Initial import of compcert

git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@1 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e

1 files changed, 1032 insertions, 0 deletions

diff --git a/backend/Constprop.v b/backend/Constprop.v
new file mode 100644
index 00000000..b1c5a2bb
--- /dev/null
+++ b/backend/Constprop.v
@@ -0,0 +1,1032 @@
+(** Constant propagation over RTL.  This is the first of the two
+  optimizations performed at RTL level.  It proceeds by a standard
+  dataflow analysis and the corresponding code transformation. *)
+
+Require Import Coqlib.
+Require Import Maps.
+Require Import AST.
+Require Import Integers.
+Require Import Floats.
+Require Import Values.
+Require Import Globalenvs.
+Require Import Op.
+Require Import Registers.
+Require Import RTL.
+Require Import Lattice.
+Require Import Kildall.
+
+(** * 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 : Set :=
+  | 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. *)
+  | S: ident -> int -> approx.
+                         (** The value is the address of the given global
+                             symbol plus the given integer offset. *)
+
+(** We equip this set of approximations with a semi-lattice structure.
+  The ordering is inclusion between the sets of values denoted by
+  the approximations. *)
+
+Module Approx <: SEMILATTICE_WITH_TOP.
+  Definition t := approx.
+  Lemma eq: forall (x y: t), {x=y} + {x<>y}.
+  Proof.
+    decide equality.
+    apply Int.eq_dec.
+    apply Float.eq_dec.
+    apply Int.eq_dec.
+    apply ident_eq.
+  Qed.
+  Definition ge (x y: t) : Prop :=
+    x = Unknown \/ y = Novalue \/ x = y.
+  Lemma ge_refl: forall x, ge x x.
+  Proof.
+    unfold ge; tauto.
+  Qed.
+  Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
+  Proof.
+    unfold ge; intuition congruence.
+  Qed.
+  Definition bot := Novalue.
+  Definition top := Unknown.
+  Lemma ge_bot: forall x, ge x bot.
+  Proof.
+    unfold ge, bot; tauto.
+  Qed.
+  Lemma ge_top: forall x, ge top x.
+  Proof.
+    unfold ge, bot; tauto.
+  Qed.
+  Definition lub (x y: t) : t :=
+    if eq x y then x else
+    match x, y with
+    | Novalue, _ => y
+    | _, Novalue => x
+    | _, _ => Unknown
+    end.
+  Lemma lub_commut: forall x y, lub x y = lub y x.
+  Proof.
+    unfold lub; intros.
+    case (eq x y); case (eq y x); intros; try congruence.
+    destruct x; destruct y; auto.
+  Qed.
+  Lemma ge_lub_left: forall x y, ge (lub x y) x.
+  Proof.
+    unfold lub; intros.
+    case (eq x y); intro.
+    apply ge_refl.
+    destruct x; destruct y; unfold ge; tauto.
+  Qed.
+End Approx.
+
+Module D := LPMap 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 [Cmconstr] to avoid excessive
+  duplication of cases in proofs. *)
+
+(*
+Definition 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, 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), Set :=
+  | 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 z1, vl as z2 return eval_static_condition_cases z1 z2 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_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 => S s n
+  | Ocast8signed, I n1 :: nil => I(Int.cast8signed n)
+  | Ocast16signed, I n1 :: nil => I(Int.cast16signed n)
+  | Oadd, I n1 :: I n2 :: nil => I(Int.add n1 n2)
+  | Oadd, S s1 n1 :: I n2 :: nil => S s1 (Int.add n1 n2)
+  | Oaddimm n, I n1 :: nil => I (Int.add n1 n)
+  | Oaddimm n, S s1 n1 :: nil => S s1 (Int.add n1 n)
+  | Osub, I n1 :: I n2 :: nil => I(Int.sub n1 n2)
+  | Osub, S s1 n1 :: I n2 :: nil => S s1 (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.repr 32) then I(Int.shl n1 n2) else Unknown
+  | Oshr, I n1 :: I n2 :: nil => if Int.ltu n2 (Int.repr 32) then I(Int.shr n1 n2) else Unknown
+  | Oshrimm n, I n1 :: nil => if Int.ltu n (Int.repr 32) then I(Int.shr n1 n) else Unknown
+  | Oshrximm n, I n1 :: nil => if Int.ltu n (Int.repr 32) then I(Int.shrx n1 n) else Unknown
+  | Oshru, I n1 :: I n2 :: nil => if Int.ltu n2 (Int.repr 32) then I(Int.shru n1 n2) else Unknown
+  | Orolm amount mask, I n1 :: nil => I(Int.rolm n1 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)
+  | Omuladdf, F n1 :: F n2 :: F n3 :: nil => F(Float.add (Float.mul n1 n2) n3)
+  | Omulsubf, F n1 :: F n2 :: F n3 :: nil => F(Float.sub (Float.mul n1 n2) n3)
+  | Osingleoffloat, F n1 :: nil => F(Float.singleoffloat n1)
+  | Ointoffloat, F n1 :: nil => I(Float.intoffloat n1)
+  | Ofloatofint, I n1 :: nil => F(Float.floatofint n1)
+  | Ofloatofintu, I n1 :: nil => F(Float.floatofintu n1)
+  | Ocmp c, vl =>
+      match eval_static_condition c vl with
+      | None => Unknown
+      | Some b => I(if b then Int.one else Int.zero)
+      end
+  | _, _ => Unknown
+  end.
+*)
+
+Inductive eval_static_operation_cases: forall (op: operation) (vl: list approx), Set :=
+  | 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_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) (S s1 n1 :: I n2 :: nil)
+  | eval_static_operation_case11:
+      forall n n1,
+      eval_static_operation_cases (Oaddimm n) (I n1 :: nil)
+  | eval_static_operation_case12:
+      forall n s1 n1,
+      eval_static_operation_cases (Oaddimm n) (S s1 n1 :: nil)
+  | eval_static_operation_case13:
+      forall n1 n2,
+      eval_static_operation_cases (Osub) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case14:
+      forall s1 n1 n2,
+      eval_static_operation_cases (Osub) (S s1 n1 :: I n2 :: nil)
+  | eval_static_operation_case15:
+      forall n n1,
+      eval_static_operation_cases (Osubimm n) (I n1 :: nil)
+  | eval_static_operation_case16:
+      forall n1 n2,
+      eval_static_operation_cases (Omul) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case17:
+      forall n n1,
+      eval_static_operation_cases (Omulimm n) (I n1 :: nil)
+  | eval_static_operation_case18:
+      forall n1 n2,
+      eval_static_operation_cases (Odiv) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case19:
+      forall n1 n2,
+      eval_static_operation_cases (Odivu) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case20:
+      forall n1 n2,
+      eval_static_operation_cases (Oand) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case21:
+      forall n n1,
+      eval_static_operation_cases (Oandimm n) (I n1 :: nil)
+  | eval_static_operation_case22:
+      forall n1 n2,
+      eval_static_operation_cases (Oor) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case23:
+      forall n n1,
+      eval_static_operation_cases (Oorimm n) (I n1 :: nil)
+  | eval_static_operation_case24:
+      forall n1 n2,
+      eval_static_operation_cases (Oxor) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case25:
+      forall n n1,
+      eval_static_operation_cases (Oxorimm n) (I n1 :: nil)
+  | eval_static_operation_case26:
+      forall n1 n2,
+      eval_static_operation_cases (Onand) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case27:
+      forall n1 n2,
+      eval_static_operation_cases (Onor) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case28:
+      forall n1 n2,
+      eval_static_operation_cases (Onxor) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case29:
+      forall n1 n2,
+      eval_static_operation_cases (Oshl) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case30:
+      forall n1 n2,
+      eval_static_operation_cases (Oshr) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case31:
+      forall n n1,
+      eval_static_operation_cases (Oshrimm n) (I n1 :: nil)
+  | eval_static_operation_case32:
+      forall n n1,
+      eval_static_operation_cases (Oshrximm n) (I n1 :: nil)
+  | eval_static_operation_case33:
+      forall n1 n2,
+      eval_static_operation_cases (Oshru) (I n1 :: I n2 :: nil)
+  | eval_static_operation_case34:
+      forall amount mask n1,
+      eval_static_operation_cases (Orolm amount mask) (I n1 :: nil)
+  | eval_static_operation_case35:
+      forall n1,
+      eval_static_operation_cases (Onegf) (F n1 :: nil)
+  | eval_static_operation_case36:
+      forall n1,
+      eval_static_operation_cases (Oabsf) (F n1 :: nil)
+  | eval_static_operation_case37:
+      forall n1 n2,
+      eval_static_operation_cases (Oaddf) (F n1 :: F n2 :: nil)
+  | eval_static_operation_case38:
+      forall n1 n2,
+      eval_static_operation_cases (Osubf) (F n1 :: F n2 :: nil)
+  | eval_static_operation_case39:
+      forall n1 n2,
+      eval_static_operation_cases (Omulf) (F n1 :: F n2 :: nil)
+  | eval_static_operation_case40:
+      forall n1 n2,
+      eval_static_operation_cases (Odivf) (F n1 :: F n2 :: nil)
+  | eval_static_operation_case41:
+      forall n1 n2 n3,
+      eval_static_operation_cases (Omuladdf) (F n1 :: F n2 :: F n3 :: nil)
+  | eval_static_operation_case42:
+      forall n1 n2 n3,
+      eval_static_operation_cases (Omulsubf) (F n1 :: F n2 :: F n3 :: nil)
+  | eval_static_operation_case43:
+      forall n1,
+      eval_static_operation_cases (Osingleoffloat) (F n1 :: nil)
+  | eval_static_operation_case44:
+      forall n1,
+      eval_static_operation_cases (Ointoffloat) (F n1 :: nil)
+  | eval_static_operation_case45:
+      forall n1,
+      eval_static_operation_cases (Ofloatofint) (I n1 :: nil)
+  | eval_static_operation_case46:
+      forall n1,
+      eval_static_operation_cases (Ofloatofintu) (I n1 :: nil)
+  | eval_static_operation_case47:
+      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 z1, vl as z2 return eval_static_operation_cases z1 z2 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
+  | 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, S s1 n1 :: I n2 :: nil =>
+      eval_static_operation_case9 s1 n1 n2
+  | Oaddimm n, I n1 :: nil =>
+      eval_static_operation_case11 n n1
+  | Oaddimm n, S s1 n1 :: nil =>
+      eval_static_operation_case12 n s1 n1
+  | Osub, I n1 :: I n2 :: nil =>
+      eval_static_operation_case13 n1 n2
+  | Osub, S s1 n1 :: I n2 :: nil =>
+      eval_static_operation_case14 s1 n1 n2
+  | Osubimm n, I n1 :: nil =>
+      eval_static_operation_case15 n n1
+  | Omul, I n1 :: I n2 :: nil =>
+      eval_static_operation_case16 n1 n2
+  | Omulimm n, I n1 :: nil =>
+      eval_static_operation_case17 n n1
+  | Odiv, I n1 :: I n2 :: nil =>
+      eval_static_operation_case18 n1 n2
+  | Odivu, I n1 :: I n2 :: nil =>
+      eval_static_operation_case19 n1 n2
+  | Oand, I n1 :: I n2 :: nil =>
+      eval_static_operation_case20 n1 n2
+  | Oandimm n, I n1 :: nil =>
+      eval_static_operation_case21 n n1
+  | Oor, I n1 :: I n2 :: nil =>
+      eval_static_operation_case22 n1 n2
+  | Oorimm n, I n1 :: nil =>
+      eval_static_operation_case23 n n1
+  | Oxor, I n1 :: I n2 :: nil =>
+      eval_static_operation_case24 n1 n2
+  | Oxorimm n, I n1 :: nil =>
+      eval_static_operation_case25 n n1
+  | Onand, I n1 :: I n2 :: nil =>
+      eval_static_operation_case26 n1 n2
+  | Onor, I n1 :: I n2 :: nil =>
+      eval_static_operation_case27 n1 n2
+  | Onxor, I n1 :: I n2 :: nil =>
+      eval_static_operation_case28 n1 n2
+  | Oshl, I n1 :: I n2 :: nil =>
+      eval_static_operation_case29 n1 n2
+  | Oshr, I n1 :: I n2 :: nil =>
+      eval_static_operation_case30 n1 n2
+  | Oshrimm n, I n1 :: nil =>
+      eval_static_operation_case31 n n1
+  | Oshrximm n, I n1 :: nil =>
+      eval_static_operation_case32 n n1
+  | Oshru, I n1 :: I n2 :: nil =>
+      eval_static_operation_case33 n1 n2
+  | Orolm amount mask, I n1 :: nil =>
+      eval_static_operation_case34 amount mask n1
+  | Onegf, F n1 :: nil =>
+      eval_static_operation_case35 n1
+  | Oabsf, F n1 :: nil =>
+      eval_static_operation_case36 n1
+  | Oaddf, F n1 :: F n2 :: nil =>
+      eval_static_operation_case37 n1 n2
+  | Osubf, F n1 :: F n2 :: nil =>
+      eval_static_operation_case38 n1 n2
+  | Omulf, F n1 :: F n2 :: nil =>
+      eval_static_operation_case39 n1 n2
+  | Odivf, F n1 :: F n2 :: nil =>
+      eval_static_operation_case40 n1 n2
+  | Omuladdf, F n1 :: F n2 :: F n3 :: nil =>
+      eval_static_operation_case41 n1 n2 n3
+  | Omulsubf, F n1 :: F n2 :: F n3 :: nil =>
+      eval_static_operation_case42 n1 n2 n3
+  | Osingleoffloat, F n1 :: nil =>
+      eval_static_operation_case43 n1
+  | Ointoffloat, F n1 :: nil =>
+      eval_static_operation_case44 n1
+  | Ofloatofint, I n1 :: nil =>
+      eval_static_operation_case45 n1
+  | Ofloatofintu, I n1 :: nil =>
+      eval_static_operation_case46 n1
+  | Ocmp c, vl =>
+      eval_static_operation_case47 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 =>
+      S s n
+  | eval_static_operation_case6 n1 =>
+      I(Int.cast8signed n1)
+  | eval_static_operation_case7 n1 =>
+      I(Int.cast16signed n1)
+  | eval_static_operation_case8 n1 n2 =>
+      I(Int.add n1 n2)
+  | eval_static_operation_case9 s1 n1 n2 =>
+      S s1 (Int.add n1 n2)
+  | eval_static_operation_case11 n n1 =>
+      I (Int.add n1 n)
+  | eval_static_operation_case12 n s1 n1 =>
+      S s1 (Int.add n1 n)
+  | eval_static_operation_case13 n1 n2 =>
+      I(Int.sub n1 n2)
+  | eval_static_operation_case14 s1 n1 n2 =>
+      S s1 (Int.sub n1 n2)
+  | eval_static_operation_case15 n n1 =>
+      I (Int.sub n n1)
+  | eval_static_operation_case16 n1 n2 =>
+      I(Int.mul n1 n2)
+  | eval_static_operation_case17 n n1 =>
+      I(Int.mul n1 n)
+  | eval_static_operation_case18 n1 n2 =>
+      if Int.eq n2 Int.zero then Unknown else I(Int.divs n1 n2)
+  | eval_static_operation_case19 n1 n2 =>
+      if Int.eq n2 Int.zero then Unknown else I(Int.divu n1 n2)
+  | eval_static_operation_case20 n1 n2 =>
+      I(Int.and n1 n2)
+  | eval_static_operation_case21 n n1 =>
+      I(Int.and n1 n)
+  | eval_static_operation_case22 n1 n2 =>
+      I(Int.or n1 n2)
+  | eval_static_operation_case23 n n1 =>
+      I(Int.or n1 n)
+  | eval_static_operation_case24 n1 n2 =>
+      I(Int.xor n1 n2)
+  | eval_static_operation_case25 n n1 =>
+      I(Int.xor n1 n)
+  | eval_static_operation_case26 n1 n2 =>
+      I(Int.xor (Int.and n1 n2) Int.mone)
+  | eval_static_operation_case27 n1 n2 =>
+      I(Int.xor (Int.or n1 n2) Int.mone)
+  | eval_static_operation_case28 n1 n2 =>
+      I(Int.xor (Int.xor n1 n2) Int.mone)
+  | eval_static_operation_case29 n1 n2 =>
+      if Int.ltu n2 (Int.repr 32) then I(Int.shl n1 n2) else Unknown
+  | eval_static_operation_case30 n1 n2 =>
+      if Int.ltu n2 (Int.repr 32) then I(Int.shr n1 n2) else Unknown
+  | eval_static_operation_case31 n n1 =>
+      if Int.ltu n (Int.repr 32) then I(Int.shr n1 n) else Unknown
+  | eval_static_operation_case32 n n1 =>
+      if Int.ltu n (Int.repr 32) then I(Int.shrx n1 n) else Unknown
+  | eval_static_operation_case33 n1 n2 =>
+      if Int.ltu n2 (Int.repr 32) then I(Int.shru n1 n2) else Unknown
+  | eval_static_operation_case34 amount mask n1 =>
+      I(Int.rolm n1 amount mask)
+  | eval_static_operation_case35 n1 =>
+      F(Float.neg n1)
+  | eval_static_operation_case36 n1 =>
+      F(Float.abs n1)
+  | eval_static_operation_case37 n1 n2 =>
+      F(Float.add n1 n2)
+  | eval_static_operation_case38 n1 n2 =>
+      F(Float.sub n1 n2)
+  | eval_static_operation_case39 n1 n2 =>
+      F(Float.mul n1 n2)
+  | eval_static_operation_case40 n1 n2 =>
+      F(Float.div n1 n2)
+  | eval_static_operation_case41 n1 n2 n3 =>
+      F(Float.add (Float.mul n1 n2) n3)
+  | eval_static_operation_case42 n1 n2 n3 =>
+      F(Float.sub (Float.mul n1 n2) n3)
+  | eval_static_operation_case43 n1 =>
+      F(Float.singleoffloat n1)
+  | eval_static_operation_case44 n1 =>
+      I(Float.intoffloat n1)
+  | eval_static_operation_case45 n1 =>
+      F(Float.floatofint n1)
+  | eval_static_operation_case46 n1 =>
+      F(Float.floatofintu n1)
+  | eval_static_operation_case47 c vl =>
+      match eval_static_condition c vl with
+      | None => Unknown
+      | Some b => I(if b then Int.one else Int.zero)
+      end
+  | eval_static_operation_default op vl =>
+      Unknown
+  end.
+
+(** The transfer function for the dataflow analysis is straightforward:
+  for [Iop] instructions, we set the approximation of the destination
+  register to the result of executing abstractly the operation;
+  for [Iload] and [Icall], we set the approximation of the destination
+  to [Unknown]. *)
+
+Definition approx_regs (rl: list reg) (approx: D.t) :=
+  List.map (fun r => D.get r approx) rl.
+
+Definition transfer (f: function) (pc: node) (before: D.t) :=
+  match f.(fn_code)!pc with
+  | None => before
+  | Some i =>
+      match i with
+      | Inop s =>
+          before
+      | Iop op args res s =>
+          let a := eval_static_operation op (approx_regs args before) in
+          D.set res a before
+      | Iload chunk addr args dst s =>
+          D.set dst Unknown before
+      | Istore chunk addr args src s =>
+          before
+      | Icall sig ros args res s =>
+          D.set res Unknown before
+      | Icond cond args ifso ifnot =>
+          before
+      | Ireturn optarg =>
+          before
+      end
+  end.
+
+(** The static analysis itself is then an instantiation of Kildall's
+  generic solver for forward dataflow inequations. [analyze f]
+  returns a mapping from program points to mappings of pseudo-registers
+  to approximations.  It can fail to reach a fixpoint in a reasonable
+  number of iterations, in which case [None] is returned. *)
+
+Module DS := Dataflow_Solver D.
+
+Definition analyze (f: RTL.function): option (PMap.t D.t) :=
+  DS.fixpoint (successors f) f.(fn_nextpc) (transfer f) 
+              ((f.(fn_entrypoint), D.top) :: nil).
+
+(** * Code transformation *)
+
+(** ** 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 approx: D.t.
+
+Definition intval (r: reg) : option int :=
+  match D.get r approx with I n => Some n | _ => None end.
+
+Inductive cond_strength_reduction_cases: condition -> list reg -> Set :=
+  | csr_case1:
+      forall c r1 r2,
+      cond_strength_reduction_cases (Ccomp c) (r1 :: r2 :: nil)
+  | csr_case2:
+      forall c r1 r2,
+      cond_strength_reduction_cases (Ccompu c) (r1 :: r2 :: nil)
+  | csr_default:
+      forall c rl,
+      cond_strength_reduction_cases c rl.
+
+Definition cond_strength_reduction_match (cond: condition) (rl: list reg) :=
+  match cond as x, rl as y return cond_strength_reduction_cases x y with
+  | Ccomp c, r1 :: r2 :: nil =>
+      csr_case1 c r1 r2
+  | Ccompu c, r1 :: r2 :: nil =>
+      csr_case2 c r1 r2
+  | cond, rl =>
+      csr_default cond rl
+  end.
+
+Definition cond_strength_reduction
+              (cond: condition) (args: list reg) : condition * list reg :=
+  match cond_strength_reduction_match cond args with
+  | csr_case1 c r1 r2 =>
+      match intval r1, intval r2 with
+      | Some n, _ =>
+          (Ccompimm (swap_comparison c) n, r2 :: nil)
+      | _, Some n =>
+          (Ccompimm c n, r1 :: nil)
+      | _, _ =>
+          (cond, args)
+      end
+  | csr_case2 c r1 r2 =>
+      match intval r1, intval r2 with
+      | Some n, _ =>
+          (Ccompuimm (swap_comparison c) n, r2 :: nil)
+      | _, Some n =>
+          (Ccompuimm c n, r1 :: nil)
+      | _, _ =>
+          (cond, args)
+      end
+  | csr_default cond args =>
+      (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) (r: reg) :=
+  if Int.eq n Int.zero
+  then (Omove, r :: nil)
+  else (Orolm n (Int.shl Int.mone 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 (Orolm (Int.sub (Int.repr 32) n) (Int.shru Int.mone 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).
+
+Inductive op_strength_reduction_cases: operation -> list reg -> Set :=
+  | op_strength_reduction_case1:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oadd (r1 :: r2 :: nil)
+  | op_strength_reduction_case2:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Osub (r1 :: r2 :: nil)
+  | op_strength_reduction_case3:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Omul (r1 :: r2 :: nil)
+  | op_strength_reduction_case4:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Odiv (r1 :: r2 :: nil)
+  | op_strength_reduction_case5:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Odivu (r1 :: r2 :: nil)
+  | op_strength_reduction_case6:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oand (r1 :: r2 :: nil)
+  | op_strength_reduction_case7:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oor (r1 :: r2 :: nil)
+  | op_strength_reduction_case8:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oxor (r1 :: r2 :: nil)
+  | op_strength_reduction_case9:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oshl (r1 :: r2 :: nil)
+  | op_strength_reduction_case10:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oshr (r1 :: r2 :: nil)
+  | op_strength_reduction_case11:
+      forall (r1: reg) (r2: reg),
+      op_strength_reduction_cases Oshru (r1 :: r2 :: nil)
+  | op_strength_reduction_case12:
+      forall (c: condition) (rl: list reg),
+      op_strength_reduction_cases (Ocmp c) rl
+  | op_strength_reduction_default:
+      forall (op: operation) (args: list reg),
+      op_strength_reduction_cases op args.
+
+Definition op_strength_reduction_match (op: operation) (args: list reg) :=
+  match op as z1, args as z2 return op_strength_reduction_cases z1 z2 with
+  | Oadd, r1 :: r2 :: nil =>
+      op_strength_reduction_case1 r1 r2
+  | Osub, r1 :: r2 :: nil =>
+      op_strength_reduction_case2 r1 r2
+  | Omul, r1 :: r2 :: nil =>
+      op_strength_reduction_case3 r1 r2
+  | Odiv, r1 :: r2 :: nil =>
+      op_strength_reduction_case4 r1 r2
+  | Odivu, r1 :: r2 :: nil =>
+      op_strength_reduction_case5 r1 r2
+  | Oand, r1 :: r2 :: nil =>
+      op_strength_reduction_case6 r1 r2
+  | Oor, r1 :: r2 :: nil =>
+      op_strength_reduction_case7 r1 r2
+  | Oxor, r1 :: r2 :: nil =>
+      op_strength_reduction_case8 r1 r2
+  | Oshl, r1 :: r2 :: nil =>
+      op_strength_reduction_case9 r1 r2
+  | Oshr, r1 :: r2 :: nil =>
+      op_strength_reduction_case10 r1 r2
+  | Oshru, r1 :: r2 :: nil =>
+      op_strength_reduction_case11 r1 r2
+  | Ocmp c, rl =>
+      op_strength_reduction_case12 c rl
+  | op, args =>
+      op_strength_reduction_default op args
+  end.
+
+Definition op_strength_reduction (op: operation) (args: list reg) :=
+  match op_strength_reduction_match op args with
+  | op_strength_reduction_case1 r1 r2 => (* Oadd *)
+      match intval r1, intval r2 with
+      | Some n, _ => make_addimm n r2
+      | _, Some n => make_addimm n r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case2 r1 r2 => (* Osub *)
+      match intval r1, intval r2 with
+      | Some n, _ => (Osubimm n, r2 :: nil)
+      | _, Some n => make_addimm (Int.neg n) r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case3 r1 r2 => (* Omul *)
+      match intval r1, intval r2 with
+      | Some n, _ => make_mulimm n r2
+      | _, Some n => make_mulimm n r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case4 r1 r2 => (* Odiv *)
+      match intval r2 with
+      | Some n =>
+          match Int.is_power2 n with
+          | Some l => (Oshrximm l, r1 :: nil)
+          | None   => (op, args)
+          end
+      | None =>
+          (op, args)
+      end
+  | op_strength_reduction_case5 r1 r2 => (* Odivu *)
+      match intval r2 with
+      | Some n =>
+          match Int.is_power2 n with
+          | Some l => make_shruimm l r1
+          | None   => (op, args)
+          end
+      | None =>
+          (op, args)
+      end
+  | op_strength_reduction_case6 r1 r2 => (* Oand *)
+      match intval r1, intval r2 with
+      | Some n, _ => make_andimm n r2
+      | _, Some n => make_andimm n r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case7 r1 r2 => (* Oor *)
+      match intval r1, intval r2 with
+      | Some n, _ => make_orimm n r2
+      | _, Some n => make_orimm n r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case8 r1 r2 => (* Oxor *)
+      match intval r1, intval r2 with
+      | Some n, _ => make_xorimm n r2
+      | _, Some n => make_xorimm n r1
+      | _, _ => (op, args)
+      end
+  | op_strength_reduction_case9 r1 r2 => (* Oshl *)
+      match intval r2 with
+      | Some n =>
+          if Int.ltu n (Int.repr 32)
+          then make_shlimm n r1
+          else (op, args)
+      | _ => (op, args)
+      end
+  | op_strength_reduction_case10 r1 r2 => (* Oshr *)
+      match intval r2 with
+      | Some n =>
+          if Int.ltu n (Int.repr 32)
+          then make_shrimm n r1
+          else (op, args)
+      | _ => (op, args)
+      end
+  | op_strength_reduction_case11 r1 r2 => (* Oshru *)
+      match intval r2 with
+      | Some n =>
+          if Int.ltu n (Int.repr 32)
+          then make_shruimm n r1
+          else (op, args)
+      | _ => (op, args)
+      end
+  | op_strength_reduction_case12 c args => (* Ocmp *)
+      let (c', args') := cond_strength_reduction c args in
+      (Ocmp c', args')
+  | op_strength_reduction_default op args => (* default *)
+      (op, args)
+  end.
+
+Inductive addr_strength_reduction_cases: forall (addr: addressing) (args: list reg), Set :=
+  | addr_strength_reduction_case1:
+      forall (r1: reg) (r2: reg),
+      addr_strength_reduction_cases (Aindexed2) (r1 :: r2 :: nil)
+  | addr_strength_reduction_case2:
+      forall (symb: ident) (ofs: int) (r1: reg),
+      addr_strength_reduction_cases (Abased symb ofs) (r1 :: nil)
+  | addr_strength_reduction_case3:
+      forall n r1,
+      addr_strength_reduction_cases (Aindexed n) (r1 :: nil)
+  | addr_strength_reduction_default:
+      forall (addr: addressing) (args: list reg),
+      addr_strength_reduction_cases addr args.
+
+Definition addr_strength_reduction_match (addr: addressing) (args: list reg) :=
+  match addr as z1, args as z2 return addr_strength_reduction_cases z1 z2 with
+  | Aindexed2, r1 :: r2 :: nil =>
+      addr_strength_reduction_case1 r1 r2
+  | Abased symb ofs, r1 :: nil =>
+      addr_strength_reduction_case2 symb ofs r1
+  | Aindexed n, r1 :: nil =>
+      addr_strength_reduction_case3 n r1
+  | addr, args =>
+      addr_strength_reduction_default addr args
+  end.
+
+Definition addr_strength_reduction (addr: addressing) (args: list reg) :=
+  match addr_strength_reduction_match addr args with
+  | addr_strength_reduction_case1 r1 r2 => (* Aindexed2 *)
+      match D.get r1 approx, D.get r2 approx with
+      | S symb n1, I n2 => (Aglobal symb (Int.add n1 n2), nil)
+      | S symb n1, _ => (Abased symb n1, r2 :: nil)
+      | I n1, S symb n2 => (Aglobal symb (Int.add n1 n2), nil)
+      | I n1, _ => (Aindexed n1, r2 :: nil)
+      | _, S symb n2 => (Abased symb n2, r1 :: nil)
+      | _, I n2 => (Aindexed n2, r1 :: nil)
+      | _, _ => (addr, args)
+      end
+  | addr_strength_reduction_case2 symb ofs r1 => (* Abased *)
+      match intval r1 with
+      | Some n => (Aglobal symb (Int.add ofs n), nil)
+      | _ => (addr, args)
+      end
+  | addr_strength_reduction_case3 n r1 => (* Aindexed *)
+      match D.get r1 approx with
+      | S symb ofs => (Aglobal symb (Int.add ofs n), nil)
+      | _ => (addr, args)
+      end
+  | addr_strength_reduction_default addr args => (* default *)
+      (addr, args)
+  end.
+
+End STRENGTH_REDUCTION.
+
+(** ** Code transformation *)
+
+(** The code transformation proceeds instruction by instruction.
+    Operators whose arguments are all statically known are turned
+    into ``load integer constant'', ``load float constant'' or
+    ``load symbol address'' operations.  Operators for which some
+    but not all arguments are known are subject to strength reduction,
+    and similarly for the addressing modes of load and store instructions.
+    Other instructions are unchanged. *)
+
+Definition transf_instr (approx: D.t) (instr: instruction) :=
+  match instr with
+  | Iop op args res s =>
+      match eval_static_operation op (approx_regs args approx) with
+      | I n =>
+          Iop (Ointconst n) nil res s
+      | F n =>
+          Iop (Ofloatconst n) nil res s
+      | S symb ofs =>
+          Iop (Oaddrsymbol symb ofs) nil res s
+      | _ =>
+          let (op', args') := op_strength_reduction approx op args in
+          Iop op' args' res s
+      end
+  | Iload chunk addr args dst s =>
+      let (addr', args') := addr_strength_reduction approx addr args in
+      Iload chunk addr' args' dst s      
+  | Istore chunk addr args src s =>
+      let (addr', args') := addr_strength_reduction approx addr args in
+      Istore chunk addr' args' src s      
+  | Icall sig ros args res s =>
+      let ros' :=
+        match ros with
+        | inl r =>
+            match D.get r approx with
+            | S symb ofs => if Int.eq ofs Int.zero then inr _ symb else ros
+            | _ => ros
+            end
+        | inr s => ros
+        end in
+      Icall sig ros' args res s
+  | Icond cond args s1 s2 =>
+      match eval_static_condition cond (approx_regs args approx) with
+      | Some b =>
+          if b then Inop s1 else Inop s2
+      | None =>
+          let (cond', args') := cond_strength_reduction approx cond args in
+          Icond cond' args' s1 s2
+      end
+  | _ =>
+      instr
+  end.
+
+Definition transf_code (approxs: PMap.t D.t) (instrs: code) : code :=
+  PTree.map (fun pc instr => transf_instr approxs!!pc instr) instrs.
+
+Lemma transf_code_wf:
+  forall f approxs,
+  (forall pc, Plt pc f.(fn_nextpc) \/ f.(fn_code)!pc = None) ->
+  (forall pc, Plt pc f.(fn_nextpc)
+      \/ (transf_code approxs f.(fn_code))!pc = None).
+Proof.
+  intros. 
+  elim (H pc); intro.
+  left; auto.
+  right. unfold transf_code. rewrite PTree.gmap. 
+  unfold option_map; rewrite H0. reflexivity.
+Qed.
+
+Definition transf_function (f: function) : function :=
+  match analyze f with
+  | None => f
+  | Some approxs =>
+      mkfunction
+        f.(fn_sig)
+        f.(fn_params)
+        f.(fn_stacksize)
+        (transf_code approxs f.(fn_code))
+        f.(fn_entrypoint)
+        f.(fn_nextpc)
+        (transf_code_wf f approxs f.(fn_code_wf))
+  end.
+
+Definition transf_program (p: program) : program :=
+  transform_program transf_function p.