Module Selectionproof


Correctness of instruction selection

Require Import Coqlib Maps.
Require Import AST Linking Errors Integers Values Memory Events Globalenvs Smallstep.
Require Import Switch Cminor Op CminorSel.
Require Import SelectOp SelectDiv SplitLong SelectLong Selection.
Require Import SelectOpproof SelectDivproof SplitLongproof SelectLongproof.

Local Open Scope cminorsel_scope.
Local Open Scope error_monad_scope.

Relational specification of instruction selection


Definition match_fundef (cunit: Cminor.program) (f: Cminor.fundef) (tf: CminorSel.fundef) : Prop :=
  exists hf, helper_functions_declared cunit hf /\ sel_fundef (prog_defmap cunit) hf f = OK tf.

Definition match_prog (p: Cminor.program) (tp: CminorSel.program) :=
  match_program match_fundef eq p tp.

Processing of helper functions

Lemma record_globdefs_sound:
  forall dm id gd, (record_globdefs dm)!id = Some gd -> dm!id = Some gd.
Proof.
  intros.
  set (f := fun m id gd => if globdef_of_interest gd then PTree.set id gd m else m) in *.
  set (P := fun m m' => m'!id = Some gd -> m!id = Some gd).
  assert (X: P dm (PTree.fold f dm (PTree.empty _))).
  { apply PTree_Properties.fold_rec.
  - unfold P; intros. rewrite <- H0; auto.
  - red. rewrite ! PTree.gempty. auto.
  - unfold P; intros. rewrite PTree.gsspec. unfold f in H3.
    destruct (globdef_of_interest v).
    + rewrite PTree.gsspec in H3. destruct (peq id k); auto.
    + apply H2 in H3. destruct (peq id k). congruence. auto. }
  apply X. auto.
Qed.

Lemma lookup_helper_correct_1:
  forall globs name sg id,
  lookup_helper globs name sg = OK id ->
  globs!id = Some (Gfun (External (EF_runtime name sg))).
Proof.
  intros.
  set (P := fun (m: PTree.t globdef) res => res = Some id -> m!id = Some(Gfun(External (EF_runtime name sg)))).
  assert (P globs (PTree.fold (lookup_helper_aux name sg) globs None)).
  { apply PTree_Properties.fold_rec; red; intros.
  - rewrite <- H0. apply H1; auto.
  - discriminate.
  - assert (EITHER: k = id /\ v = Gfun (External (EF_runtime name sg))
                \/ a = Some id).
    { unfold lookup_helper_aux in H3. destruct v; auto. destruct f; auto. destruct e; auto.
      destruct (String.string_dec name name0); auto.
      destruct (signature_eq sg sg0); auto.
      inversion H3. left; split; auto. repeat f_equal; auto. }
    destruct EITHER as [[X Y] | X].
    subst k v. apply PTree.gss.
    apply H2 in X. rewrite PTree.gso by congruence. auto.
  }
  red in H0. unfold lookup_helper in H.
  destruct (PTree.fold (lookup_helper_aux name sg) globs None); inv H. auto.
Qed.

Lemma lookup_helper_correct:
  forall p name sg id,
  lookup_helper (record_globdefs (prog_defmap p)) name sg = OK id ->
  helper_declared p id name sg.
Proof.
  intros. apply lookup_helper_correct_1 in H. apply record_globdefs_sound in H. auto.
Qed.

Lemma get_helpers_correct:
  forall p hf,
  get_helpers (prog_defmap p) = OK hf -> helper_functions_declared p hf.
Proof.
  intros. monadInv H. red; simpl. auto 20 using lookup_helper_correct.
Qed.

Theorem transf_program_match:
  forall p tp, sel_program p = OK tp -> match_prog p tp.
Proof.
  intros. monadInv H.
  eapply match_transform_partial_program_contextual. eexact EQ0.
  intros. exists x; split; auto. apply get_helpers_correct; auto.
Qed.

Lemma helper_functions_declared_linkorder:
  forall (p p': Cminor.program) hf,
  helper_functions_declared p hf -> linkorder p p' -> helper_functions_declared p' hf.
Proof.
  intros.
  assert (X: forall id name sg, helper_declared p id name sg -> helper_declared p' id name sg).
  { unfold helper_declared; intros.
    destruct (prog_defmap_linkorder _ _ _ _ H0 H1) as (gd & P & Q).
    inv Q. inv H3. auto. }
  red in H. decompose [Logic.and] H; clear H. red; auto 20.
Qed.

Correctness of the instruction selection functions for expressions


Section PRESERVATION.

Variable prog: Cminor.program.
Variable tprog: CminorSel.program.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.
Hypothesis TRANSF: match_prog prog tprog.

Lemma symbols_preserved:
  forall (s: ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
Proof (Genv.find_symbol_match TRANSF).

Lemma senv_preserved:
  Senv.equiv (Genv.to_senv ge) (Genv.to_senv tge).
Proof (Genv.senv_match TRANSF).

Lemma function_ptr_translated:
  forall (b: block) (f: Cminor.fundef),
  Genv.find_funct_ptr ge b = Some f ->
  exists cu tf, Genv.find_funct_ptr tge b = Some tf /\ match_fundef cu f tf /\ linkorder cu prog.
Proof (Genv.find_funct_ptr_match TRANSF).

Lemma functions_translated:
  forall (v v': val) (f: Cminor.fundef),
  Genv.find_funct ge v = Some f ->
  Val.lessdef v v' ->
  exists cu tf, Genv.find_funct tge v' = Some tf /\ match_fundef cu f tf /\ linkorder cu prog.
Proof.
  intros. inv H0.
  eapply Genv.find_funct_match; eauto.
  discriminate.
Qed.

Lemma sig_function_translated:
  forall cu f tf, match_fundef cu f tf -> funsig tf = Cminor.funsig f.
Proof.
  intros. destruct H as (hf & P & Q). destruct f; monadInv Q; auto. monadInv EQ; auto.
Qed.

Lemma stackspace_function_translated:
  forall dm hf f tf, sel_function dm hf f = OK tf -> fn_stackspace tf = Cminor.fn_stackspace f.
Proof.
  intros. monadInv H. auto.
Qed.

Lemma helper_functions_preserved:
  forall hf, helper_functions_declared prog hf -> helper_functions_declared tprog hf.
Proof.
  assert (X: forall id name sg, helper_declared prog id name sg -> helper_declared tprog id name sg).
  { unfold helper_declared; intros.
    generalize (match_program_defmap _ _ _ _ _ TRANSF id).
    unfold Cminor.fundef; rewrite H; intros R; inv R. inv H2.
    destruct H4 as (cu & A & B). monadInv B. auto. }
  unfold helper_functions_declared; intros. decompose [Logic.and] H; clear H. auto 20.
Qed.

Section CMCONSTR.

Variable cunit: Cminor.program.
Variable hf: helper_functions.
Hypothesis LINK: linkorder cunit prog.
Hypothesis HF: helper_functions_declared cunit hf.

Let HF': helper_functions_declared tprog hf.
Proof.
  apply helper_functions_preserved. eapply helper_functions_declared_linkorder; eauto.
Qed.

Variable sp: val.
Variable e: env.
Variable m: mem.

Lemma eval_condexpr_of_expr:
  forall a le v b,
  eval_expr tge sp e m le a v ->
  Val.bool_of_val v b ->
  eval_condexpr tge sp e m le (condexpr_of_expr a) b.
Proof.
  intros until a. functional induction (condexpr_of_expr a); intros.
 compare *)  inv H. econstructor; eauto.
  simpl in H6. inv H6. apply Val.bool_of_val_of_optbool. auto.
 condition *)  inv H. econstructor; eauto. destruct va; eauto.
 let *)  inv H. econstructor; eauto.
 default *)  econstructor. constructor. eauto. constructor.
  simpl. inv H0. auto.
Qed.

Lemma eval_load:
  forall le a v chunk v',
  eval_expr tge sp e m le a v ->
  Mem.loadv chunk m v = Some v' ->
  eval_expr tge sp e m le (load chunk a) v'.
Proof.
  intros. generalize H0; destruct v; simpl; intro; try discriminate.
  unfold load.
  generalize (eval_addressing _ _ _ _ _ chunk _ _ _ _ H (refl_equal _)).
  destruct (addressing chunk a). intros [vl [EV EQ]].
  eapply eval_Eload; eauto.
Qed.

Lemma eval_store:
  forall chunk a1 a2 v1 v2 f k m',
  eval_expr tge sp e m nil a1 v1 ->
  eval_expr tge sp e m nil a2 v2 ->
  Mem.storev chunk m v1 v2 = Some m' ->
  step tge (State f (store chunk a1 a2) k sp e m)
        E0 (State f Sskip k sp e m').
Proof.
  intros. generalize H1; destruct v1; simpl; intro; try discriminate.
  unfold store.
  generalize (eval_addressing _ _ _ _ _ chunk _ _ _ _ H (refl_equal _)).
  destruct (addressing chunk a1). intros [vl [EV EQ]].
  eapply step_store; eauto.
Qed.

Correctness of instruction selection for operators

Lemma eval_sel_unop:
  forall le op a1 v1 v,
  eval_expr tge sp e m le a1 v1 ->
  eval_unop op v1 = Some v ->
  exists v', eval_expr tge sp e m le (sel_unop op a1) v' /\ Val.lessdef v v'.
Proof.
  destruct op; simpl; intros; FuncInv; try subst v.
  apply eval_cast8unsigned; auto.
  apply eval_cast8signed; auto.
  apply eval_cast16unsigned; auto.
  apply eval_cast16signed; auto.
  apply eval_negint; auto.
  apply eval_notint; auto.
  apply eval_negf; auto.
  apply eval_absf; auto.
  apply eval_negfs; auto.
  apply eval_absfs; auto.
  apply eval_singleoffloat; auto.
  apply eval_floatofsingle; auto.
  eapply eval_intoffloat; eauto.
  eapply eval_intuoffloat; eauto.
  eapply eval_floatofint; eauto.
  eapply eval_floatofintu; eauto.
  eapply eval_intofsingle; eauto.
  eapply eval_intuofsingle; eauto.
  eapply eval_singleofint; eauto.
  eapply eval_singleofintu; eauto.
  eapply eval_negl; eauto.
  eapply eval_notl; eauto.
  eapply eval_intoflong; eauto.
  eapply eval_longofint; eauto.
  eapply eval_longofintu; eauto.
  eapply eval_longoffloat; eauto.
  eapply eval_longuoffloat; eauto.
  eapply eval_floatoflong; eauto.
  eapply eval_floatoflongu; eauto.
  eapply eval_longofsingle; eauto.
  eapply eval_longuofsingle; eauto.
  eapply eval_singleoflong; eauto.
  eapply eval_singleoflongu; eauto.
Qed.

Lemma eval_sel_binop:
  forall le op a1 a2 v1 v2 v,
  eval_expr tge sp e m le a1 v1 ->
  eval_expr tge sp e m le a2 v2 ->
  eval_binop op v1 v2 m = Some v ->
  exists v', eval_expr tge sp e m le (sel_binop op a1 a2) v' /\ Val.lessdef v v'.
Proof.
  destruct op; simpl; intros; FuncInv; try subst v.
  apply eval_add; auto.
  apply eval_sub; auto.
  apply eval_mul; auto.
  eapply eval_divs; eauto.
  eapply eval_divu; eauto.
  eapply eval_mods; eauto.
  eapply eval_modu; eauto.
  apply eval_and; auto.
  apply eval_or; auto.
  apply eval_xor; auto.
  apply eval_shl; auto.
  apply eval_shr; auto.
  apply eval_shru; auto.
  apply eval_addf; auto.
  apply eval_subf; auto.
  apply eval_mulf; auto.
  apply eval_divf; auto.
  apply eval_addfs; auto.
  apply eval_subfs; auto.
  apply eval_mulfs; auto.
  apply eval_divfs; auto.
  eapply eval_addl; eauto.
  eapply eval_subl; eauto.
  eapply eval_mull; eauto.
  eapply eval_divls; eauto.
  eapply eval_divlu; eauto.
  eapply eval_modls; eauto.
  eapply eval_modlu; eauto.
  eapply eval_andl; eauto.
  eapply eval_orl; eauto.
  eapply eval_xorl; eauto.
  eapply eval_shll; eauto.
  eapply eval_shrl; eauto.
  eapply eval_shrlu; eauto.
  apply eval_comp; auto.
  apply eval_compu; auto.
  apply eval_compf; auto.
  apply eval_compfs; auto.
  exists v; split; auto. eapply eval_cmpl; eauto.
  exists v; split; auto. eapply eval_cmplu; eauto.
Qed.

End CMCONSTR.

Recognition of calls to built-in functions

Lemma expr_is_addrof_ident_correct:
  forall e id,
  expr_is_addrof_ident e = Some id ->
  e = Cminor.Econst (Cminor.Oaddrsymbol id Ptrofs.zero).
Proof.
  intros e id. unfold expr_is_addrof_ident.
  destruct e; try congruence.
  destruct c; try congruence.
  predSpec Ptrofs.eq Ptrofs.eq_spec i0 Ptrofs.zero; congruence.
Qed.

Lemma classify_call_correct:
  forall unit sp e m a v fd,
  linkorder unit prog ->
  Cminor.eval_expr ge sp e m a v ->
  Genv.find_funct ge v = Some fd ->
  match classify_call (prog_defmap unit) a with
  | Call_default => True
  | Call_imm id => exists b, Genv.find_symbol ge id = Some b /\ v = Vptr b Ptrofs.zero
  | Call_builtin ef => fd = External ef
  end.
Proof.
  unfold classify_call; intros.
  destruct (expr_is_addrof_ident a) as [id|] eqn:EA; auto.
  exploit expr_is_addrof_ident_correct; eauto. intros EQ; subst a.
  inv H0. inv H3. unfold Genv.symbol_address in *.
  destruct (Genv.find_symbol ge id) as [b|] eqn:FS; try discriminate.
  rewrite Genv.find_funct_find_funct_ptr in H1.
  assert (DFL: exists b1, Genv.find_symbol ge id = Some b1 /\ Vptr b Ptrofs.zero = Vptr b1 Ptrofs.zero) by (exists b; auto).
  unfold globdef; destruct (prog_defmap unit)!id as [[[f|ef] |gv] |] eqn:G; auto.
  destruct (ef_inline ef) eqn:INLINE; auto.
  destruct (prog_defmap_linkorder _ _ _ _ H G) as (gd & P & Q).
  inv Q. inv H2.
- apply Genv.find_def_symbol in P. destruct P as (b' & X & Y). fold ge in X, Y.
  rewrite <- Genv.find_funct_ptr_iff in Y. congruence.
- simpl in INLINE. discriminate.
Qed.

Translation of switch statements

Inductive Rint: Z -> val -> Prop :=
  | Rint_intro: forall n, Rint (Int.unsigned n) (Vint n).

Inductive Rlong: Z -> val -> Prop :=
  | Rlong_intro: forall n, Rlong (Int64.unsigned n) (Vlong n).

Section SEL_SWITCH.

Variable make_cmp_eq: expr -> Z -> expr.
Variable make_cmp_ltu: expr -> Z -> expr.
Variable make_sub: expr -> Z -> expr.
Variable make_to_int: expr -> expr.
Variable modulus: Z.
Variable R: Z -> val -> Prop.

Hypothesis eval_make_cmp_eq: forall sp e m le a v i n,
  eval_expr tge sp e m le a v -> R i v -> 0 <= n < modulus ->
  eval_expr tge sp e m le (make_cmp_eq a n) (Val.of_bool (zeq i n)).
Hypothesis eval_make_cmp_ltu: forall sp e m le a v i n,
  eval_expr tge sp e m le a v -> R i v -> 0 <= n < modulus ->
  eval_expr tge sp e m le (make_cmp_ltu a n) (Val.of_bool (zlt i n)).
Hypothesis eval_make_sub: forall sp e m le a v i n,
  eval_expr tge sp e m le a v -> R i v -> 0 <= n < modulus ->
  exists v', eval_expr tge sp e m le (make_sub a n) v' /\ R ((i - n) mod modulus) v'.
Hypothesis eval_make_to_int: forall sp e m le a v i,
  eval_expr tge sp e m le a v -> R i v ->
  exists v', eval_expr tge sp e m le (make_to_int a) v' /\ Rint (i mod Int.modulus) v'.

Lemma sel_switch_correct_rec:
  forall sp e m varg i x,
  R i varg ->
  forall t arg le,
  wf_comptree modulus t ->
  nth_error le arg = Some varg ->
  comptree_match modulus i t = Some x ->
  eval_exitexpr tge sp e m le (sel_switch make_cmp_eq make_cmp_ltu make_sub make_to_int arg t) x.
Proof.
  intros until x; intros Ri. induction t; simpl; intros until le; intros WF ARG MATCH.
- (* base case *)
  inv MATCH. constructor.
- (* eq test *)
  inv WF.
  assert (eval_expr tge sp e m le (make_cmp_eq (Eletvar arg) key) (Val.of_bool (zeq i key))).
  { eapply eval_make_cmp_eq; eauto. constructor; auto. }
  eapply eval_XEcondition with (va := zeq i key).
  eapply eval_condexpr_of_expr; eauto. destruct (zeq i key); constructor; auto.
  destruct (zeq i key); simpl.
  + inv MATCH. constructor.
  + eapply IHt; eauto.
- (* lt test *)
  inv WF.
  assert (eval_expr tge sp e m le (make_cmp_ltu (Eletvar arg) key) (Val.of_bool (zlt i key))).
  { eapply eval_make_cmp_ltu; eauto. constructor; auto. }
  eapply eval_XEcondition with (va := zlt i key).
  eapply eval_condexpr_of_expr; eauto. destruct (zlt i key); constructor; auto.
  destruct (zlt i key); simpl.
  + eapply IHt1; eauto.
  + eapply IHt2; eauto.
- (* jump table *)
  inv WF.
  exploit (eval_make_sub sp e m le). eapply eval_Eletvar. eauto. eauto.
  instantiate (1 := ofs). auto.
  intros (v' & A & B).
  set (i' := (i - ofs) mod modulus) in *.
  assert (eval_expr tge sp e m (v' :: le) (make_cmp_ltu (Eletvar O) sz) (Val.of_bool (zlt i' sz))).
  { eapply eval_make_cmp_ltu; eauto. constructor; auto. }
  econstructor. eauto.
  eapply eval_XEcondition with (va := zlt i' sz).
  eapply eval_condexpr_of_expr; eauto. destruct (zlt i' sz); constructor; auto.
  destruct (zlt i' sz); simpl.
  + exploit (eval_make_to_int sp e m (v' :: le) (Eletvar O)).
    constructor. simpl; eauto. eauto. intros (v'' & C & D). inv D.
    econstructor; eauto. congruence.
  + eapply IHt; eauto.
Qed.

Lemma sel_switch_correct:
  forall dfl cases arg sp e m varg i t le,
  validate_switch modulus dfl cases t = true ->
  eval_expr tge sp e m le arg varg ->
  R i varg ->
  0 <= i < modulus ->
  eval_exitexpr tge sp e m le
     (XElet arg (sel_switch make_cmp_eq make_cmp_ltu make_sub make_to_int O t))
     (switch_target i dfl cases).
Proof.
  intros. exploit validate_switch_correct; eauto. omega. intros [A B].
  econstructor. eauto. eapply sel_switch_correct_rec; eauto.
Qed.

End SEL_SWITCH.

Lemma sel_switch_int_correct:
  forall dfl cases arg sp e m i t le,
  validate_switch Int.modulus dfl cases t = true ->
  eval_expr tge sp e m le arg (Vint i) ->
  eval_exitexpr tge sp e m le (XElet arg (sel_switch_int O t)) (switch_target (Int.unsigned i) dfl cases).
Proof.
  assert (INTCONST: forall n sp e m le,
            eval_expr tge sp e m le (Eop (Ointconst n) Enil) (Vint n)).
  { intros. econstructor. constructor. auto. }
  intros. eapply sel_switch_correct with (R := Rint); eauto.
- intros until n; intros EVAL R RANGE.
  exploit eval_comp. eexact EVAL. apply (INTCONST (Int.repr n)).
  instantiate (1 := Ceq). intros (vb & A & B).
  inv R. unfold Val.cmp in B. simpl in B. revert B.
  predSpec Int.eq Int.eq_spec n0 (Int.repr n); intros B; inv B.
  rewrite Int.unsigned_repr. unfold proj_sumbool; rewrite zeq_true; auto.
  unfold Int.max_unsigned; omega.
  unfold proj_sumbool; rewrite zeq_false; auto.
  red; intros; elim H1. rewrite <- (Int.repr_unsigned n0). congruence.
- intros until n; intros EVAL R RANGE.
  exploit eval_compu. eexact EVAL. apply (INTCONST (Int.repr n)).
  instantiate (1 := Clt). intros (vb & A & B).
  inv R. unfold Val.cmpu in B. simpl in B.
  unfold Int.ltu in B. rewrite Int.unsigned_repr in B.
  destruct (zlt (Int.unsigned n0) n); inv B; auto.
  unfold Int.max_unsigned; omega.
- intros until n; intros EVAL R RANGE.
  exploit eval_sub. eexact EVAL. apply (INTCONST (Int.repr n)). intros (vb & A & B).
  inv R. simpl in B. inv B. econstructor; split; eauto.
  replace ((Int.unsigned n0 - n) mod Int.modulus)
     with (Int.unsigned (Int.sub n0 (Int.repr n))).
  constructor.
  unfold Int.sub. rewrite Int.unsigned_repr_eq. f_equal. f_equal.
  apply Int.unsigned_repr. unfold Int.max_unsigned; omega.
- intros until i0; intros EVAL R. exists v; split; auto.
  inv R. rewrite Zmod_small by (apply Int.unsigned_range). constructor.
- constructor.
- apply Int.unsigned_range.
Qed.

Lemma sel_switch_long_correct:
  forall dfl cases arg sp e m i t le,
  validate_switch Int64.modulus dfl cases t = true ->
  eval_expr tge sp e m le arg (Vlong i) ->
  eval_exitexpr tge sp e m le (XElet arg (sel_switch_long O t)) (switch_target (Int64.unsigned i) dfl cases).
Proof.
  intros. eapply sel_switch_correct with (R := Rlong); eauto.
- intros until n; intros EVAL R RANGE.
  eapply eval_cmpl. eexact EVAL. apply eval_longconst with (n := Int64.repr n).
  inv R. unfold Val.cmpl. simpl. f_equal; f_equal. unfold Int64.eq.
  rewrite Int64.unsigned_repr. destruct (zeq (Int64.unsigned n0) n); auto.
  unfold Int64.max_unsigned; omega.
- intros until n; intros EVAL R RANGE.
  eapply eval_cmplu; auto. eexact EVAL. apply eval_longconst with (n := Int64.repr n).
  inv R. unfold Val.cmplu. simpl. f_equal; f_equal. unfold Int64.ltu.
  rewrite Int64.unsigned_repr. destruct (zlt (Int64.unsigned n0) n); auto.
  unfold Int64.max_unsigned; omega.
- intros until n; intros EVAL R RANGE.
  exploit eval_subl; auto. eexact EVAL. apply eval_longconst with (n := Int64.repr n).
  intros (vb & A & B).
  inv R. simpl in B. inv B. econstructor; split; eauto.
  replace ((Int64.unsigned n0 - n) mod Int64.modulus)
     with (Int64.unsigned (Int64.sub n0 (Int64.repr n))).
  constructor.
  unfold Int64.sub. rewrite Int64.unsigned_repr_eq. f_equal. f_equal.
  apply Int64.unsigned_repr. unfold Int64.max_unsigned; omega.
- intros until i0; intros EVAL R.
  exploit eval_lowlong. eexact EVAL. intros (vb & A & B).
  inv R. simpl in B. inv B. econstructor; split; eauto.
  replace (Int64.unsigned n mod Int.modulus) with (Int.unsigned (Int64.loword n)).
  constructor.
  unfold Int64.loword. apply Int.unsigned_repr_eq.
- constructor.
- apply Int64.unsigned_range.
Qed.

Compatibility of evaluation functions with the "less defined than" relation.

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

Lemma eval_unop_lessdef:
  forall op v1 v1' v,
  eval_unop op v1 = Some v -> Val.lessdef v1 v1' ->
  exists v', eval_unop op v1' = Some v' /\ Val.lessdef v v'.
Proof.
  intros until v; intros EV LD. inv LD.
  exists v; auto.
  destruct op; simpl in *; inv EV; TrivialExists.
Qed.

Lemma eval_binop_lessdef:
  forall op v1 v1' v2 v2' v m m',
  eval_binop op v1 v2 m = Some v ->
  Val.lessdef v1 v1' -> Val.lessdef v2 v2' -> Mem.extends m m' ->
  exists v', eval_binop op v1' v2' m' = Some v' /\ Val.lessdef v v'.
Proof.
  intros until m'; intros EV LD1 LD2 ME.
  assert (exists v', eval_binop op v1' v2' m = Some v' /\ Val.lessdef v v').
  { inv LD1. inv LD2. exists v; auto.
    destruct op; destruct v1'; simpl in *; inv EV; TrivialExists.
    destruct op; simpl in *; inv EV; TrivialExists. }
  assert (CMPU: forall c,
    eval_binop (Ocmpu c) v1 v2 m = Some v ->
    exists v' : val, eval_binop (Ocmpu c) v1' v2' m' = Some v' /\ Val.lessdef v v').
  { intros c A. simpl in *. inv A. econstructor; split. eauto.
    apply Val.of_optbool_lessdef.
    intros. apply Val.cmpu_bool_lessdef with (Mem.valid_pointer m) v1 v2; auto.
    intros; eapply Mem.valid_pointer_extends; eauto. }
  assert (CMPLU: forall c,
    eval_binop (Ocmplu c) v1 v2 m = Some v ->
    exists v' : val, eval_binop (Ocmplu c) v1' v2' m' = Some v' /\ Val.lessdef v v').
  { intros c A. simpl in *. unfold Val.cmplu in *.
    destruct (Val.cmplu_bool (Mem.valid_pointer m) c v1 v2) as [b|] eqn:C; simpl in A; inv A.
    eapply Val.cmplu_bool_lessdef with (valid_ptr' := (Mem.valid_pointer m')) in C;
    eauto using Mem.valid_pointer_extends.
    rewrite C. exists (Val.of_bool b); auto. }
  destruct op; auto.
Qed.

Semantic preservation for instruction selection.


Relationship between the local environments.

Definition env_lessdef (e1 e2: env) : Prop :=
  forall id v1, e1!id = Some v1 -> exists v2, e2!id = Some v2 /\ Val.lessdef v1 v2.

Lemma set_var_lessdef:
  forall e1 e2 id v1 v2,
  env_lessdef e1 e2 -> Val.lessdef v1 v2 ->
  env_lessdef (PTree.set id v1 e1) (PTree.set id v2 e2).
Proof.
  intros; red; intros. rewrite PTree.gsspec in *. destruct (peq id0 id).
  exists v2; split; congruence.
  auto.
Qed.

Lemma set_optvar_lessdef:
  forall e1 e2 optid v1 v2,
  env_lessdef e1 e2 -> Val.lessdef v1 v2 ->
  env_lessdef (set_optvar optid v1 e1) (set_optvar optid v2 e2).
Proof.
  unfold set_optvar; intros. destruct optid; auto. apply set_var_lessdef; auto.
Qed.

Lemma set_params_lessdef:
  forall il vl1 vl2,
  Val.lessdef_list vl1 vl2 ->
  env_lessdef (set_params vl1 il) (set_params vl2 il).
Proof.
  induction il; simpl; intros.
  red; intros. rewrite PTree.gempty in H0; congruence.
  inv H; apply set_var_lessdef; auto.
Qed.

Lemma set_locals_lessdef:
  forall e1 e2, env_lessdef e1 e2 ->
  forall il, env_lessdef (set_locals il e1) (set_locals il e2).
Proof.
  induction il; simpl. auto. apply set_var_lessdef; auto.
Qed.

Semantic preservation for expressions.

Section EXPRESSIONS.

Variable cunit: Cminor.program.
Variable hf: helper_functions.
Hypothesis LINK: linkorder cunit prog.
Hypothesis HF: helper_functions_declared cunit hf.

Lemma sel_expr_correct:
  forall sp e m a v,
  Cminor.eval_expr ge sp e m a v ->
  forall e' le m',
  env_lessdef e e' -> Mem.extends m m' ->
  exists v', eval_expr tge sp e' m' le (sel_expr a) v' /\ Val.lessdef v v'.
Proof.
  induction 1; intros; simpl.
 Evar *)  exploit H0; eauto. intros [v' [A B]]. exists v'; split; auto. constructor; auto.
 Econst *)  destruct cst; simpl in *; inv H.
  exists (Vint i); split; auto. econstructor. constructor. auto.
  exists (Vfloat f); split; auto. econstructor. constructor. auto.
  exists (Vsingle f); split; auto. econstructor. constructor. auto.
  exists (Vlong i); split; auto. apply eval_longconst.
  unfold Genv.symbol_address; rewrite <- symbols_preserved; fold (Genv.symbol_address tge i i0). apply eval_addrsymbol.
  apply eval_addrstack.
 Eunop *)  exploit IHeval_expr; eauto. intros [v1' [A B]].
  exploit eval_unop_lessdef; eauto. intros [v' [C D]].
  exploit eval_sel_unop; eauto. intros [v'' [E F]].
  exists v''; split; eauto. eapply Val.lessdef_trans; eauto.
 Ebinop *)  exploit IHeval_expr1; eauto. intros [v1' [A B]].
  exploit IHeval_expr2; eauto. intros [v2' [C D]].
  exploit eval_binop_lessdef; eauto. intros [v' [E F]].
  assert (G: exists v'', eval_expr tge sp e' m' le (sel_binop op (sel_expr a1) (sel_expr a2)) v'' /\ Val.lessdef v' v'')
  by (eapply eval_sel_binop; eauto).
  destruct G as [v'' [P Q]].
  exists v''; split; eauto. eapply Val.lessdef_trans; eauto.
 Eload *)  exploit IHeval_expr; eauto. intros [vaddr' [A B]].
  exploit Mem.loadv_extends; eauto. intros [v' [C D]].
  exists v'; split; auto. eapply eval_load; eauto.
Qed.

Lemma sel_exprlist_correct:
  forall sp e m a v,
  Cminor.eval_exprlist ge sp e m a v ->
  forall e' le m',
  env_lessdef e e' -> Mem.extends m m' ->
  exists v', eval_exprlist tge sp e' m' le (sel_exprlist a) v' /\ Val.lessdef_list v v'.
Proof.
  induction 1; intros; simpl.
  exists (@nil val); split; auto. constructor.
  exploit sel_expr_correct; eauto. intros [v1' [A B]].
  exploit IHeval_exprlist; eauto. intros [vl' [C D]].
  exists (v1' :: vl'); split; auto. constructor; eauto.
Qed.

Lemma sel_builtin_arg_correct:
  forall sp e e' m m' a v c,
  env_lessdef e e' -> Mem.extends m m' ->
  Cminor.eval_expr ge sp e m a v ->
  exists v',
     CminorSel.eval_builtin_arg tge sp e' m' (sel_builtin_arg a c) v'
  /\ Val.lessdef v v'.
Proof.
  intros. unfold sel_builtin_arg.
  exploit sel_expr_correct; eauto. intros (v1 & A & B).
  exists v1; split; auto.
  destruct (builtin_arg_ok (builtin_arg (sel_expr a)) c).
  apply eval_builtin_arg; eauto.
  constructor; auto.
Qed.

Lemma sel_builtin_args_correct:
  forall sp e e' m m',
  env_lessdef e e' -> Mem.extends m m' ->
  forall al vl,
  Cminor.eval_exprlist ge sp e m al vl ->
  forall cl,
  exists vl',
     list_forall2 (CminorSel.eval_builtin_arg tge sp e' m')
                  (sel_builtin_args al cl)
                  vl'
  /\ Val.lessdef_list vl vl'.
Proof.
  induction 3; intros; simpl.
- exists (@nil val); split; constructor.
- exploit sel_builtin_arg_correct; eauto. intros (v1' & A & B).
  edestruct IHeval_exprlist as (vl' & C & D).
  exists (v1' :: vl'); split; auto. constructor; eauto.
Qed.

Lemma sel_builtin_res_correct:
  forall oid v e v' e',
  env_lessdef e e' -> Val.lessdef v v' ->
  env_lessdef (set_optvar oid v e) (set_builtin_res (sel_builtin_res oid) v' e').
Proof.
  intros. destruct oid; simpl; auto. apply set_var_lessdef; auto.
Qed.

End EXPRESSIONS.

Semantic preservation for functions and statements.

Inductive match_cont: Cminor.program -> helper_functions -> Cminor.cont -> CminorSel.cont -> Prop :=
  | match_cont_stop: forall cunit hf,
      match_cont cunit hf Cminor.Kstop Kstop
  | match_cont_seq: forall cunit hf s s' k k',
      sel_stmt (prog_defmap cunit) s = OK s' ->
      match_cont cunit hf k k' ->
      match_cont cunit hf (Cminor.Kseq s k) (Kseq s' k')
  | match_cont_block: forall cunit hf k k',
      match_cont cunit hf k k' ->
      match_cont cunit hf (Cminor.Kblock k) (Kblock k')
  | match_cont_call: forall cunit' hf' cunit hf id f sp e k f' e' k',
      linkorder cunit prog ->
      helper_functions_declared cunit hf ->
      sel_function (prog_defmap cunit) hf f = OK f' ->
      match_cont cunit hf k k' -> env_lessdef e e' ->
      match_cont cunit' hf' (Cminor.Kcall id f sp e k) (Kcall id f' sp e' k').

Definition match_call_cont (k: Cminor.cont) (k': CminorSel.cont) : Prop :=
  forall cunit hf, match_cont cunit hf k k'.

Inductive match_states: Cminor.state -> CminorSel.state -> Prop :=
  | match_state: forall cunit hf f f' s k s' k' sp e m e' m'
        (LINK: linkorder cunit prog)
        (HF: helper_functions_declared cunit hf)
        (TF: sel_function (prog_defmap cunit) hf f = OK f')
        (TS: sel_stmt (prog_defmap cunit) s = OK s')
        (MC: match_cont cunit hf k k')
        (LD: env_lessdef e e')
        (ME: Mem.extends m m'),
      match_states
        (Cminor.State f s k sp e m)
        (State f' s' k' sp e' m')
  | match_callstate: forall cunit f f' args args' k k' m m'
        (LINK: linkorder cunit prog)
        (TF: match_fundef cunit f f')
        (MC: match_call_cont k k')
        (LD: Val.lessdef_list args args')
        (ME: Mem.extends m m'),
      match_states
        (Cminor.Callstate f args k m)
        (Callstate f' args' k' m')
  | match_returnstate: forall v v' k k' m m'
        (MC: match_call_cont k k')
        (LD: Val.lessdef v v')
        (ME: Mem.extends m m'),
      match_states
        (Cminor.Returnstate v k m)
        (Returnstate v' k' m')
  | match_builtin_1: forall cunit hf ef args args' optid f sp e k m al f' e' k' m'
        (LINK: linkorder cunit prog)
        (HF: helper_functions_declared cunit hf)
        (TF: sel_function (prog_defmap cunit) hf f = OK f')
        (MC: match_cont cunit hf k k')
        (LDA: Val.lessdef_list args args')
        (LDE: env_lessdef e e')
        (ME: Mem.extends m m')
        (EA: list_forall2 (CminorSel.eval_builtin_arg tge sp e' m') al args'),
      match_states
        (Cminor.Callstate (External ef) args (Cminor.Kcall optid f sp e k) m)
        (State f' (Sbuiltin (sel_builtin_res optid) ef al) k' sp e' m')
  | match_builtin_2: forall cunit hf v v' optid f sp e k m f' e' m' k'
        (LINK: linkorder cunit prog)
        (HF: helper_functions_declared cunit hf)
        (TF: sel_function (prog_defmap cunit) hf f = OK f')
        (MC: match_cont cunit hf k k')
        (LDV: Val.lessdef v v')
        (LDE: env_lessdef e e')
        (ME: Mem.extends m m'),
      match_states
        (Cminor.Returnstate v (Cminor.Kcall optid f sp e k) m)
        (State f' Sskip k' sp (set_builtin_res (sel_builtin_res optid) v' e') m').

Remark call_cont_commut:
  forall cunit hf k k', match_cont cunit hf k k' -> match_call_cont (Cminor.call_cont k) (call_cont k').
Proof.
  induction 1; simpl; auto; red; intros.
- constructor.
- eapply match_cont_call with (hf := hf); eauto.
Qed.

Remark match_is_call_cont:
  forall cunit hf k k', match_cont cunit hf k k' -> Cminor.is_call_cont k -> match_call_cont k k'.
Proof.
  destruct 1; intros; try contradiction; red; intros.
- constructor.
- eapply match_cont_call with (hf := hf); eauto.
Qed.

Remark match_call_cont_cont:
  forall k k', match_call_cont k k' -> exists cunit hf, match_cont cunit hf k k'.
Proof.
  intros. simple refine (let cunit : Cminor.program := _ in _).
  econstructor. apply nil. apply nil. apply xH.
  simple refine (let hf : helper_functions := _ in _).
  econstructor; apply xH.
  exists cunit, hf; auto.
Qed.

Remark find_label_commut:
  forall cunit hf lbl s k s' k',
  match_cont cunit hf k k' ->
  sel_stmt (prog_defmap cunit) s = OK s' ->
  match Cminor.find_label lbl s k, find_label lbl s' k' with
  | None, None => True
  | Some(s1, k1), Some(s1', k1') => sel_stmt (prog_defmap cunit) s1 = OK s1' /\ match_cont cunit hf k1 k1'
  | _, _ => False
  end.
Proof.
  induction s; intros until k'; simpl; intros MC SE; try (monadInv SE); simpl; auto.
 store *)  unfold store. destruct (addressing m (sel_expr e)); simpl; auto.
 call *)  destruct (classify_call (prog_defmap cunit) e); simpl; auto.
 tailcall *)  destruct (classify_call (prog_defmap cunit) e); simpl; auto.
 seq *)  exploit (IHs1 (Cminor.Kseq s2 k)). constructor; eauto. eauto.
  destruct (Cminor.find_label lbl s1 (Cminor.Kseq s2 k)) as [[sx kx] | ];
  destruct (find_label lbl x (Kseq x0 k')) as [[sy ky] | ];
  intuition. apply IHs2; auto.
 ifthenelse *)  exploit (IHs1 k); eauto.
  destruct (Cminor.find_label lbl s1 k) as [[sx kx] | ];
  destruct (find_label lbl x k') as [[sy ky] | ];
  intuition. apply IHs2; auto.
 loop *)  apply IHs. constructor; auto. simpl; rewrite EQ; auto. auto.
 block *)  apply IHs. constructor; auto. auto.
 switch *)  destruct b.
  destruct (validate_switch Int64.modulus n l (compile_switch Int64.modulus n l)); inv SE.
  simpl; auto.
  destruct (validate_switch Int.modulus n l (compile_switch Int.modulus n l)); inv SE.
  simpl; auto.
 return *)  destruct o; inv SE; simpl; auto.
 label *)  destruct (ident_eq lbl l). auto. apply IHs; auto.
Qed.

Definition measure (s: Cminor.state) : nat :=
  match s with
  | Cminor.Callstate _ _ _ _ => 0%nat
  | Cminor.State _ _ _ _ _ _ => 1%nat
  | Cminor.Returnstate _ _ _ => 2%nat
  end.

Lemma sel_step_correct:
  forall S1 t S2, Cminor.step ge S1 t S2 ->
  forall T1, match_states S1 T1 ->
  (exists T2, step tge T1 t T2 /\ match_states S2 T2)
  \/ (measure S2 < measure S1 /\ t = E0 /\ match_states S2 T1)%nat.
Proof.
  induction 1; intros T1 ME; inv ME; try (monadInv TS).
- (* skip seq *)
  inv MC. left; econstructor; split. econstructor. econstructor; eauto.
- (* skip block *)
  inv MC. left; econstructor; split. econstructor. econstructor; eauto.
- (* skip call *)
  exploit Mem.free_parallel_extends; eauto. intros [m2' [A B]].
  left; econstructor; split.
  econstructor. inv MC; simpl in H; simpl; auto.
  eauto.
  erewrite stackspace_function_translated; eauto.
  econstructor; eauto. eapply match_is_call_cont; eauto.
- (* assign *)
  exploit sel_expr_correct; eauto. intros [v' [A B]].
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto. apply set_var_lessdef; auto.
- (* store *)
  exploit sel_expr_correct. eauto. eauto. eexact H. eauto. eauto. intros [vaddr' [A B]].
  exploit sel_expr_correct. eauto. eauto. eexact H0. eauto. eauto. intros [v' [C D]].
  exploit Mem.storev_extends; eauto. intros [m2' [P Q]].
  left; econstructor; split.
  eapply eval_store; eauto.
  econstructor; eauto.
- (* Scall *)
  exploit classify_call_correct; eauto.
  destruct (classify_call (prog_defmap cunit) a) as [ | id | ef].
+ (* indirect *)
  exploit sel_expr_correct; eauto. intros [vf' [A B]].
  exploit sel_exprlist_correct; eauto. intros [vargs' [C D]].
  exploit functions_translated; eauto. intros (cunit' & fd' & U & V & W).
  left; econstructor; split.
  econstructor; eauto. econstructor; eauto.
  eapply sig_function_translated; eauto.
  eapply match_callstate with (cunit := cunit'); eauto.
  red; intros. eapply match_cont_call with (cunit := cunit) (hf := hf); eauto.
+ (* direct *)
  intros [b [U V]].
  exploit sel_exprlist_correct; eauto. intros [vargs' [C D]].
  exploit functions_translated; eauto. intros (cunit' & fd' & X & Y & Z).
  left; econstructor; split.
  econstructor; eauto.
  subst vf. econstructor; eauto. rewrite symbols_preserved; eauto.
  eapply sig_function_translated; eauto.
  eapply match_callstate with (cunit := cunit'); eauto.
  red; intros; eapply match_cont_call with (cunit := cunit) (hf := hf); eauto.
+ (* turned into Sbuiltin *)
  intros EQ. subst fd.
  exploit sel_builtin_args_correct; eauto. intros [vargs' [C D]].
  right; split. simpl. omega. split. auto.
  econstructor; eauto.
- (* Stailcall *)
  exploit Mem.free_parallel_extends; eauto. intros [m2' [P Q]].
  erewrite <- stackspace_function_translated in P by eauto.
  exploit sel_expr_correct; eauto. intros [vf' [A B]].
  exploit sel_exprlist_correct; eauto. intros [vargs' [C D]].
  exploit functions_translated; eauto. intros (cunit' & fd' & E & F & G).
  left; econstructor; split.
  exploit classify_call_correct. eexact LINK. eauto. eauto.
  destruct (classify_call (prog_defmap cunit)) as [ | id | ef]; intros.
  econstructor; eauto. econstructor; eauto. eapply sig_function_translated; eauto.
  destruct H2 as [b [U V]]. subst vf. inv B.
  econstructor; eauto. econstructor; eauto. rewrite symbols_preserved; eauto. eapply sig_function_translated; eauto.
  econstructor; eauto. econstructor; eauto. eapply sig_function_translated; eauto.
  eapply match_callstate with (cunit := cunit'); eauto.
  eapply call_cont_commut; eauto.
- (* Sbuiltin *)
  exploit sel_builtin_args_correct; eauto. intros [vargs' [P Q]].
  exploit external_call_mem_extends; eauto.
  intros [vres' [m2 [A [B [C D]]]]].
  left; econstructor; split.
  econstructor. eauto. eapply external_call_symbols_preserved; eauto. apply senv_preserved.
  econstructor; eauto. apply sel_builtin_res_correct; auto.
- (* Seq *)
  left; econstructor; split.
  constructor.
  econstructor; eauto. constructor; auto.
- (* Sifthenelse *)
  exploit sel_expr_correct; eauto. intros [v' [A B]].
  assert (Val.bool_of_val v' b). inv B. auto. inv H0.
  left; exists (State f' (if b then x else x0) k' sp e' m'); split.
  econstructor; eauto. eapply eval_condexpr_of_expr; eauto.
  econstructor; eauto. destruct b; auto.
- (* Sloop *)
  left; econstructor; split. constructor. econstructor; eauto.
  constructor; auto. simpl; rewrite EQ; auto.
- (* Sblock *)
  left; econstructor; split. constructor. econstructor; eauto. constructor; auto.
- (* Sexit seq *)
  inv MC. left; econstructor; split. constructor. econstructor; eauto.
- (* Sexit0 block *)
  inv MC. left; econstructor; split. constructor. econstructor; eauto.
- (* SexitS block *)
  inv MC. left; econstructor; split. constructor. econstructor; eauto.
- (* Sswitch *)
  inv H0; simpl in TS.
+ set (ct := compile_switch Int.modulus default cases) in *.
  destruct (validate_switch Int.modulus default cases ct) eqn:VALID; inv TS.
  exploit sel_expr_correct; eauto. intros [v' [A B]]. inv B.
  left; econstructor; split.
  econstructor. eapply sel_switch_int_correct; eauto.
  econstructor; eauto.
+ set (ct := compile_switch Int64.modulus default cases) in *.
  destruct (validate_switch Int64.modulus default cases ct) eqn:VALID; inv TS.
  exploit sel_expr_correct; eauto. intros [v' [A B]]. inv B.
  left; econstructor; split.
  econstructor. eapply sel_switch_long_correct; eauto.
  econstructor; eauto.
- (* Sreturn None *)
  exploit Mem.free_parallel_extends; eauto. intros [m2' [P Q]].
  erewrite <- stackspace_function_translated in P by eauto.
  left; econstructor; split.
  econstructor. simpl; eauto.
  econstructor; eauto. eapply call_cont_commut; eauto.
- (* Sreturn Some *)
  exploit Mem.free_parallel_extends; eauto. intros [m2' [P Q]].
  erewrite <- stackspace_function_translated in P by eauto.
  exploit sel_expr_correct; eauto. intros [v' [A B]].
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto. eapply call_cont_commut; eauto.
- (* Slabel *)
  left; econstructor; split. constructor. econstructor; eauto.
- (* Sgoto *)
  assert (sel_stmt (prog_defmap cunit) (Cminor.fn_body f) = OK (fn_body f')).
  { monadInv TF; simpl; auto. }
  exploit (find_label_commut cunit hf lbl (Cminor.fn_body f) (Cminor.call_cont k)).
    eapply call_cont_commut; eauto. eauto.
  rewrite H.
  destruct (find_label lbl (fn_body f') (call_cont k'0))
  as [[s'' k'']|] eqn:?; intros; try contradiction.
  destruct H1.
  left; econstructor; split.
  econstructor; eauto.
  econstructor; eauto.
- (* internal function *)
  destruct TF as (hf & HF & TF). specialize (MC cunit hf).
  monadInv TF. generalize EQ; intros TF; monadInv TF.
  exploit Mem.alloc_extends. eauto. eauto. apply Zle_refl. apply Zle_refl.
  intros [m2' [A B]].
  left; econstructor; split.
  econstructor; simpl; eauto.
  econstructor; simpl; eauto. apply set_locals_lessdef. apply set_params_lessdef; auto.
- (* external call *)
  destruct TF as (hf & HF & TF).
  monadInv TF.
  exploit external_call_mem_extends; eauto.
  intros [vres' [m2 [A [B [C D]]]]].
  left; econstructor; split.
  econstructor. eapply external_call_symbols_preserved; eauto. apply senv_preserved.
  econstructor; eauto.
- (* external call turned into a Sbuiltin *)
  exploit external_call_mem_extends; eauto.
  intros [vres' [m2 [A [B [C D]]]]].
  left; econstructor; split.
  econstructor. eauto. eapply external_call_symbols_preserved; eauto. apply senv_preserved.
  econstructor; eauto.
- (* return *)
  apply match_call_cont_cont in MC. destruct MC as (cunit0 & hf0 & MC).
  inv MC.
  left; econstructor; split.
  econstructor.
  econstructor; eauto. destruct optid; simpl; auto. apply set_var_lessdef; auto.
- (* return of an external call turned into a Sbuiltin *)
  right; split. simpl; omega. split. auto. econstructor; eauto.
  apply sel_builtin_res_correct; auto.
Qed.

Lemma sel_initial_states:
  forall S, Cminor.initial_state prog S ->
  exists R, initial_state tprog R /\ match_states S R.
Proof.
  destruct 1.
  exploit function_ptr_translated; eauto. intros (cu & f' & A & B & C).
  econstructor; split.
  econstructor.
  eapply (Genv.init_mem_match TRANSF); eauto.
  rewrite (match_program_main TRANSF). fold tge. rewrite symbols_preserved. eauto.
  eexact A.
  rewrite <- H2. eapply sig_function_translated; eauto.
  econstructor; eauto. red; intros; constructor. apply Mem.extends_refl.
Qed.

Lemma sel_final_states:
  forall S R r,
  match_states S R -> Cminor.final_state S r -> final_state R r.
Proof.
  intros. inv H0. inv H.
  apply match_call_cont_cont in MC. destruct MC as (cunit0 & hf0 & MC).
  inv MC. inv LD. constructor.
Qed.

Theorem transf_program_correct:
  forward_simulation (Cminor.semantics prog) (CminorSel.semantics tprog).
Proof.
  apply forward_simulation_opt with (match_states := match_states) (measure := measure).
  apply senv_preserved.
  apply sel_initial_states; auto.
  apply sel_final_states; auto.
  apply sel_step_correct; auto.
Qed.

End PRESERVATION.

Commutation with linking


Instance TransfSelectionLink : TransfLink match_prog.
Proof.
  red; intros. destruct (link_linkorder _ _ _ H) as [LO1 LO2].
  eapply link_match_program; eauto.
  intros. elim H3; intros hf1 [A1 B1]. elim H4; intros hf2 [A2 B2].
Local Transparent Linker_fundef.
  simpl in *. destruct f1, f2; simpl in *; monadInv B1; monadInv B2; simpl.
- discriminate.
- destruct e; inv H2. econstructor; eauto.
- destruct e; inv H2. econstructor; eauto.
- destruct (external_function_eq e e0); inv H2. econstructor; eauto.
Qed.