Correctness proof for x86 generation: main proof.

Require Import Coqlib.
Require Import Errors.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Op.
Require Import Locations.
Require Import Mach.
Require Import Conventions.
Require Import Asm.
Require Import Asmgen.
Require Import Asmgenproof0.
Require Import Asmgenproof1.

Section PRESERVATION.

Variable prog: Mach.program.
Variable tprog: Asm.program.
Hypothesis TRANSF: transf_program prog = Errors.OK tprog.

Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.

Lemma symbols_preserved:
forall id, Genv.find_symbol tge id = Genv.find_symbol ge id.

Proof.

  intros. unfold ge, tge.
  apply Genv.find_symbol_transf_partial with transf_fundef.
  exact TRANSF.
Qed.

Lemma public_preserved:
forall id, Genv.public_symbol tge id = Genv.public_symbol ge id.

Proof.

  intros. unfold ge, tge.
  apply Genv.public_symbol_transf_partial with transf_fundef.
  exact TRANSF.
Qed.

Lemma functions_translated:
  forall b f,
  Genv.find_funct_ptr ge b = Some f ->
  exists tf, Genv.find_funct_ptr tge b = Some tf /\ transf_fundef f = Errors.OK tf.
Proof
  (Genv.find_funct_ptr_transf_partial transf_fundef _ TRANSF).

Lemma functions_transl:
  forall fb f tf,
  Genv.find_funct_ptr ge fb = Some (Internal f) ->
  transf_function f = OK tf ->
  Genv.find_funct_ptr tge fb = Some (Internal tf).

Proof.

intros. exploit functions_translated; eauto. intros [tf' [A B]].
monadInv B. rewrite H0 in EQ; inv EQ; auto.
Qed.

Lemma varinfo_preserved:
forall b, Genv.find_var_info tge b = Genv.find_var_info ge b.

Proof.

  intros. unfold ge, tge.
  apply Genv.find_var_info_transf_partial with transf_fundef.
  exact TRANSF.
Qed.

Properties of control flow

Lemma transf_function_no_overflow:
forall f tf,
transf_function f = OK tf -> list_length_z (fn_code tf) <= Int.max_unsigned.

Proof.

intros. monadInv H. destruct (zlt Int.max_unsigned (list_length_z (fn_code x))); monadInv EQ0.
omega.
Qed.

Lemma exec_straight_exec:
  forall fb f c ep tf tc c' rs m rs' m',
  transl_code_at_pc ge (rs PC) fb f c ep tf tc ->
  exec_straight tge tf tc rs m c' rs' m' ->
  plus step tge (State rs m) E0 (State rs' m').

Proof.

  intros. inv H.
  eapply exec_straight_steps_1; eauto.
  eapply transf_function_no_overflow; eauto.
  eapply functions_transl; eauto.
Qed.

Lemma exec_straight_at:
  forall fb f c ep tf tc c' ep' tc' rs m rs' m',
  transl_code_at_pc ge (rs PC) fb f c ep tf tc ->
  transl_code f c' ep' = OK tc' ->
  exec_straight tge tf tc rs m tc' rs' m' ->
  transl_code_at_pc ge (rs' PC) fb f c' ep' tf tc'.

Proof.

  intros. inv H.
  exploit exec_straight_steps_2; eauto.
  eapply transf_function_no_overflow; eauto.
  eapply functions_transl; eauto.
  intros [ofs' [PC' CT']].
  rewrite PC'. constructor; auto.
Qed.

The following lemmas show that the translation from Mach to Asm preserves labels, in the sense that the following diagram commutes:

                          translation
        Mach code ------------------------ Asm instr sequence
            |                                          |
            | Mach.find_label lbl       find_label lbl |
            |                                          |
            v                                          v
        Mach code tail ------------------- Asm instr seq tail
                          translation

The proof demands many boring lemmas showing that Asm constructor functions do not introduce new labels. In passing, we also prove a "is tail" property of the generated Asm code.

Section TRANSL_LABEL.

Remark mk_mov_label:
forall rd rs k c, mk_mov rd rs k = OK c -> tail_nolabel k c.

Proof.

unfold mk_mov; intros.
destruct rd; try discriminate; destruct rs; TailNoLabel.
Qed.

Hint Resolve mk_mov_label: labels.

Remark mk_shrximm_label:
forall n k c, mk_shrximm n k = OK c -> tail_nolabel k c.

Proof.

intros. monadInv H; TailNoLabel.
Qed.

Hint Resolve mk_shrximm_label: labels.

Remark mk_intconv_label:
  forall f r1 r2 k c, mk_intconv f r1 r2 k = OK c ->
  (forall r r', nolabel (f r r')) ->
  tail_nolabel k c.

Proof.

unfold mk_intconv; intros. TailNoLabel.
Qed.

Hint Resolve mk_intconv_label: labels.

Remark mk_smallstore_label:
  forall f addr r k c, mk_smallstore f addr r k = OK c ->
  (forall r addr, nolabel (f r addr)) ->
  tail_nolabel k c.

Proof.

unfold mk_smallstore; intros. TailNoLabel.
Qed.

Hint Resolve mk_smallstore_label: labels.

Remark loadind_label:
  forall base ofs ty dst k c,
  loadind base ofs ty dst k = OK c ->
  tail_nolabel k c.

Proof.

unfold loadind; intros. destruct ty; try discriminate; destruct (preg_of dst); TailNoLabel.
Qed.

Remark storeind_label:
  forall base ofs ty src k c,
  storeind src base ofs ty k = OK c ->
  tail_nolabel k c.

Proof.

unfold storeind; intros. destruct ty; try discriminate; destruct (preg_of src); TailNoLabel.
Qed.

Remark mk_setcc_base_label:
forall xc rd k,
tail_nolabel k (mk_setcc_base xc rd k).

Proof.

intros. destruct xc; simpl; destruct (ireg_eq rd EAX); TailNoLabel.
Qed.

Remark mk_setcc_label:
forall xc rd k,
tail_nolabel k (mk_setcc xc rd k).

Proof.

  intros. unfold mk_setcc. destruct (low_ireg rd).
  apply mk_setcc_base_label.
  eapply tail_nolabel_trans. apply mk_setcc_base_label. TailNoLabel.
Qed.

Remark mk_jcc_label:
forall xc lbl' k,
tail_nolabel k (mk_jcc xc lbl' k).

Proof.

intros. destruct xc; simpl; TailNoLabel.
Qed.

Remark transl_cond_label:
  forall cond args k c,
  transl_cond cond args k = OK c ->
  tail_nolabel k c.

Proof.

  unfold transl_cond; intros.
  destruct cond; TailNoLabel.
  destruct (Int.eq_dec i Int.zero); TailNoLabel.
  destruct c0; simpl; TailNoLabel.
  destruct c0; simpl; TailNoLabel.
  destruct c0; simpl; TailNoLabel.
  destruct c0; simpl; TailNoLabel.
Qed.

Remark transl_op_label:
  forall op args r k c,
  transl_op op args r k = OK c ->
  tail_nolabel k c.

Proof.

  unfold transl_op; intros. destruct op; TailNoLabel.
  destruct (Int.eq_dec i Int.zero); TailNoLabel.
  destruct (Float.eq_dec f Float.zero); TailNoLabel.
  destruct (Float32.eq_dec f Float32.zero); TailNoLabel.
  eapply tail_nolabel_trans. eapply transl_cond_label; eauto. eapply mk_setcc_label.
Qed.

Remark transl_load_label:
  forall chunk addr args dest k c,
  transl_load chunk addr args dest k = OK c ->
  tail_nolabel k c.

Proof.

intros. monadInv H. destruct chunk; TailNoLabel.
Qed.

Remark transl_store_label:
  forall chunk addr args src k c,
  transl_store chunk addr args src k = OK c ->
  tail_nolabel k c.

Proof.

intros. monadInv H. destruct chunk; TailNoLabel.
Qed.

Lemma transl_instr_label:
  forall f i ep k c,
  transl_instr f i ep k = OK c ->
  match i with Mlabel lbl => c = Plabel lbl :: k | _ => tail_nolabel k c end.

Proof.

Opaque loadind.
  unfold transl_instr; intros; destruct i; TailNoLabel.
  eapply loadind_label; eauto.
  eapply storeind_label; eauto.
  eapply loadind_label; eauto.
  eapply tail_nolabel_trans; eapply loadind_label; eauto.
  eapply transl_op_label; eauto.
  eapply transl_load_label; eauto.
  eapply transl_store_label; eauto.
  destruct s0; TailNoLabel.
  destruct s0; TailNoLabel.
  eapply tail_nolabel_trans. eapply transl_cond_label; eauto. eapply mk_jcc_label.
Qed.

Lemma transl_instr_label':
  forall lbl f i ep k c,
  transl_instr f i ep k = OK c ->
  find_label lbl c = if Mach.is_label lbl i then Some k else find_label lbl k.

Proof.

  intros. exploit transl_instr_label; eauto.
  destruct i; try (intros [A B]; apply B).
  intros. subst c. simpl. auto.
Qed.

Lemma transl_code_label:
  forall lbl f c ep tc,
  transl_code f c ep = OK tc ->
  match Mach.find_label lbl c with
  | None => find_label lbl tc = None
  | Some c' => exists tc', find_label lbl tc = Some tc' /\ transl_code f c' false = OK tc'
  end.

Proof.

  induction c; simpl; intros.
  inv H. auto.
  monadInv H. rewrite (transl_instr_label' lbl _ _ _ _ _ EQ0).
  generalize (Mach.is_label_correct lbl a).
  destruct (Mach.is_label lbl a); intros.
  subst a. simpl in EQ. exists x; auto.
  eapply IHc; eauto.
Qed.

Lemma transl_find_label:
  forall lbl f tf,
  transf_function f = OK tf ->
  match Mach.find_label lbl f.(Mach.fn_code) with
  | None => find_label lbl tf.(fn_code) = None
  | Some c => exists tc, find_label lbl tf.(fn_code) = Some tc /\ transl_code f c false = OK tc
  end.

Proof.

intros. monadInv H. destruct (zlt Int.max_unsigned (list_length_z (fn_code x))); inv EQ0.
monadInv EQ. simpl. eapply transl_code_label; eauto. rewrite transl_code'_transl_code in EQ0; eauto.
Qed.

End TRANSL_LABEL.

A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated PPC code.

Lemma find_label_goto_label:
  forall f tf lbl rs m c' b ofs,
  Genv.find_funct_ptr ge b = Some (Internal f) ->
  transf_function f = OK tf ->
  rs PC = Vptr b ofs ->
  Mach.find_label lbl f.(Mach.fn_code) = Some c' ->
  exists tc', exists rs',
    goto_label tf lbl rs m = Next rs' m
  /\ transl_code_at_pc ge (rs' PC) b f c' false tf tc'
  /\ forall r, r <> PC -> rs'#r = rs#r.

Proof.

  intros. exploit (transl_find_label lbl f tf); eauto. rewrite H2.
  intros [tc [A B]].
  exploit label_pos_code_tail; eauto. instantiate (1 := 0).
  intros [pos' [P [Q R]]].
  exists tc; exists (rs#PC <- (Vptr b (Int.repr pos'))).
  split. unfold goto_label. rewrite P. rewrite H1. auto.
  split. rewrite Pregmap.gss. constructor; auto.
  rewrite Int.unsigned_repr. replace (pos' - 0) with pos' in Q.
  auto. omega.
  generalize (transf_function_no_overflow _ _ H0). omega.
  intros. apply Pregmap.gso; auto.
Qed.

Existence of return addresses

Lemma return_address_exists:
forall f sg ros c, is_tail (Mcall sg ros :: c) f.(Mach.fn_code) ->
exists ra, return_address_offset f c ra.

Proof.

  intros. eapply Asmgenproof0.return_address_exists; eauto.
- intros. exploit transl_instr_label; eauto.
  destruct i; try (intros [A B]; apply A). intros. subst c0. repeat constructor.
- intros. monadInv H0.
  destruct (zlt Int.max_unsigned (list_length_z (fn_code x))); inv EQ0.
  monadInv EQ. rewrite transl_code'_transl_code in EQ0.
  exists x; exists true; split; auto. unfold fn_code. repeat constructor.
- exact transf_function_no_overflow.
Qed.

Proof of semantic preservation

Semantic preservation is proved using simulation diagrams of the following form.

           st1 --------------- st2
            |                   |
           t|                  *|t
            |                   |
            v                   v
           st1'--------------- st2'

The invariant is the match_states predicate below, which includes:

The PPC code pointed by the PC register is the translation of the current Mach code sequence.
Mach register values and PPC register values agree.

Inductive match_states: Mach.state -> Asm.state -> Prop :=
  | match_states_intro:
      forall s fb sp c ep ms m m' rs f tf tc
        (STACKS: match_stack ge s)
        (FIND: Genv.find_funct_ptr ge fb = Some (Internal f))
        (MEXT: Mem.extends m m')
        (AT: transl_code_at_pc ge (rs PC) fb f c ep tf tc)
        (AG: agree ms sp rs)
        (DXP: ep = true -> rs#EDX = parent_sp s),
      match_states (Mach.State s fb sp c ms m)
                   (Asm.State rs m')
  | match_states_call:
      forall s fb ms m m' rs
        (STACKS: match_stack ge s)
        (MEXT: Mem.extends m m')
        (AG: agree ms (parent_sp s) rs)
        (ATPC: rs PC = Vptr fb Int.zero)
        (ATLR: rs RA = parent_ra s),
      match_states (Mach.Callstate s fb ms m)
                   (Asm.State rs m')
  | match_states_return:
      forall s ms m m' rs
        (STACKS: match_stack ge s)
        (MEXT: Mem.extends m m')
        (AG: agree ms (parent_sp s) rs)
        (ATPC: rs PC = parent_ra s),
      match_states (Mach.Returnstate s ms m)
                   (Asm.State rs m').

Lemma exec_straight_steps:
  forall s fb f rs1 i c ep tf tc m1' m2 m2' sp ms2,
  match_stack ge s ->
  Mem.extends m2 m2' ->
  Genv.find_funct_ptr ge fb = Some (Internal f) ->
  transl_code_at_pc ge (rs1 PC) fb f (i :: c) ep tf tc ->
  (forall k c (TR: transl_instr f i ep k = OK c),
   exists rs2,
       exec_straight tge tf c rs1 m1' k rs2 m2'
    /\ agree ms2 sp rs2
    /\ (it1_is_parent ep i = true -> rs2#EDX = parent_sp s)) ->
  exists st',
  plus step tge (State rs1 m1') E0 st' /\
  match_states (Mach.State s fb sp c ms2 m2) st'.

Proof.

  intros. inversion H2. subst. monadInv H7.
  exploit H3; eauto. intros [rs2 [A [B C]]].
  exists (State rs2 m2'); split.
  eapply exec_straight_exec; eauto.
  econstructor; eauto. eapply exec_straight_at; eauto.
Qed.

Lemma exec_straight_steps_goto:
  forall s fb f rs1 i c ep tf tc m1' m2 m2' sp ms2 lbl c',
  match_stack ge s ->
  Mem.extends m2 m2' ->
  Genv.find_funct_ptr ge fb = Some (Internal f) ->
  Mach.find_label lbl f.(Mach.fn_code) = Some c' ->
  transl_code_at_pc ge (rs1 PC) fb f (i :: c) ep tf tc ->
  it1_is_parent ep i = false ->
  (forall k c (TR: transl_instr f i ep k = OK c),
   exists jmp, exists k', exists rs2,
       exec_straight tge tf c rs1 m1' (jmp :: k') rs2 m2'
    /\ agree ms2 sp rs2
    /\ exec_instr tge tf jmp rs2 m2' = goto_label tf lbl rs2 m2') ->
  exists st',
  plus step tge (State rs1 m1') E0 st' /\
  match_states (Mach.State s fb sp c' ms2 m2) st'.

Proof.

  intros. inversion H3. subst. monadInv H9.
  exploit H5; eauto. intros [jmp [k' [rs2 [A [B C]]]]].
  generalize (functions_transl _ _ _ H7 H8); intro FN.
  generalize (transf_function_no_overflow _ _ H8); intro NOOV.
  exploit exec_straight_steps_2; eauto.
  intros [ofs' [PC2 CT2]].
  exploit find_label_goto_label; eauto.
  intros [tc' [rs3 [GOTO [AT' OTH]]]].
  exists (State rs3 m2'); split.
  eapply plus_right'.
  eapply exec_straight_steps_1; eauto.
  econstructor; eauto.
  eapply find_instr_tail. eauto.
  rewrite C. eexact GOTO.
  traceEq.
  econstructor; eauto.
  apply agree_exten with rs2; auto with asmgen.
  congruence.
Qed.

We need to show that, in the simulation diagram, we cannot take infinitely many Mach transitions that correspond to zero transitions on the PPC side. Actually, all Mach transitions correspond to at least one Asm transition, except the transition from Mach.Returnstate to Mach.State. So, the following integer measure will suffice to rule out the unwanted behaviour.

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

This is the simulation diagram. We prove it by case analysis on the Mach transition.

Theorem step_simulation:
  forall S1 t S2, Mach.step return_address_offset ge S1 t S2 ->
  forall S1' (MS: match_states S1 S1'),
  (exists S2', plus step tge S1' t S2' /\ match_states S2 S2')
  \/ (measure S2 < measure S1 /\ t = E0 /\ match_states S2 S1')%nat.

Proof.

  induction 1; intros; inv MS.

- (* Mlabel *)
  left; eapply exec_straight_steps; eauto; intros.
  monadInv TR. econstructor; split. apply exec_straight_one. simpl; eauto. auto.
  split. apply agree_nextinstr; auto. simpl; congruence.

- (* Mgetstack *)
  unfold load_stack in H.
  exploit Mem.loadv_extends; eauto. intros [v' [A B]].
  rewrite (sp_val _ _ _ AG) in A.
  left; eapply exec_straight_steps; eauto. intros. simpl in TR.
  exploit loadind_correct; eauto. intros [rs' [P [Q R]]].
  exists rs'; split. eauto.
  split. eapply agree_set_mreg; eauto. congruence.
  simpl; congruence.

- (* Msetstack *)
  unfold store_stack in H.
  assert (Val.lessdef (rs src) (rs0 (preg_of src))). eapply preg_val; eauto.
  exploit Mem.storev_extends; eauto. intros [m2' [A B]].
  left; eapply exec_straight_steps; eauto.
  rewrite (sp_val _ _ _ AG) in A. intros. simpl in TR.
  exploit storeind_correct; eauto. intros [rs' [P Q]].
  exists rs'; split. eauto.
  split. eapply agree_undef_regs; eauto.
  simpl; intros. rewrite Q; auto with asmgen.
Local Transparent destroyed_by_setstack.
  destruct ty; simpl; intuition congruence.

- (* Mgetparam *)
  assert (f0 = f) by congruence; subst f0.
  unfold load_stack in *.
  exploit Mem.loadv_extends. eauto. eexact H0. auto.
  intros [parent' [A B]]. rewrite (sp_val _ _ _ AG) in A.
  exploit lessdef_parent_sp; eauto. clear B; intros B; subst parent'.
  exploit Mem.loadv_extends. eauto. eexact H1. auto.
  intros [v' [C D]].
Opaque loadind.
  left; eapply exec_straight_steps; eauto; intros.
  assert (DIFF: negb (mreg_eq dst DX) = true -> IR EDX <> preg_of dst).
    intros. change (IR EDX) with (preg_of DX). red; intros.
    unfold proj_sumbool in H1. destruct (mreg_eq dst DX); try discriminate.
    elim n. eapply preg_of_injective; eauto.
  destruct ep; simpl in TR.
EDX contains parent *)  exploit loadind_correct. eexact TR.
  instantiate (2 := rs0). rewrite DXP; eauto.
  intros [rs1 [P [Q R]]].
  exists rs1; split. eauto.
  split. eapply agree_set_mreg. eapply agree_set_mreg; eauto. congruence. auto.
  simpl; intros. rewrite R; auto.
EDX does not contain parent *)  monadInv TR.
  exploit loadind_correct. eexact EQ0. eauto. intros [rs1 [P [Q R]]]. simpl in Q.
  exploit loadind_correct. eexact EQ. instantiate (2 := rs1). rewrite Q. eauto.
  intros [rs2 [S [T U]]].
  exists rs2; split. eapply exec_straight_trans; eauto.
  split. eapply agree_set_mreg. eapply agree_set_mreg; eauto. congruence. auto.
  simpl; intros. rewrite U; auto.

- (* Mop *)
  assert (eval_operation tge sp op rs##args m = Some v).
    rewrite <- H. apply eval_operation_preserved. exact symbols_preserved.
  exploit eval_operation_lessdef. eapply preg_vals; eauto. eauto. eexact H0.
  intros [v' [A B]]. rewrite (sp_val _ _ _ AG) in A.
  left; eapply exec_straight_steps; eauto; intros. simpl in TR.
  exploit transl_op_correct; eauto. intros [rs2 [P [Q R]]].
  assert (S: Val.lessdef v (rs2 (preg_of res))) by (eapply Val.lessdef_trans; eauto).
  exists rs2; split. eauto.
  split. eapply agree_set_undef_mreg; eauto.
  simpl; congruence.

- (* Mload *)
  assert (eval_addressing tge sp addr rs##args = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  exploit eval_addressing_lessdef. eapply preg_vals; eauto. eexact H1.
  intros [a' [A B]]. rewrite (sp_val _ _ _ AG) in A.
  exploit Mem.loadv_extends; eauto. intros [v' [C D]].
  left; eapply exec_straight_steps; eauto; intros. simpl in TR.
  exploit transl_load_correct; eauto. intros [rs2 [P [Q R]]].
  exists rs2; split. eauto.
  split. eapply agree_set_undef_mreg; eauto. congruence.
  simpl; congruence.

- (* Mstore *)
  assert (eval_addressing tge sp addr rs##args = Some a).
    rewrite <- H. apply eval_addressing_preserved. exact symbols_preserved.
  exploit eval_addressing_lessdef. eapply preg_vals; eauto. eexact H1.
  intros [a' [A B]]. rewrite (sp_val _ _ _ AG) in A.
  assert (Val.lessdef (rs src) (rs0 (preg_of src))). eapply preg_val; eauto.
  exploit Mem.storev_extends; eauto. intros [m2' [C D]].
  left; eapply exec_straight_steps; eauto.
  intros. simpl in TR.
  exploit transl_store_correct; eauto. intros [rs2 [P Q]].
  exists rs2; split. eauto.
  split. eapply agree_undef_regs; eauto.
  simpl; congruence.

- (* Mcall *)
  assert (f0 = f) by congruence. subst f0.
  inv AT.
  assert (NOOV: list_length_z tf.(fn_code) <= Int.max_unsigned).
    eapply transf_function_no_overflow; eauto.
  destruct ros as [rf|fid]; simpl in H; monadInv H5.
+ (* Indirect call *)
  assert (rs rf = Vptr f' Int.zero).
    destruct (rs rf); try discriminate.
    revert H; predSpec Int.eq Int.eq_spec i Int.zero; intros; congruence.
  assert (rs0 x0 = Vptr f' Int.zero).
    exploit ireg_val; eauto. rewrite H5; intros LD; inv LD; auto.
  generalize (code_tail_next_int _ _ _ _ NOOV H6). intro CT1.
  assert (TCA: transl_code_at_pc ge (Vptr fb (Int.add ofs Int.one)) fb f c false tf x).
    econstructor; eauto.
  exploit return_address_offset_correct; eauto. intros; subst ra.
  left; econstructor; split.
  apply plus_one. eapply exec_step_internal. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. eauto.
  econstructor; eauto.
  econstructor; eauto.
  eapply agree_sp_def; eauto.
  simpl. eapply agree_exten; eauto. intros. Simplifs.
  Simplifs. rewrite <- H2. auto.
+ (* Direct call *)
  generalize (code_tail_next_int _ _ _ _ NOOV H6). intro CT1.
  assert (TCA: transl_code_at_pc ge (Vptr fb (Int.add ofs Int.one)) fb f c false tf x).
    econstructor; eauto.
  exploit return_address_offset_correct; eauto. intros; subst ra.
  left; econstructor; split.
  apply plus_one. eapply exec_step_internal. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. unfold Genv.symbol_address. rewrite symbols_preserved. rewrite H. eauto.
  econstructor; eauto.
  econstructor; eauto.
  eapply agree_sp_def; eauto.
  simpl. eapply agree_exten; eauto. intros. Simplifs.
  Simplifs. rewrite <- H2. auto.

- (* Mtailcall *)
  assert (f0 = f) by congruence. subst f0.
  inv AT.
  assert (NOOV: list_length_z tf.(fn_code) <= Int.max_unsigned).
    eapply transf_function_no_overflow; eauto.
  rewrite (sp_val _ _ _ AG) in *. unfold load_stack in *.
  exploit Mem.loadv_extends. eauto. eexact H1. auto. simpl. intros [parent' [A B]].
  exploit Mem.loadv_extends. eauto. eexact H2. auto. simpl. intros [ra' [C D]].
  exploit lessdef_parent_sp; eauto. intros. subst parent'. clear B.
  exploit lessdef_parent_ra; eauto. intros. subst ra'. clear D.
  exploit Mem.free_parallel_extends; eauto. intros [m2' [E F]].
  destruct ros as [rf|fid]; simpl in H; monadInv H7.
+ (* Indirect call *)
  assert (rs rf = Vptr f' Int.zero).
    destruct (rs rf); try discriminate.
    revert H; predSpec Int.eq Int.eq_spec i Int.zero; intros; congruence.
  assert (rs0 x0 = Vptr f' Int.zero).
    exploit ireg_val; eauto. rewrite H7; intros LD; inv LD; auto.
  generalize (code_tail_next_int _ _ _ _ NOOV H8). intro CT1.
  left; econstructor; split.
  eapply plus_left. eapply exec_step_internal. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. rewrite C. rewrite A. rewrite <- (sp_val _ _ _ AG). rewrite E. eauto.
  apply star_one. eapply exec_step_internal.
  transitivity (Val.add rs0#PC Vone). auto. rewrite <- H4. simpl. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. eauto. traceEq.
  econstructor; eauto.
  apply agree_set_other; auto. apply agree_nextinstr. apply agree_set_other; auto.
  eapply agree_change_sp; eauto. eapply parent_sp_def; eauto.
  Simplifs. rewrite Pregmap.gso; auto.
  generalize (preg_of_not_SP rf). rewrite (ireg_of_eq _ _ EQ1). congruence.
+ (* Direct call *)
  generalize (code_tail_next_int _ _ _ _ NOOV H8). intro CT1.
  left; econstructor; split.
  eapply plus_left. eapply exec_step_internal. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. rewrite C. rewrite A. rewrite <- (sp_val _ _ _ AG). rewrite E. eauto.
  apply star_one. eapply exec_step_internal.
  transitivity (Val.add rs0#PC Vone). auto. rewrite <- H4. simpl. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. eauto. traceEq.
  econstructor; eauto.
  apply agree_set_other; auto. apply agree_nextinstr. apply agree_set_other; auto.
  eapply agree_change_sp; eauto. eapply parent_sp_def; eauto.
  rewrite Pregmap.gss. unfold Genv.symbol_address. rewrite symbols_preserved. rewrite H. auto.

- (* Mbuiltin *)
  inv AT. monadInv H4.
  exploit functions_transl; eauto. intro FN.
  generalize (transf_function_no_overflow _ _ H3); intro NOOV.
  exploit builtin_args_match; eauto. intros [vargs' [P Q]].
  exploit external_call_mem_extends; eauto.
  intros [vres' [m2' [A [B [C D]]]]].
  left. econstructor; split. apply plus_one.
  eapply exec_step_builtin. eauto. eauto.
  eapply find_instr_tail; eauto.
  erewrite <- sp_val by eauto.
  eapply eval_builtin_args_preserved with (ge1 := ge); eauto. exact symbols_preserved.
  eapply external_call_symbols_preserved; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  eauto.
  econstructor; eauto.
  instantiate (2 := tf); instantiate (1 := x).
  unfold nextinstr_nf, nextinstr. rewrite Pregmap.gss.
  rewrite undef_regs_other. rewrite set_res_other. rewrite undef_regs_other_2.
  rewrite <- H1. simpl. econstructor; eauto.
  eapply code_tail_next_int; eauto.
  rewrite preg_notin_charact. intros. auto with asmgen.
  auto with asmgen.
  simpl; intros. intuition congruence.
  apply agree_nextinstr_nf. eapply agree_set_res; auto.
  eapply agree_undef_regs; eauto. intros; apply undef_regs_other_2; auto.
  congruence.

- (* Mgoto *)
  assert (f0 = f) by congruence. subst f0.
  inv AT. monadInv H4.
  exploit find_label_goto_label; eauto. intros [tc' [rs' [GOTO [AT2 INV]]]].
  left; exists (State rs' m'); split.
  apply plus_one. econstructor; eauto.
  eapply functions_transl; eauto.
  eapply find_instr_tail; eauto.
  simpl; eauto.
  econstructor; eauto.
  eapply agree_exten; eauto with asmgen.
  congruence.

- (* Mcond true *)
  assert (f0 = f) by congruence. subst f0.
  exploit eval_condition_lessdef. eapply preg_vals; eauto. eauto. eauto. intros EC.
  left; eapply exec_straight_steps_goto; eauto.
  intros. simpl in TR.
  destruct (transl_cond_correct tge tf cond args _ _ rs0 m' TR)
  as [rs' [A [B C]]].
  rewrite EC in B.
  destruct (testcond_for_condition cond); simpl in *.
simple jcc *)  exists (Pjcc c1 lbl); exists k; exists rs'.
  split. eexact A.
  split. eapply agree_exten; eauto.
  simpl. rewrite B. auto.
jcc; jcc *)  destruct (eval_testcond c1 rs') as [b1|] eqn:TC1;
  destruct (eval_testcond c2 rs') as [b2|] eqn:TC2; inv B.
  destruct b1.
first jcc jumps *)  exists (Pjcc c1 lbl); exists (Pjcc c2 lbl :: k); exists rs'.
  split. eexact A.
  split. eapply agree_exten; eauto.
  simpl. rewrite TC1. auto.
second jcc jumps *)  exists (Pjcc c2 lbl); exists k; exists (nextinstr rs').
  split. eapply exec_straight_trans. eexact A.
  eapply exec_straight_one. simpl. rewrite TC1. auto. auto.
  split. eapply agree_exten; eauto.
  intros; Simplifs.
  simpl. rewrite eval_testcond_nextinstr. rewrite TC2.
  destruct b2; auto || discriminate.
jcc2 *)  destruct (eval_testcond c1 rs') as [b1|] eqn:TC1;
  destruct (eval_testcond c2 rs') as [b2|] eqn:TC2; inv B.
  destruct (andb_prop _ _ H3). subst.
  exists (Pjcc2 c1 c2 lbl); exists k; exists rs'.
  split. eexact A.
  split. eapply agree_exten; eauto.
  simpl. rewrite TC1; rewrite TC2; auto.

- (* Mcond false *)
  exploit eval_condition_lessdef. eapply preg_vals; eauto. eauto. eauto. intros EC.
  left; eapply exec_straight_steps; eauto. intros. simpl in TR.
  destruct (transl_cond_correct tge tf cond args _ _ rs0 m' TR)
  as [rs' [A [B C]]].
  rewrite EC in B.
  destruct (testcond_for_condition cond); simpl in *.
simple jcc *)  econstructor; split.
  eapply exec_straight_trans. eexact A.
  apply exec_straight_one. simpl. rewrite B. eauto. auto.
  split. apply agree_nextinstr. eapply agree_exten; eauto.
  simpl; congruence.
jcc ; jcc *)  destruct (eval_testcond c1 rs') as [b1|] eqn:TC1;
  destruct (eval_testcond c2 rs') as [b2|] eqn:TC2; inv B.
  destruct (orb_false_elim _ _ H1); subst.
  econstructor; split.
  eapply exec_straight_trans. eexact A.
  eapply exec_straight_two. simpl. rewrite TC1. eauto. auto.
  simpl. rewrite eval_testcond_nextinstr. rewrite TC2. eauto. auto. auto.
  split. apply agree_nextinstr. apply agree_nextinstr. eapply agree_exten; eauto.
  simpl; congruence.
jcc2 *)  destruct (eval_testcond c1 rs') as [b1|] eqn:TC1;
  destruct (eval_testcond c2 rs') as [b2|] eqn:TC2; inv B.
  exists (nextinstr rs'); split.
  eapply exec_straight_trans. eexact A.
  apply exec_straight_one. simpl.
  rewrite TC1; rewrite TC2.
  destruct b1. simpl in *. subst b2. auto. auto.
  auto.
  split. apply agree_nextinstr. eapply agree_exten; eauto.
  rewrite H1; congruence.

- (* Mjumptable *)
  assert (f0 = f) by congruence. subst f0.
  inv AT. monadInv H6.
  exploit functions_transl; eauto. intro FN.
  generalize (transf_function_no_overflow _ _ H5); intro NOOV.
  exploit find_label_goto_label; eauto.
  intros [tc' [rs' [A [B C]]]].
  exploit ireg_val; eauto. rewrite H. intros LD; inv LD.
  left; econstructor; split.
  apply plus_one. econstructor; eauto.
  eapply find_instr_tail; eauto.
  simpl. rewrite <- H9. unfold Mach.label in H0; unfold label; rewrite H0. eauto.
  econstructor; eauto.
Transparent destroyed_by_jumptable.
  simpl. eapply agree_exten; eauto. intros. rewrite C; auto with asmgen.
  congruence.

- (* Mreturn *)
  assert (f0 = f) by congruence. subst f0.
  inv AT.
  assert (NOOV: list_length_z tf.(fn_code) <= Int.max_unsigned).
    eapply transf_function_no_overflow; eauto.
  rewrite (sp_val _ _ _ AG) in *. unfold load_stack in *.
  exploit Mem.loadv_extends. eauto. eexact H0. auto. simpl. intros [parent' [A B]].
  exploit lessdef_parent_sp; eauto. intros. subst parent'. clear B.
  exploit Mem.loadv_extends. eauto. eexact H1. auto. simpl. intros [ra' [C D]].
  exploit lessdef_parent_ra; eauto. intros. subst ra'. clear D.
  exploit Mem.free_parallel_extends; eauto. intros [m2' [E F]].
  monadInv H6.
  exploit code_tail_next_int; eauto. intro CT1.
  left; econstructor; split.
  eapply plus_left. eapply exec_step_internal. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. rewrite C. rewrite A. rewrite <- (sp_val _ _ _ AG). rewrite E. eauto.
  apply star_one. eapply exec_step_internal.
  transitivity (Val.add rs0#PC Vone). auto. rewrite <- H3. simpl. eauto.
  eapply functions_transl; eauto. eapply find_instr_tail; eauto.
  simpl. eauto. traceEq.
  constructor; auto.
  apply agree_set_other; auto. apply agree_nextinstr. apply agree_set_other; auto.
  eapply agree_change_sp; eauto. eapply parent_sp_def; eauto.

- (* internal function *)
  exploit functions_translated; eauto. intros [tf [A B]]. monadInv B.
  generalize EQ; intros EQ'. monadInv EQ'.
  destruct (zlt Int.max_unsigned (list_length_z (fn_code x0))); inv EQ1.
  monadInv EQ0. rewrite transl_code'_transl_code in EQ1.
  unfold store_stack in *.
  exploit Mem.alloc_extends. eauto. eauto. apply Zle_refl. apply Zle_refl.
  intros [m1' [C D]].
  exploit Mem.storev_extends. eexact D. eexact H1. eauto. eauto.
  intros [m2' [F G]].
  exploit Mem.storev_extends. eexact G. eexact H2. eauto. eauto.
  intros [m3' [P Q]].
  left; econstructor; split.
  apply plus_one. econstructor; eauto.
  simpl. rewrite Int.unsigned_zero. simpl. eauto.
  simpl. rewrite C. simpl in F. rewrite (sp_val _ _ _ AG) in F. rewrite F.
  simpl in P. rewrite ATLR. rewrite P. eauto.
  econstructor; eauto.
  unfold nextinstr. rewrite Pregmap.gss. repeat rewrite Pregmap.gso; auto with asmgen.
  rewrite ATPC. simpl. constructor; eauto.
  unfold fn_code. eapply code_tail_next_int. simpl in g. omega.
  constructor.
  apply agree_nextinstr. eapply agree_change_sp; eauto.
Transparent destroyed_at_function_entry.
  apply agree_undef_regs with rs0; eauto.
  simpl; intros. apply Pregmap.gso; auto with asmgen. tauto.
  congruence.
  intros. Simplifs. eapply agree_sp; eauto.

- (* external function *)
  exploit functions_translated; eauto.
  intros [tf [A B]]. simpl in B. inv B.
  exploit extcall_arguments_match; eauto.
  intros [args' [C D]].
  exploit external_call_mem_extends'; eauto.
  intros [res' [m2' [P [Q [R S]]]]].
  left; econstructor; split.
  apply plus_one. eapply exec_step_external; eauto.
  eapply external_call_symbols_preserved'; eauto.
  exact symbols_preserved. exact public_preserved. exact varinfo_preserved.
  econstructor; eauto.
  unfold loc_external_result.
  apply agree_set_other; auto. apply agree_set_mregs; auto.

- (* return *)
  inv STACKS. simpl in *.
  right. split. omega. split. auto.
  econstructor; eauto. rewrite ATPC; eauto. congruence.
Qed.

Lemma transf_initial_states:
forall st1, Mach.initial_state prog st1 ->
exists st2, Asm.initial_state tprog st2 /\ match_states st1 st2.

Proof.

  intros. inversion H. unfold ge0 in *.
  econstructor; split.
  econstructor.
  eapply Genv.init_mem_transf_partial; eauto.
  replace (Genv.symbol_address (Genv.globalenv tprog) (prog_main tprog) Int.zero)
     with (Vptr fb Int.zero).
  econstructor; eauto.
  constructor.
  apply Mem.extends_refl.
  split. auto. simpl. unfold Vzero; congruence. intros. rewrite Regmap.gi. auto.
  unfold Genv.symbol_address.
  rewrite (transform_partial_program_main _ _ TRANSF).
  rewrite symbols_preserved.
  unfold ge; rewrite H1. auto.
Qed.

Lemma transf_final_states:
forall st1 st2 r,
match_states st1 st2 -> Mach.final_state st1 r -> Asm.final_state st2 r.

Proof.

  intros. inv H0. inv H. constructor. auto.
  compute in H1. inv H1.
  generalize (preg_val _ _ _ AX AG). rewrite H2. intros LD; inv LD. auto.
Qed.

Theorem transf_program_correct:
forward_simulation (Mach.semantics return_address_offset prog) (Asm.semantics tprog).

Proof.

  eapply forward_simulation_star with (measure := measure).
  eexact public_preserved.
  eexact transf_initial_states.
  eexact transf_final_states.
  exact step_simulation.
Qed.

End PRESERVATION.

Module Asmgenproof

Properties of control flow

Proof of semantic preservation