Module ConstpropOp


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
  | Unknown: approx
  | I: int -> approx
  | F: float -> approx
  | G: ident -> int -> approx
The value is the address of the given global symbol plus the given integer offset.
  | S: int -> approx.

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 =>
      Some(Int.cmp c n1 n2)
  | eval_static_condition_case2 c n1 n2 =>
      Some(Int.cmpu c n1 n2)
  | eval_static_condition_case3 c n n1 =>
      Some(Int.cmp c n1 n)
  | eval_static_condition_case4 c n n1 =>
      Some(Int.cmpu c n1 n)
  | eval_static_condition_case5 c n1 n2 =>
      Some(Float.cmp c n1 n2)
  | eval_static_condition_case6 c n1 n2 =>
      Some(negb(Float.cmp c n1 n2))
  | eval_static_condition_case7 n n1 =>
      Some(Int.eq (Int.and n1 n) Int.zero)
  | eval_static_condition_case8 n n1 =>
      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_operation (op: operation) (vl: list approx) :=
  match op, vl with
  | Omove, v1::nil => v1
  | Ointconst n, nil => I n
  | Ofloatconst n, nil => F n
  | Oaddrsymbol s n, nil => G s n
  | Oaddrstack n, nil => S n
  | Ocast8signed, I n1 :: nil => I(Int.sign_ext 8 n1)
  | Ocast16signed, I n1 :: nil => I(Int.sign_ext 16 n1)
  | Oadd, I n1 :: I n2 :: nil => I(Int.add n1 n2)
  | Oadd, G s1 n1 :: I n2 :: nil => G s1 (Int.add n1 n2)
  | Oadd, I n1 :: G s2 n2 :: nil => G s2 (Int.add n1 n2)
  | Oadd, S n1 :: I n2 :: nil => S (Int.add n1 n2)
  | Oadd, I n1 :: S n2 :: nil => S (Int.add n1 n2)
  | Oaddimm n, I n1 :: nil => I (Int.add n1 n)
  | Oaddimm n, G s1 n1 :: nil => G s1 (Int.add n1 n)
  | Oaddimm n, S n1 :: nil => S (Int.add n1 n)
  | Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2)
  | Osub, G s1 n1 :: I n2 :: nil => G s1 (Int.sub n1 n2)
  | Osub, S n1 :: I n2 :: nil => S (Int.sub n1 n2)
  | Osubimm n, I n1 :: nil => I (Int.sub n n1)
  | 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)
  | 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)
  | Onand, I n1 :: I n2 :: nil => I(Int.xor (Int.and n1 n2) Int.mone)
  | Onor, I n1 :: I n2 :: nil => I(Int.xor (Int.or n1 n2) Int.mone)
  | Onxor, I n1 :: I n2 :: nil => I(Int.xor (Int.xor n1 n2) Int.mone)
  | Oshl, I n1 :: I n2 :: nil => if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) 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
  | Orolm amount mask, I n1 :: nil => I(Int.rolm n1 amount mask)
  | Oroli amount mask, I n1 :: I n2 :: nil => I(Int.or (Int.and n1 (Int.not mask)) (Int.rolm n2 amount mask))
  | 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
  | Ofloatofwords, I n1 :: I n2 :: nil => F(Float.from_words n1 n2)
  | 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 s n, eval_static_operation_cases (Oaddrsymbol s n) (nil)
  | eval_static_operation_case5: forall n, eval_static_operation_cases (Oaddrstack n) (nil)
  | eval_static_operation_case6: forall n1, eval_static_operation_cases (Ocast8signed) (I n1 :: nil)
  | eval_static_operation_case7: forall n1, eval_static_operation_cases (Ocast16signed) (I n1 :: nil)
  | eval_static_operation_case8: forall n1 n2, eval_static_operation_cases (Oadd) (I n1 :: I n2 :: nil)
  | eval_static_operation_case9: forall s1 n1 n2, eval_static_operation_cases (Oadd) (G s1 n1 :: I n2 :: nil)
  | eval_static_operation_case10: forall n1 s2 n2, eval_static_operation_cases (Oadd) (I n1 :: G s2 n2 :: nil)
  | eval_static_operation_case11: forall n1 n2, eval_static_operation_cases (Oadd) (S n1 :: I n2 :: nil)
  | eval_static_operation_case12: forall n1 n2, eval_static_operation_cases (Oadd) (I n1 :: S n2 :: nil)
  | eval_static_operation_case13: forall n n1, eval_static_operation_cases (Oaddimm n) (I n1 :: nil)
  | eval_static_operation_case14: forall n s1 n1, eval_static_operation_cases (Oaddimm n) (G s1 n1 :: nil)
  | eval_static_operation_case15: forall n n1, eval_static_operation_cases (Oaddimm n) (S n1 :: nil)
  | eval_static_operation_case16: forall n1 n2, eval_static_operation_cases (Osub) (I n1 :: I n2 :: nil)
  | eval_static_operation_case17: forall s1 n1 n2, eval_static_operation_cases (Osub) (G s1 n1 :: I n2 :: nil)
  | eval_static_operation_case18: forall n1 n2, eval_static_operation_cases (Osub) (S n1 :: I n2 :: nil)
  | eval_static_operation_case19: forall n n1, eval_static_operation_cases (Osubimm n) (I n1 :: nil)
  | eval_static_operation_case20: forall n1 n2, eval_static_operation_cases (Omul) (I n1 :: I n2 :: nil)
  | eval_static_operation_case21: forall n n1, eval_static_operation_cases (Omulimm n) (I n1 :: nil)
  | eval_static_operation_case22: forall n1 n2, eval_static_operation_cases (Odiv) (I n1 :: I n2 :: nil)
  | eval_static_operation_case23: forall n1 n2, eval_static_operation_cases (Odivu) (I n1 :: I n2 :: nil)
  | eval_static_operation_case24: forall n1 n2, eval_static_operation_cases (Oand) (I n1 :: I n2 :: nil)
  | eval_static_operation_case25: forall n n1, eval_static_operation_cases (Oandimm n) (I n1 :: nil)
  | eval_static_operation_case26: forall n1 n2, eval_static_operation_cases (Oor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case27: forall n n1, eval_static_operation_cases (Oorimm n) (I n1 :: nil)
  | eval_static_operation_case28: forall n1 n2, eval_static_operation_cases (Oxor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case29: forall n n1, eval_static_operation_cases (Oxorimm n) (I n1 :: nil)
  | eval_static_operation_case30: forall n1 n2, eval_static_operation_cases (Onand) (I n1 :: I n2 :: nil)
  | eval_static_operation_case31: forall n1 n2, eval_static_operation_cases (Onor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case32: forall n1 n2, eval_static_operation_cases (Onxor) (I n1 :: I n2 :: nil)
  | eval_static_operation_case33: forall n1 n2, eval_static_operation_cases (Oshl) (I n1 :: I n2 :: nil)
  | eval_static_operation_case34: forall n1 n2, eval_static_operation_cases (Oshr) (I n1 :: I n2 :: nil)
  | eval_static_operation_case35: forall n n1, eval_static_operation_cases (Oshrimm n) (I n1 :: nil)
  | eval_static_operation_case36: forall n n1, eval_static_operation_cases (Oshrximm n) (I n1 :: nil)
  | eval_static_operation_case37: forall n1 n2, eval_static_operation_cases (Oshru) (I n1 :: I n2 :: nil)
  | eval_static_operation_case38: forall amount mask n1, eval_static_operation_cases (Orolm amount mask) (I n1 :: nil)
  | eval_static_operation_case39: forall amount mask n1 n2, eval_static_operation_cases (Oroli amount mask) (I n1 :: I n2 :: nil)
  | eval_static_operation_case40: forall n1, eval_static_operation_cases (Onegf) (F n1 :: nil)
  | eval_static_operation_case41: forall n1, eval_static_operation_cases (Oabsf) (F n1 :: nil)
  | eval_static_operation_case42: forall n1 n2, eval_static_operation_cases (Oaddf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case43: forall n1 n2, eval_static_operation_cases (Osubf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case44: forall n1 n2, eval_static_operation_cases (Omulf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case45: forall n1 n2, eval_static_operation_cases (Odivf) (F n1 :: F n2 :: nil)
  | eval_static_operation_case46: forall n1, eval_static_operation_cases (Osingleoffloat) (F n1 :: nil)
  | eval_static_operation_case47: forall n1, eval_static_operation_cases (Ointoffloat) (F n1 :: nil)
  | eval_static_operation_case48: forall n1 n2, eval_static_operation_cases (Ofloatofwords) (I n1 :: I n2 :: nil)
  | eval_static_operation_case49: 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
  | Oaddrsymbol s n, nil => eval_static_operation_case4 s n
  | Oaddrstack n, nil => eval_static_operation_case5 n
  | Ocast8signed, I n1 :: nil => eval_static_operation_case6 n1
  | Ocast16signed, I n1 :: nil => eval_static_operation_case7 n1
  | Oadd, I n1 :: I n2 :: nil => eval_static_operation_case8 n1 n2
  | Oadd, G s1 n1 :: I n2 :: nil => eval_static_operation_case9 s1 n1 n2
  | Oadd, I n1 :: G s2 n2 :: nil => eval_static_operation_case10 n1 s2 n2
  | Oadd, S n1 :: I n2 :: nil => eval_static_operation_case11 n1 n2
  | Oadd, I n1 :: S n2 :: nil => eval_static_operation_case12 n1 n2
  | Oaddimm n, I n1 :: nil => eval_static_operation_case13 n n1
  | Oaddimm n, G s1 n1 :: nil => eval_static_operation_case14 n s1 n1
  | Oaddimm n, S n1 :: nil => eval_static_operation_case15 n n1
  | Osub, I n1 :: I n2 :: nil => eval_static_operation_case16 n1 n2
  | Osub, G s1 n1 :: I n2 :: nil => eval_static_operation_case17 s1 n1 n2
  | Osub, S n1 :: I n2 :: nil => eval_static_operation_case18 n1 n2
  | Osubimm n, I n1 :: nil => eval_static_operation_case19 n n1
  | Omul, I n1 :: I n2 :: nil => eval_static_operation_case20 n1 n2
  | Omulimm n, I n1 :: nil => eval_static_operation_case21 n n1
  | Odiv, I n1 :: I n2 :: nil => eval_static_operation_case22 n1 n2
  | Odivu, I n1 :: I n2 :: nil => eval_static_operation_case23 n1 n2
  | Oand, I n1 :: I n2 :: nil => eval_static_operation_case24 n1 n2
  | Oandimm n, I n1 :: nil => eval_static_operation_case25 n n1
  | Oor, I n1 :: I n2 :: nil => eval_static_operation_case26 n1 n2
  | Oorimm n, I n1 :: nil => eval_static_operation_case27 n n1
  | Oxor, I n1 :: I n2 :: nil => eval_static_operation_case28 n1 n2
  | Oxorimm n, I n1 :: nil => eval_static_operation_case29 n n1
  | Onand, I n1 :: I n2 :: nil => eval_static_operation_case30 n1 n2
  | Onor, I n1 :: I n2 :: nil => eval_static_operation_case31 n1 n2
  | Onxor, I n1 :: I n2 :: nil => eval_static_operation_case32 n1 n2
  | Oshl, I n1 :: I n2 :: nil => eval_static_operation_case33 n1 n2
  | Oshr, I n1 :: I n2 :: nil => eval_static_operation_case34 n1 n2
  | Oshrimm n, I n1 :: nil => eval_static_operation_case35 n n1
  | Oshrximm n, I n1 :: nil => eval_static_operation_case36 n n1
  | Oshru, I n1 :: I n2 :: nil => eval_static_operation_case37 n1 n2
  | Orolm amount mask, I n1 :: nil => eval_static_operation_case38 amount mask n1
  | Oroli amount mask, I n1 :: I n2 :: nil => eval_static_operation_case39 amount mask n1 n2
  | Onegf, F n1 :: nil => eval_static_operation_case40 n1
  | Oabsf, F n1 :: nil => eval_static_operation_case41 n1
  | Oaddf, F n1 :: F n2 :: nil => eval_static_operation_case42 n1 n2
  | Osubf, F n1 :: F n2 :: nil => eval_static_operation_case43 n1 n2
  | Omulf, F n1 :: F n2 :: nil => eval_static_operation_case44 n1 n2
  | Odivf, F n1 :: F n2 :: nil => eval_static_operation_case45 n1 n2
  | Osingleoffloat, F n1 :: nil => eval_static_operation_case46 n1
  | Ointoffloat, F n1 :: nil => eval_static_operation_case47 n1
  | Ofloatofwords, I n1 :: I n2 :: nil => eval_static_operation_case48 n1 n2
  | Ocmp c, vl => eval_static_operation_case49 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 =>
      v1
  | eval_static_operation_case2 n =>
      I n
  | eval_static_operation_case3 n =>
      F n
  | eval_static_operation_case4 s n =>
      G s n
  | eval_static_operation_case5 n =>
      S n
  | eval_static_operation_case6 n1 =>
      I(Int.sign_ext 8 n1)
  | eval_static_operation_case7 n1 =>
      I(Int.sign_ext 16 n1)
  | eval_static_operation_case8 n1 n2 =>
      I(Int.add n1 n2)
  | eval_static_operation_case9 s1 n1 n2 =>
      G s1 (Int.add n1 n2)
  | eval_static_operation_case10 n1 s2 n2 =>
      G s2 (Int.add n1 n2)
  | eval_static_operation_case11 n1 n2 =>
      S (Int.add n1 n2)
  | eval_static_operation_case12 n1 n2 =>
      S (Int.add n1 n2)
  | eval_static_operation_case13 n n1 =>
      I (Int.add n1 n)
  | eval_static_operation_case14 n s1 n1 =>
      G s1 (Int.add n1 n)
  | eval_static_operation_case15 n n1 =>
      S (Int.add n1 n)
  | eval_static_operation_case16 n1 n2 =>
      I(Int.sub n1 n2)
  | eval_static_operation_case17 s1 n1 n2 =>
      G s1 (Int.sub n1 n2)
  | eval_static_operation_case18 n1 n2 =>
      S (Int.sub n1 n2)
  | eval_static_operation_case19 n n1 =>
      I (Int.sub n n1)
  | eval_static_operation_case20 n1 n2 =>
      I(Int.mul n1 n2)
  | eval_static_operation_case21 n n1 =>
      I(Int.mul n1 n)
  | eval_static_operation_case22 n1 n2 =>
      if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
  | eval_static_operation_case23 n1 n2 =>
      if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
  | eval_static_operation_case24 n1 n2 =>
      I(Int.and n1 n2)
  | eval_static_operation_case25 n n1 =>
      I(Int.and n1 n)
  | eval_static_operation_case26 n1 n2 =>
      I(Int.or n1 n2)
  | eval_static_operation_case27 n n1 =>
      I(Int.or n1 n)
  | eval_static_operation_case28 n1 n2 =>
      I(Int.xor n1 n2)
  | eval_static_operation_case29 n n1 =>
      I(Int.xor n1 n)
  | eval_static_operation_case30 n1 n2 =>
      I(Int.xor (Int.and n1 n2) Int.mone)
  | eval_static_operation_case31 n1 n2 =>
      I(Int.xor (Int.or n1 n2) Int.mone)
  | eval_static_operation_case32 n1 n2 =>
      I(Int.xor (Int.xor n1 n2) Int.mone)
  | eval_static_operation_case33 n1 n2 =>
      if Int.ltu n2 Int.iwordsize then I(Int.shl n1 n2) else Unknown
  | eval_static_operation_case34 n1 n2 =>
      if Int.ltu n2 Int.iwordsize then I(Int.shr n1 n2) else Unknown
  | eval_static_operation_case35 n n1 =>
      if Int.ltu n Int.iwordsize then I(Int.shr n1 n) else Unknown
  | eval_static_operation_case36 n n1 =>
      if Int.ltu n (Int.repr 31) then I(Int.shrx n1 n) else Unknown
  | eval_static_operation_case37 n1 n2 =>
      if Int.ltu n2 Int.iwordsize then I(Int.shru n1 n2) else Unknown
  | eval_static_operation_case38 amount mask n1 =>
      I(Int.rolm n1 amount mask)
  | eval_static_operation_case39 amount mask n1 n2 =>
      I(Int.or (Int.and n1 (Int.not mask)) (Int.rolm n2 amount mask))
  | eval_static_operation_case40 n1 =>
      F(Float.neg n1)
  | eval_static_operation_case41 n1 =>
      F(Float.abs n1)
  | eval_static_operation_case42 n1 n2 =>
      F(Float.add n1 n2)
  | eval_static_operation_case43 n1 n2 =>
      F(Float.sub n1 n2)
  | eval_static_operation_case44 n1 n2 =>
      F(Float.mul n1 n2)
  | eval_static_operation_case45 n1 n2 =>
      F(Float.div n1 n2)
  | eval_static_operation_case46 n1 =>
      F(Float.singleoffloat n1)
  | eval_static_operation_case47 n1 =>
      eval_static_intoffloat n1
  | eval_static_operation_case48 n1 n2 =>
      F(Float.from_words n1 n2)
  | eval_static_operation_case49 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.

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 =>
      (Ccompimm (swap_comparison c) n1, r2 :: nil)
  | cond_strength_reduction_case2 c r1 r2 v1 n2 =>
      (Ccompimm c n2, r1 :: nil)
  | cond_strength_reduction_case3 c r1 r2 n1 v2 =>
      (Ccompuimm (swap_comparison c) n1, r2 :: nil)
  | cond_strength_reduction_case4 c r1 r2 v1 n2 =>
      (Ccompuimm c n2, r1 :: nil)
  | cond_strength_reduction_default cond args vl =>
      (cond, 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) (r1 r2: reg) :=
  if Int.eq n Int.zero then
    (Omove, r1 :: nil)
  else if Int.ltu n Int.iwordsize then
    (Orolm n (Int.shl Int.mone n), r1 :: nil)
  else
    (Oshl, r1 :: r2 :: nil).

Definition make_shrimm (n: int) (r1 r2: reg) :=
  if Int.eq n Int.zero then
    (Omove, r1 :: nil)
  else if Int.ltu n Int.iwordsize then
    (Oshrimm n, r1 :: nil)
  else
    (Oshr, r1 :: r2 :: nil).

Definition make_shruimm (n: int) (r1 r2: reg) :=
  if Int.eq n Int.zero then
    (Omove, r1 :: nil)
  else if Int.ltu n Int.iwordsize then
    (Orolm (Int.sub Int.iwordsize n) (Int.shru Int.mone n), r1 :: nil)
  else
    (Oshru, r1 :: r2 :: nil).

Definition make_mulimm (n: int) (r1 r2: reg) :=
  if Int.eq n Int.zero then
    (Ointconst Int.zero, nil)
  else if Int.eq n Int.one then
    (Omove, r1 :: nil)
  else
    match Int.is_power2 n with
    | Some l => (Orolm l (Int.shl Int.mone l), r1 :: nil)
    | None => (Omulimm n, r1 :: nil)
    end.

Definition make_divimm (n: int) (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: int) (r1 r2: reg) :=
  match Int.is_power2 n with
  | Some l => (Orolm (Int.sub Int.iwordsize l) (Int.shru Int.mone l), r1 :: nil)
  | None => (Odivu, r1 :: r2 :: 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).

Original definition: 
Nondetfunction op_strength_reduction 
              (op: operation) (args: list reg) (vl: list approx) :=
  match op, args, vl with
  | Oadd, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_addimm n1 r2
  | Oadd, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm n2 r1
  | Osub, r1 :: r2 :: nil, I n1 :: v2 :: nil => (Osubimm n1, r2 :: nil)
  | Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_addimm (Int.neg n2) r1
  | Omul, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_mulimm n1 r2 r1
  | Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_mulimm n2 r1 r2
  | 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
  | Oand, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_andimm n1 r2
  | Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_andimm n2 r1
  | Oor, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_orimm n1 r2
  | Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_orimm n2 r1
  | Oxor, r1 :: r2 :: nil, I n1 :: v2 :: nil => make_xorimm n1 r2
  | Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_xorimm n2 r1
  | Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shlimm n2 r1 r2
  | Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shrimm n2 r1 r2
  | Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => make_shruimm n2 r1 r2
  | 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 n1 v2, op_strength_reduction_cases (Oadd) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case2: forall r1 r2 v1 n2, op_strength_reduction_cases (Oadd) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case3: forall r1 r2 n1 v2, op_strength_reduction_cases (Osub) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case4: forall r1 r2 v1 n2, op_strength_reduction_cases (Osub) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case5: forall r1 r2 n1 v2, op_strength_reduction_cases (Omul) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case6: forall r1 r2 v1 n2, op_strength_reduction_cases (Omul) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case7: forall r1 r2 v1 n2, op_strength_reduction_cases (Odiv) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case8: forall r1 r2 v1 n2, op_strength_reduction_cases (Odivu) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case9: forall r1 r2 n1 v2, op_strength_reduction_cases (Oand) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case10: forall r1 r2 v1 n2, op_strength_reduction_cases (Oand) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case11: forall r1 r2 n1 v2, op_strength_reduction_cases (Oor) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case12: forall r1 r2 v1 n2, op_strength_reduction_cases (Oor) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case13: forall r1 r2 n1 v2, op_strength_reduction_cases (Oxor) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | op_strength_reduction_case14: forall r1 r2 v1 n2, op_strength_reduction_cases (Oxor) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case15: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshl) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case16: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshr) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case17: forall r1 r2 v1 n2, op_strength_reduction_cases (Oshru) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | op_strength_reduction_case18: 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
  | Oadd, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case1 r1 r2 n1 v2
  | Oadd, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case2 r1 r2 v1 n2
  | Osub, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case3 r1 r2 n1 v2
  | Osub, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case4 r1 r2 v1 n2
  | Omul, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case5 r1 r2 n1 v2
  | Omul, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case6 r1 r2 v1 n2
  | Odiv, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case7 r1 r2 v1 n2
  | Odivu, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case8 r1 r2 v1 n2
  | Oand, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case9 r1 r2 n1 v2
  | Oand, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case10 r1 r2 v1 n2
  | Oor, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case11 r1 r2 n1 v2
  | Oor, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case12 r1 r2 v1 n2
  | Oxor, r1 :: r2 :: nil, I n1 :: v2 :: nil => op_strength_reduction_case13 r1 r2 n1 v2
  | Oxor, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case14 r1 r2 v1 n2
  | Oshl, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case15 r1 r2 v1 n2
  | Oshr, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case16 r1 r2 v1 n2
  | Oshru, r1 :: r2 :: nil, v1 :: I n2 :: nil => op_strength_reduction_case17 r1 r2 v1 n2
  | Ocmp c, args, vl => op_strength_reduction_case18 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 n1 v2 =>
      make_addimm n1 r2
  | op_strength_reduction_case2 r1 r2 v1 n2 =>
      make_addimm n2 r1
  | op_strength_reduction_case3 r1 r2 n1 v2 =>
      (Osubimm n1, r2 :: nil)
  | op_strength_reduction_case4 r1 r2 v1 n2 =>
      make_addimm (Int.neg n2) r1
  | op_strength_reduction_case5 r1 r2 n1 v2 =>
      make_mulimm n1 r2 r1
  | op_strength_reduction_case6 r1 r2 v1 n2 =>
      make_mulimm n2 r1 r2
  | op_strength_reduction_case7 r1 r2 v1 n2 =>
      make_divimm n2 r1 r2
  | op_strength_reduction_case8 r1 r2 v1 n2 =>
      make_divuimm n2 r1 r2
  | op_strength_reduction_case9 r1 r2 n1 v2 =>
      make_andimm n1 r2
  | op_strength_reduction_case10 r1 r2 v1 n2 =>
      make_andimm n2 r1
  | op_strength_reduction_case11 r1 r2 n1 v2 =>
      make_orimm n1 r2
  | op_strength_reduction_case12 r1 r2 v1 n2 =>
      make_orimm n2 r1
  | op_strength_reduction_case13 r1 r2 n1 v2 =>
      make_xorimm n1 r2
  | op_strength_reduction_case14 r1 r2 v1 n2 =>
      make_xorimm n2 r1
  | op_strength_reduction_case15 r1 r2 v1 n2 =>
      make_shlimm n2 r1 r2
  | op_strength_reduction_case16 r1 r2 v1 n2 =>
      make_shrimm n2 r1 r2
  | op_strength_reduction_case17 r1 r2 v1 n2 =>
      make_shruimm n2 r1 r2
  | op_strength_reduction_case18 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.


Original definition: 
Nondetfunction addr_strength_reduction
                (addr: addressing) (args: list reg) (vl: list approx) :=
  match addr, args, vl with
  | Aindexed2, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil =>
      (Aglobal symb (Int.add n1 n2), nil)
  | Aindexed2, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil =>
      (Aglobal symb (Int.add n1 n2), nil)
  | Aindexed2, r1 :: r2 :: nil, S n1 :: I n2 :: nil =>
      (Ainstack (Int.add n1 n2), nil)
  | Aindexed2, r1 :: r2 :: nil, I n1 :: S n2 :: nil =>
      (Ainstack (Int.add n1 n2), nil)
  | Aindexed2, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
      (Abased symb n1, r2 :: nil)
  | Aindexed2, r1 :: r2 :: nil, v1 :: G symb n2 :: nil =>
      (Abased symb n2, r1 :: nil)
  | Aindexed2, r1 :: r2 :: nil, I n1 :: v2 :: nil =>
      (Aindexed n1, r2 :: nil)
  | Aindexed2, r1 :: r2 :: nil, v1 :: I n2 :: nil =>
      (Aindexed n2, r1 :: nil)
  | Abased symb ofs, r1 :: nil, I n1 :: nil =>
      (Aglobal symb (Int.add ofs n1), nil)
  | Aindexed n, r1 :: nil, G symb n1 :: nil =>
      (Aglobal symb (Int.add n1 n), nil)
  | Aindexed n, r1 :: nil, S n1 :: nil =>
      (Ainstack (Int.add n1 n), nil)
  | _, _, _ =>
      (addr, args)
  end.

Inductive addr_strength_reduction_cases: forall (addr: addressing) (args: list reg) (vl: list approx), Type :=
  | addr_strength_reduction_case1: forall r1 r2 symb n1 n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (G symb n1 :: I n2 :: nil)
  | addr_strength_reduction_case2: forall r1 r2 n1 symb n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (I n1 :: G symb n2 :: nil)
  | addr_strength_reduction_case3: forall r1 r2 n1 n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (S n1 :: I n2 :: nil)
  | addr_strength_reduction_case4: forall r1 r2 n1 n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (I n1 :: S n2 :: nil)
  | addr_strength_reduction_case5: forall r1 r2 symb n1 v2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (G symb n1 :: v2 :: nil)
  | addr_strength_reduction_case6: forall r1 r2 v1 symb n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (v1 :: G symb n2 :: nil)
  | addr_strength_reduction_case7: forall r1 r2 n1 v2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (I n1 :: v2 :: nil)
  | addr_strength_reduction_case8: forall r1 r2 v1 n2, addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil) (v1 :: I n2 :: nil)
  | addr_strength_reduction_case9: forall symb ofs r1 n1, addr_strength_reduction_cases (Abased symb ofs) (r1 :: nil) (I n1 :: nil)
  | addr_strength_reduction_case10: forall n r1 symb n1, addr_strength_reduction_cases (Aindexed n) (r1 :: nil) (G symb n1 :: nil)
  | addr_strength_reduction_case11: forall n r1 n1, addr_strength_reduction_cases (Aindexed n) (r1 :: nil) (S 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
  | Aindexed2, r1 :: r2 :: nil, G symb n1 :: I n2 :: nil => addr_strength_reduction_case1 r1 r2 symb n1 n2
  | Aindexed2, r1 :: r2 :: nil, I n1 :: G symb n2 :: nil => addr_strength_reduction_case2 r1 r2 n1 symb n2
  | Aindexed2, r1 :: r2 :: nil, S n1 :: I n2 :: nil => addr_strength_reduction_case3 r1 r2 n1 n2
  | Aindexed2, r1 :: r2 :: nil, I n1 :: S n2 :: nil => addr_strength_reduction_case4 r1 r2 n1 n2
  | Aindexed2, r1 :: r2 :: nil, G symb n1 :: v2 :: nil => addr_strength_reduction_case5 r1 r2 symb n1 v2
  | Aindexed2, r1 :: r2 :: nil, v1 :: G symb n2 :: nil => addr_strength_reduction_case6 r1 r2 v1 symb n2
  | Aindexed2, r1 :: r2 :: nil, I n1 :: v2 :: nil => addr_strength_reduction_case7 r1 r2 n1 v2
  | Aindexed2, r1 :: r2 :: nil, v1 :: I n2 :: nil => addr_strength_reduction_case8 r1 r2 v1 n2
  | Abased symb ofs, r1 :: nil, I n1 :: nil => addr_strength_reduction_case9 symb ofs r1 n1
  | Aindexed n, r1 :: nil, G symb n1 :: nil => addr_strength_reduction_case10 n r1 symb n1
  | Aindexed n, r1 :: nil, S n1 :: nil => addr_strength_reduction_case11 n 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 r1 r2 symb n1 n2 =>
      (Aglobal symb (Int.add n1 n2), nil)
  | addr_strength_reduction_case2 r1 r2 n1 symb n2 =>
      (Aglobal symb (Int.add n1 n2), nil)
  | addr_strength_reduction_case3 r1 r2 n1 n2 =>
      (Ainstack (Int.add n1 n2), nil)
  | addr_strength_reduction_case4 r1 r2 n1 n2 =>
      (Ainstack (Int.add n1 n2), nil)
  | addr_strength_reduction_case5 r1 r2 symb n1 v2 =>
      (Abased symb n1, r2 :: nil)
  | addr_strength_reduction_case6 r1 r2 v1 symb n2 =>
      (Abased symb n2, r1 :: nil)
  | addr_strength_reduction_case7 r1 r2 n1 v2 =>
      (Aindexed n1, r2 :: nil)
  | addr_strength_reduction_case8 r1 r2 v1 n2 =>
      (Aindexed n2, r1 :: nil)
  | addr_strength_reduction_case9 symb ofs r1 n1 =>
      (Aglobal symb (Int.add ofs n1), nil)
  | addr_strength_reduction_case10 n r1 symb n1 =>
      (Aglobal symb (Int.add n1 n), nil)
  | addr_strength_reduction_case11 n r1 n1 =>
      (Ainstack (Int.add n1 n), nil)
  | addr_strength_reduction_default addr args vl =>
      (addr, args)
  end.


Original definition: 
Nondetfunction builtin_strength_reduction
               (ef: external_function) (args: list reg) (vl: list approx) :=
  match ef, args, vl with
  | EF_vload chunk, r1 :: nil, G symb n1 :: nil =>
      (EF_vload_global chunk symb n1, nil)
  | EF_vstore chunk, r1 :: r2 :: nil, G symb n1 :: v2 :: nil =>
      (EF_vstore_global chunk symb n1, r2 :: nil)
  | _, _, _ =>
      (ef, args)
  end.

Inductive builtin_strength_reduction_cases: forall (ef: external_function) (args: list reg) (vl: list approx), Type :=
  | builtin_strength_reduction_case1: forall chunk r1 symb n1, builtin_strength_reduction_cases (EF_vload chunk) (r1 :: nil) (G symb n1 :: nil)
  | builtin_strength_reduction_case2: forall chunk r1 r2 symb n1 v2, builtin_strength_reduction_cases (EF_vstore chunk) (r1 :: r2 :: nil) (G symb n1 :: v2 :: nil)
  | builtin_strength_reduction_default: forall (ef: external_function) (args: list reg) (vl: list approx), builtin_strength_reduction_cases ef args vl.

Definition builtin_strength_reduction_match (ef: external_function) (args: list reg) (vl: list approx) :=
  match ef as zz1, args as zz2, vl as zz3 return builtin_strength_reduction_cases zz1 zz2 zz3 with
  | EF_vload chunk, r1 :: nil, G symb n1 :: nil => builtin_strength_reduction_case1 chunk r1 symb n1
  | EF_vstore chunk, r1 :: r2 :: nil, G symb n1 :: v2 :: nil => builtin_strength_reduction_case2 chunk r1 r2 symb n1 v2
  | ef, args, vl => builtin_strength_reduction_default ef args vl
  end.

Definition builtin_strength_reduction (ef: external_function) (args: list reg) (vl: list approx) :=
  match builtin_strength_reduction_match ef args vl with
  | builtin_strength_reduction_case1 chunk r1 symb n1 =>
      (EF_vload_global chunk symb n1, nil)
  | builtin_strength_reduction_case2 chunk r1 r2 symb n1 v2 =>
      (EF_vstore_global chunk symb n1, r2 :: nil)
  | builtin_strength_reduction_default ef args vl =>
      (ef, args)
  end.


End STRENGTH_REDUCTION.
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