Module ConstpropOpproof


Correctness proof for operator strength reduction.

Require Import Coqlib Compopts.
Require Import Integers Floats Values Memory Globalenvs Events.
Require Import Op Registers RTL ValueDomain.
Require Import ConstpropOp.

Local Transparent Archi.ptr64.

Correctness of strength reduction


We now show that strength reduction over operators and addressing modes preserve semantics: the strength-reduced operations and addressings evaluate to the same values as the original ones if the actual arguments match the static approximations used for strength reduction.

Section STRENGTH_REDUCTION.

Variable bc: block_classification.
Variable ge: genv.
Hypothesis GENV: genv_match bc ge.
Variable sp: block.
Hypothesis STACK: bc sp = BCstack.
Variable ae: AE.t.
Variable rs: regset.
Variable m: mem.
Hypothesis MATCH: ematch bc rs ae.

Lemma match_G:
  forall r id ofs,
  AE.get r ae = Ptr(Gl id ofs) -> Val.lessdef rs#r (Genv.symbol_address ge id ofs).
Proof.
  intros. apply vmatch_ptr_gl with bc; auto. rewrite <- H. apply MATCH.
Qed.

Lemma match_S:
  forall r ofs,
  AE.get r ae = Ptr(Stk ofs) -> Val.lessdef rs#r (Vptr sp ofs).
Proof.
  intros. apply vmatch_ptr_stk with bc; auto. rewrite <- H. apply MATCH.
Qed.

Ltac InvApproxRegs :=
  match goal with
  | [ H: _ :: _ = _ :: _ |- _ ] =>
        injection H; clear H; intros; InvApproxRegs
  | [ H: ?v = AE.get ?r ae |- _ ] =>
        generalize (MATCH r); rewrite <- H; clear H; intro; InvApproxRegs
  | _ => idtac
  end.

Ltac SimplVM :=
  match goal with
  | [ H: vmatch _ ?v (I ?n) |- _ ] =>
      let E := fresh in
      assert (E: v = Vint n) by (inversion H; auto);
      rewrite E in *; clear H; SimplVM
  | [ H: vmatch _ ?v (F ?n) |- _ ] =>
      let E := fresh in
      assert (E: v = Vfloat n) by (inversion H; auto);
      rewrite E in *; clear H; SimplVM
  | [ H: vmatch _ ?v (FS ?n) |- _ ] =>
      let E := fresh in
      assert (E: v = Vsingle n) by (inversion H; auto);
      rewrite E in *; clear H; SimplVM
  | [ H: vmatch _ ?v (Ptr(Gl ?id ?ofs)) |- _ ] =>
      let E := fresh in
      assert (E: Val.lessdef v (Genv.symbol_address ge id ofs)) by (eapply vmatch_ptr_gl; eauto);
      clear H; SimplVM
  | [ H: vmatch _ ?v (Ptr(Stk ?ofs)) |- _ ] =>
      let E := fresh in
      assert (E: Val.lessdef v (Vptr sp ofs)) by (eapply vmatch_ptr_stk; eauto);
      clear H; SimplVM
  | _ => idtac
  end.

Lemma const_for_result_correct:
  forall a op v,
  const_for_result a = Some op ->
  vmatch bc v a ->
  exists v', eval_operation ge (Vptr sp Ptrofs.zero) op nil m = Some v' /\ Val.lessdef v v'.
Proof.
  unfold const_for_result; intros.
  destruct a; inv H; SimplVM.
- (* integer *)
  exists (Vint n); auto.
- (* float *)
  destruct (generate_float_constants tt); inv H2. exists (Vfloat f); auto.
- (* single *)
  destruct (generate_float_constants tt); inv H2. exists (Vsingle f); auto.
- (* pointer *)
  destruct p; try discriminate; SimplVM.
  + (* global *)
    inv H2. exists (Genv.symbol_address ge id ofs); auto.
  + (* stack *)
    inv H2. exists (Vptr sp ofs); split; auto. simpl. rewrite Ptrofs.add_zero_l; auto.
Qed.

Lemma cond_strength_reduction_correct:
  forall cond args vl,
  vl = map (fun r => AE.get r ae) args ->
  let (cond', args') := cond_strength_reduction cond args vl in
  eval_condition cond' rs##args' m = eval_condition cond rs##args m.
Proof.
  intros until vl. unfold cond_strength_reduction.
  case (cond_strength_reduction_match cond args vl); simpl; intros; InvApproxRegs; SimplVM.
- apply Val.swap_cmp_bool.
- auto.
- apply Val.swap_cmpu_bool.
- auto.
- auto.
Qed.

Lemma make_cmp_base_correct:
  forall c args vl,
  vl = map (fun r => AE.get r ae) args ->
  let (op', args') := make_cmp_base c args vl in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some v
         /\ Val.lessdef (Val.of_optbool (eval_condition c rs##args m)) v.
Proof.
  intros. unfold make_cmp_base.
  generalize (cond_strength_reduction_correct c args vl H).
  destruct (cond_strength_reduction c args vl) as [c' args']. intros EQ.
  econstructor; split. simpl; eauto. rewrite EQ. auto.
Qed.

Lemma make_cmp_correct:
  forall c args vl,
  vl = map (fun r => AE.get r ae) args ->
  let (op', args') := make_cmp c args vl in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some v
         /\ Val.lessdef (Val.of_optbool (eval_condition c rs##args m)) v.
Proof.
  intros c args vl.
  assert (Y: forall r, vincl (AE.get r ae) (Uns Ptop 1) = true ->
             rs#r = Vundef \/ rs#r = Vint Int.zero \/ rs#r = Vint Int.one).
  { intros. apply vmatch_Uns_1 with bc Ptop. eapply vmatch_ge. eapply vincl_ge; eauto. apply MATCH. }
  unfold make_cmp. case (make_cmp_match c args vl); intros.
- destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
  simpl in H; inv H. InvBooleans. subst n.
  exists (rs#r1); split; auto. simpl.
  exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
  destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
  simpl in H; inv H. InvBooleans. subst n.
  exists (Val.xor rs#r1 (Vint Int.one)); split; auto. simpl.
  exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
  apply make_cmp_base_correct; auto.
- destruct (Int.eq_dec n Int.zero && vincl v1 (Uns Ptop 1)) eqn:E0.
  simpl in H; inv H. InvBooleans. subst n.
  exists (rs#r1); split; auto. simpl.
  exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
  destruct (Int.eq_dec n Int.one && vincl v1 (Uns Ptop 1)) eqn:E1.
  simpl in H; inv H. InvBooleans. subst n.
  exists (Val.xor rs#r1 (Vint Int.one)); split; auto. simpl.
  exploit Y; eauto. intros [A | [A | A]]; rewrite A; simpl; auto.
  apply make_cmp_base_correct; auto.
- apply make_cmp_base_correct; auto.
Qed.

Lemma make_addimm_correct:
  forall n r,
  let (op, args) := make_addimm n r in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.add rs#r (Vint n)) v.
Proof.
  intros. unfold make_addimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros.
  subst. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.add_zero; auto. rewrite Ptrofs.add_zero; auto.
  exists (Val.add rs#r (Vint n)); auto.
Qed.

Lemma make_shlimm_correct:
  forall n r1 r2,
  rs#r2 = Vint n ->
  let (op, args) := make_shlimm n r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shl rs#r1 (Vint n)) v.
Proof.
  intros; unfold make_shlimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
  exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shl_zero. auto.
  destruct (Int.ltu n Int.iwordsize) eqn:?; intros.
  rewrite Val.shl_rolm; auto. econstructor; split; eauto. auto.
  econstructor; split; eauto. simpl. congruence.
Qed.

Lemma make_shrimm_correct:
  forall n r1 r2,
  rs#r2 = Vint n ->
  let (op, args) := make_shrimm n r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shr rs#r1 (Vint n)) v.
Proof.
  intros; unfold make_shrimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
  exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shr_zero. auto.
  destruct (Int.ltu n Int.iwordsize) eqn:?.
  econstructor; split; eauto. simpl. auto.
  econstructor; split; eauto. simpl. congruence.
Qed.

Lemma make_shruimm_correct:
  forall n r1 r2,
  rs#r2 = Vint n ->
  let (op, args) := make_shruimm n r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.shru rs#r1 (Vint n)) v.
Proof.
  intros; unfold make_shruimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
  exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.shru_zero. auto.
  destruct (Int.ltu n Int.iwordsize) eqn:?; intros.
  rewrite Val.shru_rolm; auto. econstructor; split; eauto. auto.
  econstructor; split; eauto. simpl. congruence.
Qed.

Lemma make_mulimm_correct:
  forall n r1 r2,
  rs#r2 = Vint n ->
  let (op, args) := make_mulimm n r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mul rs#r1 (Vint n)) v.
Proof.
  intros; unfold make_mulimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros. subst.
  exists (Vint Int.zero); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.mul_zero; auto.
  predSpec Int.eq Int.eq_spec n Int.one; intros. subst.
  exists (rs#r1); split; auto. destruct (rs#r1); simpl; auto. rewrite Int.mul_one; auto.
  destruct (Int.is_power2 n) eqn:?; intros.
  rewrite (Val.mul_pow2 rs#r1 _ _ Heqo). rewrite Val.shl_rolm.
  econstructor; split; eauto. auto.
  eapply Int.is_power2_range; eauto.
  econstructor; split; eauto. auto.
Qed.

Lemma make_divimm_correct:
  forall n r1 r2 v,
  Val.divs rs#r1 rs#r2 = Some v ->
  rs#r2 = Vint n ->
  let (op, args) := make_divimm n r1 r2 in
  exists w, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some w /\ Val.lessdef v w.
Proof.
  intros; unfold make_divimm.
  destruct (Int.is_power2 n) eqn:?.
  destruct (Int.ltu i (Int.repr 31)) eqn:?.
  exists v; split; auto. simpl. eapply Val.divs_pow2; eauto. congruence.
  exists v; auto.
  exists v; auto.
Qed.

Lemma make_divuimm_correct:
  forall n r1 r2 v,
  Val.divu rs#r1 rs#r2 = Some v ->
  rs#r2 = Vint n ->
  let (op, args) := make_divuimm n r1 r2 in
  exists w, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some w /\ Val.lessdef v w.
Proof.
  intros; unfold make_divuimm.
  destruct (Int.is_power2 n) eqn:?.
  econstructor; split. simpl; eauto.
  exploit Int.is_power2_range; eauto. intros RANGE.
  rewrite <- Val.shru_rolm; auto. rewrite H0 in H.
  destruct (rs#r1); simpl in *; inv H.
  destruct (Int.eq n Int.zero); inv H2.
  rewrite RANGE. rewrite (Int.divu_pow2 i0 _ _ Heqo). auto.
  exists v; auto.
Qed.

Lemma make_andimm_correct:
  forall n r x,
  vmatch bc rs#r x ->
  let (op, args) := make_andimm n r x in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.and rs#r (Vint n)) v.
Proof.
  intros; unfold make_andimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros.
  subst n. exists (Vint Int.zero); split; auto. destruct (rs#r); simpl; auto. rewrite Int.and_zero; auto.
  predSpec Int.eq Int.eq_spec n Int.mone; intros.
  subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.and_mone; auto.
  destruct (match x with Uns _ k => Int.eq (Int.zero_ext k (Int.not n)) Int.zero
                       | _ => false end) eqn:UNS.
  destruct x; try congruence.
  exists (rs#r); split; auto.
  inv H; auto. simpl. replace (Int.and i n) with i; auto.
  generalize (Int.eq_spec (Int.zero_ext n0 (Int.not n)) Int.zero); rewrite UNS; intro EQ.
  Int.bit_solve. destruct (zlt i0 n0).
  replace (Int.testbit n i0) with (negb (Int.testbit Int.zero i0)).
  rewrite Int.bits_zero. simpl. rewrite andb_true_r. auto.
  rewrite <- EQ. rewrite Int.bits_zero_ext by omega. rewrite zlt_true by auto.
  rewrite Int.bits_not by auto. apply negb_involutive.
  rewrite H6 by auto. auto.
  econstructor; split; eauto. auto.
Qed.

Lemma make_orimm_correct:
  forall n r,
  let (op, args) := make_orimm n r in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.or rs#r (Vint n)) v.
Proof.
  intros; unfold make_orimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros.
  subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.or_zero; auto.
  predSpec Int.eq Int.eq_spec n Int.mone; intros.
  subst n. exists (Vint Int.mone); split; auto. destruct (rs#r); simpl; auto. rewrite Int.or_mone; auto.
  econstructor; split; eauto. auto.
Qed.

Lemma make_xorimm_correct:
  forall n r,
  let (op, args) := make_xorimm n r in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.xor rs#r (Vint n)) v.
Proof.
  intros; unfold make_xorimm.
  predSpec Int.eq Int.eq_spec n Int.zero; intros.
  subst n. exists (rs#r); split; auto. destruct (rs#r); simpl; auto. rewrite Int.xor_zero; auto.
  predSpec Int.eq Int.eq_spec n Int.mone; intros.
  subst n. exists (Val.notint rs#r); split; auto.
  econstructor; split; eauto. auto.
Qed.

Lemma make_mulfimm_correct:
  forall n r1 r2,
  rs#r2 = Vfloat n ->
  let (op, args) := make_mulfimm n r1 r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulf rs#r1 rs#r2) v.
Proof.
  intros; unfold make_mulfimm.
  destruct (Float.eq_dec n (Float.of_int (Int.repr 2))); intros.
  simpl. econstructor; split. eauto. rewrite H; subst n.
  destruct (rs#r1); simpl; auto. rewrite Float.mul2_add; auto.
  simpl. econstructor; split; eauto.
Qed.

Lemma make_mulfimm_correct_2:
  forall n r1 r2,
  rs#r1 = Vfloat n ->
  let (op, args) := make_mulfimm n r2 r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulf rs#r1 rs#r2) v.
Proof.
  intros; unfold make_mulfimm.
  destruct (Float.eq_dec n (Float.of_int (Int.repr 2))); intros.
  simpl. econstructor; split. eauto. rewrite H; subst n.
  destruct (rs#r2); simpl; auto. rewrite Float.mul2_add; auto.
  rewrite Float.mul_commut; auto.
  simpl. econstructor; split; eauto.
Qed.

Lemma make_mulfsimm_correct:
  forall n r1 r2,
  rs#r2 = Vsingle n ->
  let (op, args) := make_mulfsimm n r1 r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulfs rs#r1 rs#r2) v.
Proof.
  intros; unfold make_mulfsimm.
  destruct (Float32.eq_dec n (Float32.of_int (Int.repr 2))); intros.
  simpl. econstructor; split. eauto. rewrite H; subst n.
  destruct (rs#r1); simpl; auto. rewrite Float32.mul2_add; auto.
  simpl. econstructor; split; eauto.
Qed.

Lemma make_mulfsimm_correct_2:
  forall n r1 r2,
  rs#r1 = Vsingle n ->
  let (op, args) := make_mulfsimm n r2 r1 r2 in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.mulfs rs#r1 rs#r2) v.
Proof.
  intros; unfold make_mulfsimm.
  destruct (Float32.eq_dec n (Float32.of_int (Int.repr 2))); intros.
  simpl. econstructor; split. eauto. rewrite H; subst n.
  destruct (rs#r2); simpl; auto. rewrite Float32.mul2_add; auto.
  rewrite Float32.mul_commut; auto.
  simpl. econstructor; split; eauto.
Qed.

Lemma make_cast8signed_correct:
  forall r x,
  vmatch bc rs#r x ->
  let (op, args) := make_cast8signed r x in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.sign_ext 8 rs#r) v.
Proof.
  intros; unfold make_cast8signed. destruct (vincl x (Sgn Ptop 8)) eqn:INCL.
  exists rs#r; split; auto.
  assert (V: vmatch bc rs#r (Sgn Ptop 8)).
  { eapply vmatch_ge; eauto. apply vincl_ge; auto. }
  inv V; simpl; auto. rewrite is_sgn_sign_ext in H4 by auto. rewrite H4; auto.
  econstructor; split; simpl; eauto.
Qed.

Lemma make_cast16signed_correct:
  forall r x,
  vmatch bc rs#r x ->
  let (op, args) := make_cast16signed r x in
  exists v, eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v /\ Val.lessdef (Val.sign_ext 16 rs#r) v.
Proof.
  intros; unfold make_cast16signed. destruct (vincl x (Sgn Ptop 16)) eqn:INCL.
  exists rs#r; split; auto.
  assert (V: vmatch bc rs#r (Sgn Ptop 16)).
  { eapply vmatch_ge; eauto. apply vincl_ge; auto. }
  inv V; simpl; auto. rewrite is_sgn_sign_ext in H4 by auto. rewrite H4; auto.
  econstructor; split; simpl; eauto.
Qed.

Lemma op_strength_reduction_correct:
  forall op args vl v,
  vl = map (fun r => AE.get r ae) args ->
  eval_operation ge (Vptr sp Ptrofs.zero) op rs##args m = Some v ->
  let (op', args') := op_strength_reduction op args vl in
  exists w, eval_operation ge (Vptr sp Ptrofs.zero) op' rs##args' m = Some w /\ Val.lessdef v w.
Proof.
  intros until v; unfold op_strength_reduction;
  case (op_strength_reduction_match op args vl); simpl; intros.
 cast8signed *)  InvApproxRegs; SimplVM; inv H0. apply make_cast8signed_correct; auto.
 cast8signed *)  InvApproxRegs; SimplVM; inv H0. apply make_cast16signed_correct; auto.
 add *)  InvApproxRegs; SimplVM; inv H0.
  change (let (op', args') := make_addimm n1 r2 in
          exists w : val,
          eval_operation ge (Vptr sp Ptrofs.zero) op' rs ## args' m = Some w /\
          Val.lessdef (Val.add (Vint n1) rs#r2) w).
  rewrite Val.add_commut. apply make_addimm_correct.
  InvApproxRegs; SimplVM; inv H0. apply make_addimm_correct.
  InvApproxRegs; SimplVM; inv H0. econstructor; split; eauto. apply Val.add_lessdef; auto.
  InvApproxRegs; SimplVM; inv H0. econstructor; split; eauto. rewrite Val.add_commut. apply Val.add_lessdef; auto.
 sub *)  InvApproxRegs; SimplVM; inv H0. fold (Val.sub (Vint n1) rs#r2). econstructor; split; eauto.
  InvApproxRegs; SimplVM; inv H0. rewrite Val.sub_add_opp. apply make_addimm_correct.
 mul *)  InvApproxRegs; SimplVM; inv H0. fold (Val.mul (Vint n1) rs#r2). rewrite Val.mul_commut. apply make_mulimm_correct; auto.
  InvApproxRegs; SimplVM; inv H0. apply make_mulimm_correct; auto.
 divs *)  assert (rs#r2 = Vint n2). clear H0. InvApproxRegs; SimplVM; auto.
  apply make_divimm_correct; auto.
 divu *)  assert (rs#r2 = Vint n2). clear H0. InvApproxRegs; SimplVM; auto.
  apply make_divuimm_correct; auto.
 and *)  InvApproxRegs; SimplVM; inv H0. fold (Val.and (Vint n1) rs#r2). rewrite Val.and_commut. apply make_andimm_correct; auto.
  InvApproxRegs; SimplVM; inv H0. apply make_andimm_correct. auto.
  inv H; inv H0. apply make_andimm_correct. auto.
 or *)  InvApproxRegs; SimplVM; inv H0. fold (Val.or (Vint n1) rs#r2). rewrite Val.or_commut. apply make_orimm_correct.
  InvApproxRegs; SimplVM; inv H0. apply make_orimm_correct.
 xor *)  InvApproxRegs; SimplVM; inv H0. fold (Val.xor (Vint n1) rs#r2). rewrite Val.xor_commut. apply make_xorimm_correct.
  InvApproxRegs; SimplVM; inv H0. apply make_xorimm_correct.
 shl *)  InvApproxRegs; SimplVM; inv H0. apply make_shlimm_correct; auto.
 shr *)  InvApproxRegs; SimplVM; inv H0. apply make_shrimm_correct; auto.
 shru *)  InvApproxRegs; SimplVM; inv H0. apply make_shruimm_correct; auto.
 cmp *)  inv H0. apply make_cmp_correct; auto.
 mulf *)  InvApproxRegs; SimplVM; inv H0. rewrite <- H2. apply make_mulfimm_correct; auto.
  InvApproxRegs; SimplVM; inv H0. fold (Val.mulf (Vfloat n1) rs#r2).
  rewrite <- H2. apply make_mulfimm_correct_2; auto.
 mulfs *)  InvApproxRegs; SimplVM; inv H0. rewrite <- H2. apply make_mulfsimm_correct; auto.
  InvApproxRegs; SimplVM; inv H0. fold (Val.mulfs (Vsingle n1) rs#r2).
  rewrite <- H2. apply make_mulfsimm_correct_2; auto.
 default *)  exists v; auto.
Qed.

Remark shift_symbol_address:
  forall id ofs delta,
  Genv.symbol_address ge id (Ptrofs.add ofs (Ptrofs.of_int delta)) = Val.add (Genv.symbol_address ge id ofs) (Vint delta).
Proof.
  intros. unfold Genv.symbol_address. destruct (Genv.find_symbol ge id); auto.
Qed.

Lemma addr_strength_reduction_correct:
  forall addr args vl res,
  vl = map (fun r => AE.get r ae) args ->
  eval_addressing ge (Vptr sp Ptrofs.zero) addr rs##args = Some res ->
  let (addr', args') := addr_strength_reduction addr args vl in
  exists res', eval_addressing ge (Vptr sp Ptrofs.zero) addr' rs##args' = Some res' /\ Val.lessdef res res'.
Proof.
  intros until res. unfold addr_strength_reduction.
  destruct (addr_strength_reduction_match addr args vl); simpl;
  intros VL EA; InvApproxRegs; SimplVM; try (inv EA).
- rewrite shift_symbol_address. econstructor; split; eauto. apply Val.add_lessdef; auto.
- econstructor; split; eauto.
  change (Val.lessdef (Val.add (Vint n1) rs#r2) (Genv.symbol_address ge symb (Ptrofs.add (Ptrofs.of_int n1) n2))).
  rewrite Ptrofs.add_commut. rewrite shift_symbol_address. rewrite Val.add_commut.
  apply Val.add_lessdef; auto.
- rewrite Ptrofs.add_zero_l.
  change (Vptr sp (Ptrofs.add n1 (Ptrofs.of_int n2))) with (Val.add (Vptr sp n1) (Vint n2)).
  econstructor; split; eauto. apply Val.add_lessdef; auto.
- econstructor; split; eauto.
  change (Val.lessdef (Val.add (Vint n1) rs#r2) (Vptr sp (Ptrofs.add Ptrofs.zero (Ptrofs.add (Ptrofs.of_int n1) n2)))).
  rewrite Ptrofs.add_zero_l. rewrite Ptrofs.add_commut.
  change (Val.lessdef (Val.add (Vint n1) rs#r2) (Val.add (Vptr sp n2) (Vint n1))).
  rewrite Val.add_commut. apply Val.add_lessdef; auto.
- econstructor; split; eauto. apply Val.add_lessdef; auto.
- rewrite Val.add_commut. econstructor; split; eauto. apply Val.add_lessdef; auto.
- econstructor; split; eauto.
  change (Val.lessdef (Val.add (Vint n1) rs#r2) (Val.add rs#r2 (Vint n1))).
  rewrite Val.add_commut. auto.
- econstructor; split; eauto.
- rewrite shift_symbol_address. econstructor; split; eauto.
- rewrite shift_symbol_address. econstructor; split; eauto. apply Val.add_lessdef; auto.
- rewrite Ptrofs.add_zero_l.
  change (Vptr sp (Ptrofs.add n1 (Ptrofs.of_int n))) with (Val.add (Vptr sp n1) (Vint n)).
  econstructor; split; eauto. apply Val.add_lessdef; auto.
- exists res; auto.
Qed.

End STRENGTH_REDUCTION.