomp c, I n1 :: I n2 :: nil *) 
      Some(Int.cmp c n1 n2)
  | eval_static_condition_case2 c n1 n2 => (* Ccompu c, I n1 :: I n2 :: nil *) 
      Some(Int.cmpu c n1 n2)
  | eval_static_condition_case3 c n n1 => (* Ccompimm c n, I n1 :: nil *) 
      Some(Int.cmp c n1 n)
  | eval_static_condition_case4 c n n1 => (* Ccompuimm c n, I n1 :: nil *) 
      Some(Int.cmpu c n1 n)
  | eval_static_condition_case5 c n1 n2 => (* Ccompf c, F n1 :: F n2 :: nil *) 
      Some(Float.cmp c n1 n2)
  | eval_static_condition_case6 c n1 n2 => (* Cnotcompf c, F n1 :: F n2 :: nil *) 
      Some(negb(Float.cmp c n1 n2))
  | eval_static_condition_case7 n n1 => (* Cmaskzero n, I n1 :: nil *) 
      Some(Int.eq (Int.and n1 n) Int.zero)
  | eval_static_condition_case8 n n1 => (* Cmasknotzero n, I n1::nil *) 
      Some(negb(Int.eq (Int.and n1 n) Int.zero))
  | eval_static_condition_default cond vl =>
      None
  end.


Definition eval_static_condition_val (cond: condition) (vl: list approx) :=
  match eval_static_condition cond vl with
  | None => Unknown
  | Some b => I(if b then Int.one else Int.zero)
  end.

Definition eval_static_intoffloat (f: float) :=
  match Float.intoffloat f with Some x => I x | None => Unknown end.

(** Original definition:
<<
Nondetfunction eval_static_addressing (addr: addressing) (vl: list approx) :=
  match addr, vl with
  | Aindexed n, I n1::nil => I (Int.add n1 n)
  | Aindexed n, G id ofs::nil => G id (Int.add ofs n)
  | Aindexed n, S ofs::nil => S (Int.add ofs n)
  | Aindexed2 n, I n1::I n2::nil => I (Int.add (Int.add n1 n2) n)
  | Aindexed2 n, G id ofs::I n2::nil => G id (Int.add (Int.add ofs n2) n)
  | Aindexed2 n, I n1::G id ofs::nil => G id (Int.add (Int.add ofs n1) n)
  | Aindexed2 n, S ofs::I n2::nil => S (Int.add (Int.add ofs n2) n)
  | Aindexed2 n, I n1::S ofs::nil => S (Int.add (Int.add ofs n1) n)
  | Ascaled sc n, I n1::nil => I (Int.add (Int.mul n1 sc) n)
  | Aindexed2scaled sc n, I n1::I n2::nil => I (Int.add n1 (Int.add (Int.mul n2 sc) n))
  | Aindexed2scaled sc n, G id ofs::I n2::nil => G id (Int.add ofs (Int.add (Int.mul n2 sc) n))
  | Aindexed2scaled sc n, S ofs::I n2::nil => S (Int.add ofs (Int.add (Int.mul n2 sc) n))
  | Aglobal id ofs, nil => G id ofs
  | Abased id ofs, I n1::nil => G id (Int.add ofs n1)
  | Abasedscaled sc id ofs, I n1::nil => G id (Int.add ofs (Int.mul sc n1))
  | Ainstack ofs, nil => S ofs
  | _, _ => Unknown
  end.
>>
*)

Inductive eval_static_addressing_cases: forall (addr: addressing) (vl: list approx), Type :=
  | eval_static_addressing_case1: forall n n1, eval_static_addressing_cases (Aindexed n) (I n1::nil)
  | eval_static_addressing_case2: forall n id ofs, eval_static_addressing_cases (Aindexed n) (G id ofs::nil)
  | eval_static_addressing_case3: forall n ofs, eval_static_addressing_cases (Aindexed n) (S ofs::nil)
  | eval_static_addressing_case4: forall n n1 n2, eval_static_addressing_cases (Aindexed2 n) (I n1::I n2::nil)
  | eval_static_addressing_case5: forall n id ofs n2, eval_static_addressing_cases (Aindexed2 n) (G id ofs::I n2::nil)
  | eval_static_addressing_case6: forall n n1 id ofs, eval_static_addressing_cases (Aindexed2 n) (I n1::G id ofs::nil)
  | eval_static_addressing_case7: forall n ofs n2, eval_static_addressing_cases (Aindexed2 n) (S ofs::I n2::nil)
  | eval_static_addressing_case8: forall n n1 ofs, eval_static_addressing_cases (Aindexed2 n) (I n1::S ofs::nil)
  | eval_static_addressing_case9: forall sc n n1, eval_static_addressing_cases (Ascaled sc n) (I n1::nil)
  | eval_static_addressing_case10: forall sc n n1 n2, eval_static_addressing_cases (Aindexed2scaled sc n) (I n1::I n2::nil)
  | eval_static_addressing_case11: forall sc n id ofs n2, eval_static_addressing_cases (Aindexed2scaled sc n) (G id ofs::I n2::nil)
  | eval_static_addressing_case12: forall sc n ofs n2, eval_static_addressing_cases (Aindexed2scaled sc n) (S ofs::I n2::nil)
  | eval_static_addressing_case13: forall id ofs, eval_static_addressing_cases (Aglobal id ofs) (nil)
  | eval_static_addressing_case14: forall id ofs n1, eval_static_addressing_cases (Abased id ofs) (I n1::nil)
  | eval_static_addressing_case15: forall sc id ofs n1, eval_static_addressing_cases (Abasedscaled sc id ofs) (I n1::nil)
  | eval_static_addressing_case16: forall ofs, eval_static_addressing_cases (Ainstack ofs) (nil)
  | eval_static_addressing_default: forall (addr: addressing) (vl: list approx), eval_static_addressing_cases addr vl.

Definition eval_static_addressing_match (addr: addressing) (vl: list approx) :=
  match addr as zz1, vl as zz2 return eval_static_addressing_cases zz1 zz2 with
  | Aindexed n, I n1::nil => eval_static_addressing_case1 n n1
  | Aindexed n, G id ofs::nil => eval_static_addressing_case2 n id ofs
  | Aindexed n, S ofs::nil => eval_static_addressing_case3 n ofs
  | Aindexed2 n, I n1::I n2::nil => eval_static_addressing_case4 n n1 n2
  | Aindexed2 n, G id ofs::I n2::nil => eval_static_addressing_case5 n id ofs n2
  | Aindexed2 n, I n1::G id ofs::nil => eval_static_addressing_case6 n n1 id ofs
  | Aindexed2 n, S ofs::I n2::nil => eval_static_addressing_case7 n ofs n2
  | Aindexed2 n, I n1::S ofs::nil => eval_static_addressing_case8 n n1 ofs
  | Ascaled sc n, I n1::nil => eval_static_addressing_case9 sc n n1
  | Aindexed2scaled sc n, I n1::I n2::nil => eval_static_addressing_case10 sc n n1 n2
  | Aindexed2scaled sc n, G id ofs::I n2::nil => eval_static_addressing_case11 sc n id ofs n2
  | Aindexed2scaled sc n, S ofs::I n2::nil => eval_static_addressing_case12 sc n ofs n2
  | Aglobal id ofs, nil => eval_static_addressing_case13 id ofs
  | Abased id ofs, I n1::nil => eval_static_addressing_case14 id ofs n1
  | Abasedscaled sc id ofs, I n1::nil => eval_static_addressing_case15 sc id ofs n1
  | Ainstack ofs, nil => eval_static_addressing_case16 ofs
  | addr, vl => eval_static_addressing_default addr vl
  end.

Definition eval_static_addressing (addr: addressing) (vl: list approx) :=
  match eval_static_addressing_match addr vl with
  | eval_static_addressing_case1 n n1 => (* Aindexed n, I n1::nil *) 
      I (Int.add n1 n)
  | eval_static_addressing_case2 n id ofs => (* Aindexed n, G id ofs::nil *) 
      G id (Int.add ofs n)
  | eval_static_addressing_case3 n ofs => (* Aindexed n, S ofs::nil *) 
      S (Int.add ofs n)
  | eval_static_addressing_case4 n n1 n2 => (* Aindexed2 n, I n1::I n2::nil *) 
      I (Int.add (Int.add n1 n2) n)
  | eval_static_addressing_case5 n id ofs n2 => (* Aindexed2 n, G id ofs::I n2::nil *) 
      G id (Int.add (Int.add ofs n2) n)
  | eval_static_addressing_case6 n n1 id ofs => (* Aindexed2 n, I n1::G id ofs::nil *) 
      G id (Int.add (Int.add ofs n1) n)
  | eval_static_addressing_case7 n ofs n2 => (* Aindexed2 n, S ofs::I n2::nil *) 
      S (Int.add (Int.add ofs n2) n)
  | eval_static_addressing_case8 n n1 ofs => (* Aindexed2 n, I n1::S ofs::nil *) 
      S (Int.add (Int.add ofs n1) n)
  | eval_static_addressing_case9 sc n n1 => (* Ascaled sc n, I n1::nil *) 
      I (Int.add (Int.mul n1 sc) n)
  | eval_static_addressing_case10 sc n n1 n2 => (* Aindexed2scaled sc n, I n1::I n2::nil *) 
      I (Int.add n1 (Int.add (Int.mul n2 sc) n))
  | eval_static_addressing_case11 sc n id ofs n2 => (* Aindexed2scaled sc n, G id ofs::I n2::nil *) 
      G id (Int.add ofs (Int.add (Int.mul n2 sc) n))
  | eval_static_addressing_case12 sc n ofs n2 => (* Aindexed2scaled sc n, S ofs::I n2::nil *) 
      S (Int.add ofs (Int.add (Int.mul n2 sc) n))
  | eval_static_addressing_case13 id ofs => (* Aglobal id ofs, nil *) 
      G id ofs
  | eval_static_addressing_case14 id ofs n1 => (* Abased id ofs, I n1::nil *) 
      G id (Int.add ofs n1)
  | eval_static_addressing_case15 sc id ofs n1 => (* Abasedscaled sc id ofs, I n1::nil *) 
      G id (Int.add ofs (Int.mul sc n1))
  | eval_static_addressing_case16 ofs => (* Ainstack ofs, nil *) 
      S ofs
  | eval_static_addressing_default addr vl =>
      Unknown
  end.


(** Original definition:
<<
Nondetfunction eval_static_operation (op: operation) (vl: list approx) :=
  match op, vl with
  | Omove, v1::nil => v1
  | Ointconst n, nil => I n
  | Ofloatconst n, nil => F n
  | Ocast8signed, I n1 :: nil => I(Int.sign_ext 8 n1)
  | Ocast8unsigned, I n1 :: nil => I(Int.zero_ext 8 n1)
  | Ocast16signed, I n1 :: nil => I(Int.sign_ext 16 n1)
  | Ocast16unsigned, I n1 :: nil => I(Int.zero_ext 16 n1)
  | Oneg, I n1 :: nil => I(Int.neg n1)
  | Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2)
  | Osub, G s1 n1 :: I n2 :: nil => G s1 (Int.sub n1 n2)
  | Omul, I n1 :: I n2 :: nil => I(Int.mul n1 n2)
  | Omulimm n, I n1 :: nil => I(Int.mul n1 n)
  | Odiv, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
  | Odivu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
  | Omod, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.mods n1 n2)
  | Omodu, I n1 :: I n2 :: nil => if Int.eq n2 Int.zero then Unknown else I(Int.modu n1 n2)
  | Oand, I n1 :: I n2 :: nil => I(Int.and n1 n2)
  | Oandimm n, I n1 :: nil => I(Int.and n1 n)
  | Oor, I n1 :: I n2 :: nil => I(Int.or n1 n2)
  | Oorimm n, I n1 :: nil => I(Int.or n1 n)
  | Oxor, I n1 :: I n2 :: nil => I(Int.xor n1 n2)
  | Oxorimm n, I n1 :: nil => I(Int.xor n1 n)
  | Oshl, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown
  | Oshlimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shl n1 n) else Unknown
  | Oshr, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown
  | Oshrimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown
  | Oshrximm n, I n1 :: nil => if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown
  | Oshru, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown
  | Oshruimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.shru n1 n) else Unknown
  | Ororimm n, I n1 :: nil => if Int.ltu n Int.iwordsize then I(Int.ror n1 n) else Unknown
  | Olea mode, vl => eval_static_addressing mode vl
  | Onegf, F n1 :: nil => F(Float.neg n1)
  | Oabsf, F n1 :: nil => F(Float.abs n1)
  | Oaddf, F n1 :: F n2 :: nil => F(Float.add n1 n2)
  | Osubf, F n1 :: F n2 :: nil => F(Float.sub n1 n2)
  | Omulf, F n1 :: F n2 :: nil => F(Float.mul n1 n2)
  | Odivf, F n1 :: F n2 :: nil => F(Float.div n1 n2)
  | Osingleoffloat, F n1 :: nil => F(Float.singleoffloat n1)
  | Ointoffloat, F n1 :: nil => eval_static_intoffloat n1
  | Ofloatofint, I n1 :: nil => F(Float.floatofint n1)
  | Ocmp c, vl => eval_static_condition_val c vl
  | _, _ => Unknown
  end.
>>
*)

Inductive eval_static_operation_cases: forall (op: operation) (vl: list approx), Type :=
  | eval_static_operation_case1: forall v1, eval_static_operation_cases (Omove) (v1::nil)
  | eval_static_operation_case2: forall n, eval_static_operation_cases (Ointconst n) (nil)
  | eval_static_operation_case3: forall n, eval_static_operation_cases (Ofloatconst n) (nil)
  | eval_static_operation_case4: forall n1, eval_static_operation_cases (Ocast8signed) (I n1 :: nil)
  | eval_static_operation_case5: forall n1, eval_static_operation_cases (Ocast8unsigned) (I n1 :: nil)
  | eval_static_operation_case6: forall n1, eval_static_operation_cases (Ocast16signed) (I n1 :: nil)
  | eval_static_operation_case7: forall n1, eval_static_operation_cases (Ocast16unsigned) (I n1 :: nil)
  | eval_static_operation_case8: forall n1, eval_static_operation_cases (Oneg) (I n1 :: nil)
  | eval_static_operation_case9: forall n1 n2, eval_static_operation_cases (Osub) (I n1 :: I n2 :: nil)
  | eval_static_operation_case10: forall s1 n1 n2, eval_static_operation_cases (Osub) (G s1 n1 :: I n2 :: nil)
  | eval_static_operation_case11: forall n1 n2, eval_static_operation_cases (Omul) (I n1 :: I n2 :: nil)
  | eval_static_operation_case12: forall n n1, eval_static_operation_cases (Omulimm n) (I n1 :: nil)
  | eval_static_operation_case13: forall n1 n2, eval_static_operation_cases (Odiv) (I n1 :: I n2 :: nil)
  | eval_static_operation_case14: forall n1 n2, eval_static_operation_cases (Odivu) (I n1 :: I n2 :: nil)
  | eval_static_operation_case15: forall n1 n2, eval_static_operation_cases (Omod) (I n1 :: I n2 :: nil)
  | eval_static_operation_case16: forall n1 n2, eval_static_operation_cases (Omodu) (I n1 :: I n2 :: nil)
  | eval_static_operation_case17: forall n1 n2, eval_static_operation_cases (Oand) (I n1 :: I n2 :: nil)
  | eval_static_operation_case18: forall n n1, eval_static_operation_cases (Oandimm n) (I n1 :: nil)
  | eval_static_operation_case19: forall n1 n2, eval_static_operation_cases (Oor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case20: forall n n1, eval_static_operation_cases (Oorimm n) (I n1 :: nil)
  | eval_static_operation_case21: forall n1 n2, eval_static_operation_cases (Oxor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case22: forall n n1, eval_static_operation_cases (Oxorimm n) (I n1 :: nil)
  | eval_static_operation_case23: forall n1 n2, eval_static_operation_cases (Oshl) (I n1 :: I n2 :: nil)
  | eval_static_operation_case24: forall n n1, eval_static_operation_cases (Oshlimm n) (I n1 :: nil)
  | eval_static_operation_case25: forall n1 n2, eval_static_operation_cases (Oshr) (I n1 :: I n2 :: nil)
  | eval_static_operation_case26: forall n n1, eval_static_operation_cases (Oshrimm n) (I n1 :: nil)
  | eval_static_operation_case27: forall n n1, eval_static_operation_cases (Oshrximm n) (I n1 :: nil)
  | eval_static_operation_case28: forall n1 n2, eval_static_operation_cases (Oshru) (I n1 :: I n2 :: nil)
  | eval_static_operation_case29: forall n n1, eval_static_operation_cases (Oshruimm n) (I n1 :: nil)
  | eval_static_operation_case30: forall n n1, eval_static_operation_cases (Ororimm n) (I n1 :: nil)
  | eval_static_operation_case31: forall mode vl, eval_static_operation_cases (Olea mode) (vl)
  | eval_static_operation_case32: forall n1, eval_static_operation_cases (Onegf) (F n1 :: nil)
  | eval_static_operation_case33: forall n1, eval_static_operation_cases (Oabsf) (F n1 :: nil)
  | eval_static_operation_case34: forall n1 n2, eval_static_operation_cases (Oaddf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case35: forall n1 n2, eval_static_operation_cases (Osubf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case36: forall n1 n2, eval_static_operation_cases (Omulf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case37: forall n1 n2, eval_static_operation_cases (Odivf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case38: forall n1, eval_static_operation_cases (Osingleoffloat) (F n1 :: nil)
  | eval_static_operation_case39: forall n1, eval_static_operation_cases (Ointoffloat) (F n1 :: nil)
  | eval_static_operation_case40: forall n1, eval_static_operation_cases (Ofloatofint) (I n1 :: nil)
  | eval_static_operation_case41: forall c vl, eval_static_operation_cases (Ocmp c) (vl)
  | eval_static_operation_default: forall (op: operation) (vl: list approx), eval_static_operation_cases op vl.

Definition eval_static_operation_match (op: operation) (vl: list approx) :=
  match op as zz1, vl as zz2 return eval_static_operation_cases zz1 zz2 with
  | Omove, v1::nil => eval_static_operation_case1 v1
  | Ointconst n, nil => eval_static_operation_case2 n
  | Ofloatconst n, nil => eval_static_operation_case3 n
  | Ocast8signed, I n1 :: nil => eval_static_operation_case4 n1
  | Ocast8unsigned, I n1 :: nil => eval_static_operation_case5 n1
  | Ocast16signed, I n1 :: nil => eval_static_operation_case6 n1
  | Ocast16unsigned, I n1 :: nil => eval_static_operation_case7 n1
  | Oneg, I n1 :: nil => eval_static_operation_case8 n1
  | Osub, I n1 :: I n2 :: nil => eval_static_operation_case9 n1 n2
  | Osub, G s1 n1 :: I n2 :: nil => eval_static_operation_case10 s1 n1 n2
  | Omul, I n1 :: I n2 :: nil => eval_static_operation_case11 n1 n2
  | Omulimm n, I n1 :: nil => eval_static_operation_case12 n n1
  | Odiv, I n1 :: I n2 :: nil => eval_static_operation_case13 n1 n2
  | Odivu, I n1 :: I n2 :: nil => eval_static_operation_case14 n1 n2
  | Omod, I n1 :: I n2 :: nil => eval_static_operation_case15 n1 n2
  | Omodu, I n1 :: I n2 :: nil => eval_static_operation_case16 n1 n2
  | Oand, I n1 :: I n2 :: nil => eval_static_operation_case17 n1 n2
  | Oandimm n, I n1 :: nil => eval_static_operation_case18 n n1
  | Oor, I n1 :: I n2 :: nil => eval_static_operation_case19 n1 n2
  | Oorimm n, I n1 :: nil => eval_static_operation_case20 n n1
  | Oxor, I n1 :: I n2 :: nil => eval_static_operation_case21 n1 n2
  | Oxorimm n, I n1 :: nil => eval_static_operation_case22 n n1
  | Oshl, I n1 :: I n2 :: nil => eval_static_operation_case23 n1 n2
  | Oshlimm n, I n1 :: nil => eval_static_operation_case24 n n1
  | Oshr, I n1 :: I n2 :: nil => eval_static_operation_case25 n1 n2
  | Oshrimm n, I n1 :: nil => eval_static_operation_case26 n n1
  | Oshrximm n, I n1 :: nil => eval_static_operation_case27 n n1
  | Oshru, I n1 :: I n2 :: nil => eval_static_operation_case28 n1 n2
  | Oshruimm n, I n1 :: nil => eval_static_operation_case29 n n1
  | Ororimm n, I n1 :: nil => eval_static_operation_case30 n n1
  | Olea mode, vl => eval_static_operation_case31 mode vl
  | Onegf, F n1 :: nil => eval_static_operation_case32 n1
  | Oabsf, F n1 :: nil => eval_static_operation_case33 n1
  | Oaddf, F n1 :: F n2 :: nil => eval_static_operation_case34 n1 n2
  | Osubf, F n1 :: F n2 :: nil => eval_static_operation_case35 n1 n2
  | Omulf, F n1 :: F n2 :: nil => eval_static_operation_case36 n1 n2
  | Odivf, F n1 :: F n2 :: nil => eval_static_operation_case37 n1 n2
  | Osingleoffloat, F n1 :: nil => eval_static_operation_case38 n1
  | Ointoffloat, F n1 :: nil => eval_static_operation_case39 n1
  | Ofloatofint, I n1 :: nil => eval_static_operation_case40 n1
  | Ocmp c, vl => eval_static_operation_case41 c vl
  | op, vl => eval_static_operation_default op vl
  end.

Definition eval_static_operation (op: operation) (vl: list approx) :=
  match eval_static_operation_match op vl with
  | eval_static_operation_case1 v1 => (* Omove, v1::nil *) 
      v1
  | eval_static_operation_case2 n => (* Ointconst n, nil *) 
      I n
  | eval_static_operation_case3 n => (* Ofloatconst n, nil *) 
      F n
  | eval_static_operation_case4 n1 => (* Ocast8signed, I n1 :: nil *) 
      I(Int.sign_ext 8 n1)
  | eval_static_operation_case5 n1 => (* Ocast8unsigned, I n1 :: nil *) 
      I(Int.zero_ext 8 n1)
  | eval_static_operation_case6 n1 => (* Ocast16signed, I n1 :: nil *) 
      I(Int.sign_ext 16 n1)
  | eval_static_operation_case7 n1 => (* Ocast16unsigned, I n1 :: nil *) 
      I(Int.zero_ext 16 n1)
  | eval_static_operation_case8 n1 => (* Oneg, I n1 :: nil *) 
      I(Int.neg n1)
  | eval_static_operation_case9 n1 n2 => (* Osub, I n1 :: I n2 :: nil *) 
      I(Int.sub n1 n2)
  | eval_static_operation_case10 s1 n1 n2 => (* Osub, G s1 n1 :: I n2 :: nil *) 
      G s1 (Int.sub n1 n2)
  | eval_static_operation_case11 n1 n2 => (* Omul, I n1 :: I n2 :: nil *) 
      I(Int.mul n1 n2)
  | eval_static_operation_case12 n n1 => (* Omulimm n, I n1 :: nil *) 
      I(Int.mul n1 n)
  | eval_static_operation_case13 n1 n2 => (* Odiv, I n1 :: I n2 :: nil *) 
      if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
  | eval_static_operation_case14 n1 n2 => (* Odivu, I n1 :: I n2 :: nil *) 
      if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
  | eval_static_operation_case15 n1 n2 => (* Omod, I n1 :: I n2 :: nil *) 
      if Int.eq n2 Int.zero then Unknown else I(Int.mods n1 n2)
  | eval_static_operation_case16 n1 n2 => (* Omodu, I n1 :: I n2 :: nil *) 
      if Int.eq n2 Int.zero then Unknown else I(Int.modu n1 n2)
  | eval_static_operation_case17 n1 n2 => (* Oand, I n1 :: I n2 :: nil *) 
      I(Int.and n1 n2)
  | eval_static_operation_case18 n n1 => (* Oandimm n, I n1 :: nil *) 
      I(Int.and n1 n)
  | eval_static_operation_case19 n1 n2 => (* Oor, I n1 :: I n2 :: nil *) 
      I(Int.or n1 n2)
  | eval_static_operation_case20 n n1 => (* Oorimm n, I n1 :: nil *) 
      I(Int.or n1 n)
  | eval_static_operation_case21 n1 n2 => (* Oxor, I n1 :: I n2 :: nil *) 
      I(Int.xor n1 n2)
  | eval_static_operation_case22 n n1 => (* Oxorimm n, I n1 :: nil *) 
      I(Int.xor n1 n)
  | eval_static_operation_case23 n1 n2 => (* Oshl, I n1 :: I n2 :: nil *) 
      if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown
  | eval_static_operation_case24 n n1 => (* Oshlimm n, I n1 :: nil *) 
      if Int.ltu n Int.iwordsize then I(Int.shl n1 n) else Unknown
  | eval_static_operation_case25 n1 n2 => (* Oshr, I n1 :: I n2 :: nil *) 
      if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown
  | eval_static_operation_case26 n n1 => (* Oshrimm n, I n1 :: nil *) 
      if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown
  | eval_static_operation_case27 n n1 => (* Oshrximm n, I n1 :: nil *) 
      if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown
  | eval_static_operation_case28 n1 n2 => (* Oshru, I n1 :: I n2 :: nil *) 
      if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown
  | eval_static_operation_case29 n n1 => (* Oshruimm n, I n1 :: nil *) 
      if Int.ltu n Int.iwordsize then I(Int.shru n1 n) else Unknown
  | eval_static_operation_case30 n n1 => (* Ororimm n, I n1 :: nil *) 
      if Int.ltu n Int.iwordsize then I(Int.ror n1 n) else Unknown
  | eval_static_operation_case31 mode vl => (* Olea mode, vl *) 
      eval_static_addressing mode vl
  | eval_static_operation_case32 n1 => (* Onegf, F n1 :: nil *) 
      F(Float.neg n1)
  | eval_static_operation_case33 n1 => (* Oabsf, F n1 :: nil *) 
      F(Float.abs n1)
  | eval_static_operation_case34 n1 n2 => (* Oaddf, F n1 :: F n2 :: nil *) 
      F(Float.add n1 n2)
  | eval_static_operation_case35 n1 n2 => (* Osubf, F n1 :: F n2 :: nil *) 
      F(Float.sub n1 n2)
  | eval_static_operation_case36 n1 n2 => (* Omulf, F n1 :: F n2 :: nil *) 
      F(Float.mul n1 n2)
  | eval_static_operation_case37 n1 n2 => (* Odivf, F n1 :: F n2 :: nil *) 
      F(Float.div n1 n2)
  | eval_static_operation_case38 n1 => (* Osingleoffloat, F n1 :: nil *) 
      F(Float.singleoffloat n1)
  | eval_static_operation_case39 n1 => (* Ointoffloat, F n1 :: nil *) 
      eval_static_intoffloat n1
  | eval_static_operation_case40 n1 => (* Ofloatofint, I n1 :: nil *) 
      F(Float.floatofint n1)
  | eval_static_operation_case41 c vl => (* Ocmp c, vl *) 
      eval_static_condition_val c vl
  | eval_static_operation_default op vl =>
      Unknown
  end.


(** * Operator strength reduction *)

(** We now define auxiliary functions for strength reduction of
  operators and addressing modes: replacing an operator with a cheaper
  one if some of its arguments are statically known.  These are again
  large pattern-matchings expressed in indirect style. *)

Section STRENGTH_REDUCTION.

Variable app: reg -> approx.

(** Original definition:
<<
Nondetfunction cond_strength_reduction 
              (cond: condition) (args: list reg) (vl: list approx) :=
  match cond, args, vl with
  | Ccomp c, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
      (Ccompimm (swap_comparison c) n1, r2 :: nil)
  | Ccomp c, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
      (Ccompimm c n2, r1 :: nil)
  | Ccompu c, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
      (Ccompuimm (swap_comparison c) n1, r2 :: nil)
  | Ccompu c, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
      (Ccompuimm c n2, r1 :: nil)
  | _, _, _ => 
      (cond, args)
  end.
>>
*)

Inductive cond_strength_reduction_cases: forall (cond: condition) (args: list reg) (vl: list approx), Type :=
  | cond_strength_reduction_case1: forall c r1 r2 n1 v2, cond_strength_reduction_cases (Ccomp c) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | cond_strength_reduction_case2: forall c r1 r2 v1 n2, cond_strength_reduction_cases (Ccomp c) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | cond_strength_reduction_case3: forall c r1 r2 n1 v2, cond_strength_reduction_cases (Ccompu c) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | cond_strength_reduction_case4: forall c r1 r2 v1 n2, cond_strength_reduction_cases (Ccompu c) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | cond_strength_reduction_default: forall (cond: condition) (args: list reg) (vl: list approx), cond_strength_reduction_cases cond args vl.

Definition cond_strength_reduction_match (cond: condition) (args: list reg) (vl: list approx) :=
  match cond as zz1, args as zz2, vl as zz3 return cond_strength_reduction_cases zz1 zz2 zz3 with
  | Ccomp c, r1 :: r2 :: nil, I n1 :: v2 :: nil => cond_strength_reduction_case1 c r1 r2 n1 v2
  | Ccomp c, r1 :: r2 :: nil, v1 :: I n2 :: nil => cond_strength_reduction_case2 c r1 r2 v1 n2
  | Ccompu c, r1 :: r2 :: nil, I n1 :: v2 :: nil => cond_strength_reduction_case3 c r1 r2 n1 v2
  | Ccompu c, r1 :: r2 :: nil, v1 :: I n2 :: nil => cond_strength_reduction_case4 c r1 r2 v1 n2
  | cond, args, vl => cond_strength_reduction_default cond args vl
  end.

Definition cond_strength_reduction (cond: condition) (args: list reg) (vl: list approx) :=
  match cond_strength_reduction_match cond args vl with
  | cond_strength_reduction_case1 c r1 r2 n1 v2 => (* Ccomp c, r1 :: r2 :: nil, I n1 :: v2 :: nil *) 
      (Ccompimm (swap_comparison c) n1, r2 :: nil)
  | cond_strength_reduction_case2 c r1 r2 v1 n2 => (* Ccomp c, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      (Ccompimm c n2, r1 :: nil)
  | cond_strength_reduction_case3 c r1 r2 n1 v2 => (* Ccompu c, r1 :: r2 :: nil, I n1 :: v2 :: nil *) 
      (Ccompuimm (swap_comparison c) n1, r2 :: nil)
  | cond_strength_reduction_case4 c r1 r2 v1 n2 => (* Ccompu c, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      (Ccompuimm c n2, r1 :: nil)
  | cond_strength_reduction_default cond args vl =>
      (cond, args)
  end.


(** Original definition:
<<
Nondetfunction addr_strength_reduction
                (addr: addressing) (args: list reg) (vl: list approx) :=
  match addr, args, vl with

  | Aindexed ofs, r1 :: nil, G symb n :: nil =>
      (Aglobal symb (Int.add n ofs), nil)
  | Aindexed ofs, r1 :: nil, S n :: nil =>
      (Ainstack (Int.add n ofs), nil)

  | Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil =>
      (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil =>
      (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, S n1 :: I n2 :: nil =>
      (Ainstack (Int.add (Int.add n1 n2) ofs), nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: S n2 :: nil =>
      (Ainstack (Int.add (Int.add n1 n2) ofs), nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
      (Abased symb (Int.add n1 ofs), r2 :: nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, v1 :: G symb n2 :: nil =>
      (Abased symb (Int.add n2 ofs), r1 :: nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
      (Aindexed (Int.add n1 ofs), r2 :: nil)
  | Aindexed2 ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
      (Aindexed (Int.add n2 ofs), r1 :: nil)

  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil =>
      (Aglobal symb (Int.add (Int.add n1 (Int.mul n2 sc)) ofs), nil)
  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
      (Abasedscaled sc symb (Int.add n1 ofs), r2 :: nil)
  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
      (Aindexed (Int.add (Int.mul n2 sc) ofs), r1 :: nil)

  | Abased id ofs, r1 :: nil, I n1 :: nil =>
      (Aglobal id (Int.add ofs n1), nil)

  | Abasedscaled sc id ofs, r1 :: nil, I n1 :: nil =>
      (Aglobal id (Int.add ofs (Int.mul sc n1)), nil)

  | _, _ =>
      (addr, args)
  end.
>>
*)

Inductive addr_strength_reduction_cases: forall (addr: addressing) (args: list reg) (vl: list approx), Type :=
  | addr_strength_reduction_case1: forall ofs r1 symb n, addr_strength_reduction_cases (Aindexed ofs) (r1 :: nil) (G symb n :: nil)
  | addr_strength_reduction_case2: forall ofs r1 n, addr_strength_reduction_cases (Aindexed ofs) (r1 :: nil) (S n :: nil)
  | addr_strength_reduction_case3: forall ofs r1 r2 symb n1 n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (G symb n1 :: I n2 :: nil)
  | addr_strength_reduction_case4: forall ofs r1 r2 n1 symb n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (I n1 :: G symb n2 :: nil)
  | addr_strength_reduction_case5: forall ofs r1 r2 n1 n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (S n1 :: I n2 :: nil)
  | addr_strength_reduction_case6: forall ofs r1 r2 n1 n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (I n1 :: S n2 :: nil)
  | addr_strength_reduction_case7: forall ofs r1 r2 symb n1 v2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (G symb n1 :: v2 :: nil)
  | addr_strength_reduction_case8: forall ofs r1 r2 v1 symb n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (v1 :: G symb n2 :: nil)
  | addr_strength_reduction_case9: forall ofs r1 r2 n1 v2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | addr_strength_reduction_case10: forall ofs r1 r2 v1 n2, addr_strength_reduction_cases (Aindexed2 ofs) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | addr_strength_reduction_case11: forall sc ofs r1 r2 symb n1 n2, addr_strength_reduction_cases (Aindexed2scaled sc ofs) (r1 :: r2 :: nil) (G symb n1 :: I n2 :: nil)
  | addr_strength_reduction_case12: forall sc ofs r1 r2 symb n1 v2, addr_strength_reduction_cases (Aindexed2scaled sc ofs) (r1 :: r2 :: nil) (G symb n1 :: v2 :: nil)
  | addr_strength_reduction_case13: forall sc ofs r1 r2 v1 n2, addr_strength_reduction_cases (Aindexed2scaled sc ofs) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | addr_strength_reduction_case14: forall id ofs r1 n1, addr_strength_reduction_cases (Abased id ofs) (r1 :: nil) (I n1 :: nil)
  | addr_strength_reduction_case15: forall sc id ofs r1 n1, addr_strength_reduction_cases (Abasedscaled sc id ofs) (r1 :: nil) (I n1 :: nil)
  | addr_strength_reduction_default: forall (addr: addressing) (args: list reg) (vl: list approx), addr_strength_reduction_cases addr args vl.

Definition addr_strength_reduction_match (addr: addressing) (args: list reg) (vl: list approx) :=
  match addr as zz1, args as zz2, vl as zz3 return addr_strength_reduction_cases zz1 zz2 zz3 with
  | Aindexed ofs, r1 :: nil, G symb n :: nil => addr_strength_reduction_case1 ofs r1 symb n
  | Aindexed ofs, r1 :: nil, S n :: nil => addr_strength_reduction_case2 ofs r1 n
  | Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil => addr_strength_reduction_case3 ofs r1 r2 symb n1 n2
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil => addr_strength_reduction_case4 ofs r1 r2 n1 symb n2
  | Aindexed2 ofs, r1 :: r2 :: nil, S n1 :: I n2 :: nil => addr_strength_reduction_case5 ofs r1 r2 n1 n2
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: S n2 :: nil => addr_strength_reduction_case6 ofs r1 r2 n1 n2
  | Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil => addr_strength_reduction_case7 ofs r1 r2 symb n1 v2
  | Aindexed2 ofs, r1 :: r2 :: nil, v1 :: G symb n2 :: nil => addr_strength_reduction_case8 ofs r1 r2 v1 symb n2
  | Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: v2 :: nil => addr_strength_reduction_case9 ofs r1 r2 n1 v2
  | Aindexed2 ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil => addr_strength_reduction_case10 ofs r1 r2 v1 n2
  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil => addr_strength_reduction_case11 sc ofs r1 r2 symb n1 n2
  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil => addr_strength_reduction_case12 sc ofs r1 r2 symb n1 v2
  | Aindexed2scaled sc ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil => addr_strength_reduction_case13 sc ofs r1 r2 v1 n2
  | Abased id ofs, r1 :: nil, I n1 :: nil => addr_strength_reduction_case14 id ofs r1 n1
  | Abasedscaled sc id ofs, r1 :: nil, I n1 :: nil => addr_strength_reduction_case15 sc id ofs r1 n1
  | addr, args, vl => addr_strength_reduction_default addr args vl
  end.

Definition addr_strength_reduction (addr: addressing) (args: list reg) (vl: list approx) :=
  match addr_strength_reduction_match addr args vl with
  | addr_strength_reduction_case1 ofs r1 symb n => (* Aindexed ofs, r1 :: nil, G symb n :: nil *) 
      (Aglobal symb (Int.add n ofs), nil)
  | addr_strength_reduction_case2 ofs r1 n => (* Aindexed ofs, r1 :: nil, S n :: nil *) 
      (Ainstack (Int.add n ofs), nil)
  | addr_strength_reduction_case3 ofs r1 r2 symb n1 n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil *) 
      (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
  | addr_strength_reduction_case4 ofs r1 r2 n1 symb n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil *) 
      (Aglobal symb (Int.add (Int.add n1 n2) ofs), nil)
  | addr_strength_reduction_case5 ofs r1 r2 n1 n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, S n1 :: I n2 :: nil *) 
      (Ainstack (Int.add (Int.add n1 n2) ofs), nil)
  | addr_strength_reduction_case6 ofs r1 r2 n1 n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: S n2 :: nil *) 
      (Ainstack (Int.add (Int.add n1 n2) ofs), nil)
  | addr_strength_reduction_case7 ofs r1 r2 symb n1 v2 => (* Aindexed2 ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil *) 
      (Abased symb (Int.add n1 ofs), r2 :: nil)
  | addr_strength_reduction_case8 ofs r1 r2 v1 symb n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, v1 :: G symb n2 :: nil *) 
      (Abased symb (Int.add n2 ofs), r1 :: nil)
  | addr_strength_reduction_case9 ofs r1 r2 n1 v2 => (* Aindexed2 ofs, r1 :: r2 :: nil, I n1 :: v2 :: nil *) 
      (Aindexed (Int.add n1 ofs), r2 :: nil)
  | addr_strength_reduction_case10 ofs r1 r2 v1 n2 => (* Aindexed2 ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      (Aindexed (Int.add n2 ofs), r1 :: nil)
  | addr_strength_reduction_case11 sc ofs r1 r2 symb n1 n2 => (* Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil *) 
      (Aglobal symb (Int.add (Int.add n1 (Int.mul n2 sc)) ofs), nil)
  | addr_strength_reduction_case12 sc ofs r1 r2 symb n1 v2 => (* Aindexed2scaled sc ofs, r1 :: r2 :: nil, G symb n1 :: v2 :: nil *) 
      (Abasedscaled sc symb (Int.add n1 ofs), r2 :: nil)
  | addr_strength_reduction_case13 sc ofs r1 r2 v1 n2 => (* Aindexed2scaled sc ofs, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      (Aindexed (Int.add (Int.mul n2 sc) ofs), r1 :: nil)
  | addr_strength_reduction_case14 id ofs r1 n1 => (* Abased id ofs, r1 :: nil, I n1 :: nil *) 
      (Aglobal id (Int.add ofs n1), nil)
  | addr_strength_reduction_case15 sc id ofs r1 n1 => (* Abasedscaled sc id ofs, r1 :: nil, I n1 :: nil *) 
      (Aglobal id (Int.add ofs (Int.mul sc n1)), nil)
  | addr_strength_reduction_default addr args vl =>
      (addr, args)
  end.


Definition make_addimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Omove, r :: nil)
  else (Oaddimm n, r :: nil).

Definition make_shlimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Omove, r :: nil)
  else (Oshlimm n, r :: nil).

Definition make_shrimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Omove, r :: nil)
  else (Oshrimm n, r :: nil).

Definition make_shruimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Omove, r :: nil)
  else (Oshruimm n, r :: nil).

Definition make_mulimm (n: int) (r: reg) :=
  if Int.eq n Int.zero then
    (Ointconst Int.zero, nil)
  else if Int.eq n Int.one then
    (Omove, r :: nil)
  else
    match Int.is_power2 n with
    | Some l => make_shlimm l r
    | None => (Omulimm n, r :: nil)
    end.

Definition make_andimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Ointconst Int.zero, nil)
  else if Int.eq n Int.mone then (Omove, r :: nil)
  else (Oandimm n, r :: nil).

Definition make_orimm (n: int) (r: reg) :=
  if Int.eq n Int.zero then (Omove, r :: nil)
  else if Int.eq n Int.mone then (Ointconst Int.mone, nil)
  else (Oorimm n, r :: nil).

Definition make_xorimm (n: int) (r: reg) :=
  if Int.eq n Int.zero
  then (Omove, r :: nil)
  else (Oxorimm n, r :: nil).

Definition make_divimm n (r1 r2: reg) :=
  match Int.is_power2 n with
  | Some l => if Int.ltu l (Int.repr 31)
              then (Oshrximm l, r1 :: nil)
              else (Odiv, r1 :: r2 :: nil)
  | None   => (Odiv, r1 :: r2 :: nil)
  end.

Definition make_divuimm n (r1 r2: reg) :=
  match Int.is_power2 n with
  | Some l => make_shruimm l r1
  | None   => (Odivu, r1 :: r2 :: nil)
  end.

Definition make_moduimm n (r1 r2: reg) :=
  match Int.is_power2 n with
  | Some l => (Oandimm (Int.sub n Int.one), r1 :: nil)
  | None   => (Omodu, r1 :: r2 :: nil)
  end.

(** We must be careful to preserve 2-address constraints over the
    RTL code, which means that commutative operations cannot
    be specialized if their first argument is a constant. *)

(** Original definition:
<<
Nondetfunction op_strength_reduction 
              (op: operation) (args: list reg) (vl: list approx) :=
  match op, args, vl with
  | Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm (Int.neg n2) r1
  | Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_mulimm n2 r1
  | Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divimm n2 r1 r2
  | Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_divuimm n2 r1 r2
  | Omodu, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_moduimm n2 r1 r2
  | Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_andimm n2 r1
  | Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_orimm n2 r1
  | Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_xorimm n2 r1
  | Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shlimm n2 r1
  | Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shrimm n2 r1
  | Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shruimm n2 r1
  | Olea addr, args, vl =>
      let (addr', args') := addr_strength_reduction addr args vl in
      (Olea addr', args')
  | Ocmp c, args, vl =>
      let (c', args') := cond_strength_reduction c args vl in 
      (Ocmp c', args')
  | _, _, _ => (op, args)
  end.
>>
*)

Inductive op_strength_reduction_cases: forall (op: operation) (args: list reg) (vl: list approx), Type :=
  | op_strength_reduction_case1: forall r1 r2 v1 n2, op_strength_reduction_cases (Osub) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case2: forall r1 r2 v1 n2, op_strength_reduction_cases (Omul) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case3: forall r1 r2 v1 n2, op_strength_reduction_cases (Odiv) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case4: forall r1 r2 v1 n2, op_strength_reduction_cases (Odivu) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case5: forall r1 r2 v1 n2, op_strength_reduction_cases (Omodu) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case6: forall r1 r2 v1 n2, op_strength_reduction_cases (Oand) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case7: forall r1 r2 v1 n2, op_strength_reduction_cases (Oor) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case8: forall r1 r2 v1 n2, op_strength_reduction_cases (Oxor) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case9: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshl) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case10: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshr) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case11: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshru) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case12: forall addr args vl, op_strength_reduction_cases (Olea addr) (args) (vl)
  | op_strength_reduction_case13: forall c args vl, op_strength_reduction_cases (Ocmp c) (args) (vl)
  | op_strength_reduction_default: forall (op: operation) (args: list reg) (vl: list approx), op_strength_reduction_cases op args vl.

Definition op_strength_reduction_match (op: operation) (args: list reg) (vl: list approx) :=
  match op as zz1, args as zz2, vl as zz3 return op_strength_reduction_cases zz1 zz2 zz3 with
  | Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case1 r1 r2 v1 n2
  | Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case2 r1 r2 v1 n2
  | Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case3 r1 r2 v1 n2
  | Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case4 r1 r2 v1 n2
  | Omodu, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case5 r1 r2 v1 n2
  | Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case6 r1 r2 v1 n2
  | Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case7 r1 r2 v1 n2
  | Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case8 r1 r2 v1 n2
  | Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case9 r1 r2 v1 n2
  | Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case10 r1 r2 v1 n2
  | Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case11 r1 r2 v1 n2
  | Olea addr, args, vl => op_strength_reduction_case12 addr args vl
  | Ocmp c, args, vl => op_strength_reduction_case13 c args vl
  | op, args, vl => op_strength_reduction_default op args vl
  end.

Definition op_strength_reduction (op: operation) (args: list reg) (vl: list approx) :=
  match op_strength_reduction_match op args vl with
  | op_strength_reduction_case1 r1 r2 v1 n2 => (* Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_addimm (Int.neg n2) r1
  | op_strength_reduction_case2 r1 r2 v1 n2 => (* Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_mulimm n2 r1
  | op_strength_reduction_case3 r1 r2 v1 n2 => (* Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_divimm n2 r1 r2
  | op_strength_reduction_case4 r1 r2 v1 n2 => (* Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_divuimm n2 r1 r2
  | op_strength_reduction_case5 r1 r2 v1 n2 => (* Omodu, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_moduimm n2 r1 r2
  | op_strength_reduction_case6 r1 r2 v1 n2 => (* Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_andimm n2 r1
  | op_strength_reduction_case7 r1 r2 v1 n2 => (* Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_orimm n2 r1
  | op_strength_reduction_case8 r1 r2 v1 n2 => (* Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_xorimm n2 r1
  | op_strength_reduction_case9 r1 r2 v1 n2 => (* Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_shlimm n2 r1
  | op_strength_reduction_case10 r1 r2 v1 n2 => (* Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_shrimm n2 r1
  | op_strength_reduction_case11 r1 r2 v1 n2 => (* Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil *) 
      make_shruimm n2 r1
  | op_strength_reduction_case12 addr args vl => (* Olea addr, args, vl *) 
      let (addr', args') := addr_strength_reduction addr args vl in (Olea addr', args')
  | op_strength_reduction_case13 c args vl => (* Ocmp c, args, vl *) 
      let (c', args') := cond_strength_reduction c args vl in (Ocmp c', args')
  | op_strength_reduction_default op args vl =>
      (op, args)
  end.


End STRENGTH_REDUCTION.