Correctness proof for Asm generation: machine-independent framework

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

Processor registers and register states

Hint Extern 2 (_ <> _) => congruence: asmgen.

Lemma ireg_of_eq:
forall r r', ireg_of r = OK r' -> preg_of r = IR r'.

Proof.

unfold ireg_of; intros. destruct (preg_of r); inv H; auto.
Qed.

Lemma freg_of_eq:
forall r r', freg_of r = OK r' -> preg_of r = FR r'.

Proof.

unfold freg_of; intros. destruct (preg_of r); inv H; auto.
Qed.

Lemma preg_of_injective:
forall r1 r2, preg_of r1 = preg_of r2 -> r1 = r2.

Proof.

destruct r1; destruct r2; simpl; intros; reflexivity || discriminate.
Qed.

Lemma preg_of_data:
forall r, data_preg (preg_of r) = true.

Proof.

intros. destruct r; reflexivity.
Qed.

Hint Resolve preg_of_data: asmgen.

Lemma data_diff:
forall r r',
data_preg r = true -> data_preg r' = false -> r <> r'.

Proof.

congruence.
Qed.

Hint Resolve data_diff: asmgen.

Lemma preg_of_not_SP:
forall r, preg_of r <> SP.

Proof.

intros. unfold preg_of; destruct r; simpl; congruence.
Qed.

Lemma preg_of_not_PC:
forall r, preg_of r <> PC.

Proof.

intros. apply data_diff; auto with asmgen.
Qed.

Hint Resolve preg_of_not_SP preg_of_not_PC: asmgen.

Lemma nextinstr_pc:
forall rs, (nextinstr rs)#PC = Val.add rs#PC Vone.

Proof.

intros. apply Pregmap.gss.
Qed.

Lemma nextinstr_inv:
forall r rs, r <> PC -> (nextinstr rs)#r = rs#r.

Proof.

intros. unfold nextinstr. apply Pregmap.gso. red; intro; subst. auto.
Qed.

Lemma nextinstr_inv1:
forall r rs, data_preg r = true -> (nextinstr rs)#r = rs#r.

Proof.

intros. apply nextinstr_inv. red; intro; subst; discriminate.
Qed.

Lemma nextinstr_set_preg:
forall rs m v,
(nextinstr (rs#(preg_of m) <- v))#PC = Val.add rs#PC Vone.

Proof.

intros. unfold nextinstr. rewrite Pregmap.gss.
rewrite Pregmap.gso. auto. apply sym_not_eq. apply preg_of_not_PC.
Qed.

Lemma undef_regs_other:
  forall r rl rs,
  (forall r', In r' rl -> r <> r') ->
  undef_regs rl rs r = rs r.

Proof.

induction rl; simpl; intros. auto.
rewrite IHrl by auto. rewrite Pregmap.gso; auto.
Qed.

Fixpoint preg_notin (r: preg) (rl: list mreg) : Prop :=
  match rl with
  | nil => True
  | r1 :: nil => r <> preg_of r1
  | r1 :: rl => r <> preg_of r1 /\ preg_notin r rl
  end.

Remark preg_notin_charact:
  forall r rl,
  preg_notin r rl <-> (forall mr, In mr rl -> r <> preg_of mr).

Proof.

  induction rl; simpl; intros.
  tauto.
  destruct rl.
  simpl. split. intros. intuition congruence. auto.
  rewrite IHrl. split.
  intros [A B]. intros. destruct H. congruence. auto.
  auto.
Qed.

Lemma undef_regs_other_2:
  forall r rl rs,
  preg_notin r rl ->
  undef_regs (map preg_of rl) rs r = rs r.

Proof.

  intros. apply undef_regs_other. intros.
  exploit list_in_map_inv; eauto. intros [mr [A B]]. subst.
  rewrite preg_notin_charact in H. auto.
Qed.

Lemma set_pregs_other_2:
  forall r rl vl rs,
  preg_notin r rl ->
  set_regs (map preg_of rl) vl rs r = rs r.

Proof.

  induction rl; simpl; intros.
  auto.
  destruct vl; auto.
  assert (r <> preg_of a) by (destruct rl; tauto).
  assert (preg_notin r rl) by (destruct rl; simpl; tauto).
  rewrite IHrl by auto. apply Pregmap.gso; auto.
Qed.

Agreement between Mach registers and processor registers

Record agree (ms: Mach.regset) (sp: val) (rs: Asm.regset) : Prop := mkagree {
  agree_sp: rs#SP = sp;
  agree_sp_def: sp <> Vundef;
  agree_mregs: forall r: mreg, Val.lessdef (ms r) (rs#(preg_of r))
}.

Lemma preg_val:
  forall ms sp rs r, agree ms sp rs -> Val.lessdef (ms r) rs#(preg_of r).

Proof.

intros. destruct H. auto.
Qed.

Lemma preg_vals:
forall ms sp rs, agree ms sp rs ->
forall l, Val.lessdef_list (map ms l) (map rs (map preg_of l)).

Proof.

induction l; simpl. constructor. constructor. eapply preg_val; eauto. auto.
Qed.

Lemma sp_val:
forall ms sp rs, agree ms sp rs -> sp = rs#SP.

Proof.

intros. destruct H; auto.
Qed.

Lemma ireg_val:
  forall ms sp rs r r',
  agree ms sp rs ->
  ireg_of r = OK r' ->
  Val.lessdef (ms r) rs#r'.

Proof.

intros. rewrite <- (ireg_of_eq _ _ H0). eapply preg_val; eauto.
Qed.

Lemma freg_val:
  forall ms sp rs r r',
  agree ms sp rs ->
  freg_of r = OK r' ->
  Val.lessdef (ms r) (rs#r').

Proof.

intros. rewrite <- (freg_of_eq _ _ H0). eapply preg_val; eauto.
Qed.

Lemma agree_exten:
  forall ms sp rs rs',
  agree ms sp rs ->
  (forall r, data_preg r = true -> rs'#r = rs#r) ->
  agree ms sp rs'.

Proof.

  intros. destruct H. split; auto.
  rewrite H0; auto. auto.
  intros. rewrite H0; auto. apply preg_of_data.
Qed.

Preservation of register agreement under various assignments.

Lemma agree_set_mreg:
  forall ms sp rs r v rs',
  agree ms sp rs ->
  Val.lessdef v (rs'#(preg_of r)) ->
  (forall r', data_preg r' = true -> r' <> preg_of r -> rs'#r' = rs#r') ->
  agree (Regmap.set r v ms) sp rs'.

Proof.

  intros. destruct H. split; auto.
  rewrite H1; auto. apply sym_not_equal. apply preg_of_not_SP.
  intros. unfold Regmap.set. destruct (RegEq.eq r0 r). congruence.
  rewrite H1. auto. apply preg_of_data.
  red; intros; elim n. eapply preg_of_injective; eauto.
Qed.

Lemma agree_set_other:
  forall ms sp rs r v,
  agree ms sp rs ->
  data_preg r = false ->
  agree ms sp (rs#r <- v).

Proof.

intros. apply agree_exten with rs. auto.
intros. apply Pregmap.gso. congruence.
Qed.

Lemma agree_nextinstr:
forall ms sp rs,
agree ms sp rs -> agree ms sp (nextinstr rs).

Proof.

intros. unfold nextinstr. apply agree_set_other. auto. auto.
Qed.

Lemma agree_set_mregs:
  forall sp rl vl vl' ms rs,
  agree ms sp rs ->
  Val.lessdef_list vl vl' ->
  agree (Mach.set_regs rl vl ms) sp (set_regs (map preg_of rl) vl' rs).

Proof.

  induction rl; simpl; intros.
  auto.
  inv H0. auto. apply IHrl; auto.
  eapply agree_set_mreg. eexact H.
  rewrite Pregmap.gss. auto.
  intros. apply Pregmap.gso; auto.
Qed.

Lemma agree_undef_nondata_regs:
  forall ms sp rl rs,
  agree ms sp rs ->
  (forall r, In r rl -> data_preg r = false) ->
  agree ms sp (undef_regs rl rs).

Proof.

  induction rl; simpl; intros. auto.
  apply IHrl. apply agree_exten with rs; auto.
  intros. apply Pregmap.gso. red; intros; subst.
  assert (data_preg a = false) by auto. congruence.
  intros. apply H0; auto.
Qed.

Lemma agree_undef_regs:
  forall ms sp rl rs rs',
  agree ms sp rs ->
  (forall r', data_preg r' = true -> preg_notin r' rl -> rs'#r' = rs#r') ->
  agree (Mach.undef_regs rl ms) sp rs'.

Proof.

  intros. destruct H. split; auto.
  rewrite <- agree_sp0. apply H0; auto.
  rewrite preg_notin_charact. intros. apply not_eq_sym. apply preg_of_not_SP.
  intros. destruct (In_dec mreg_eq r rl).
  rewrite Mach.undef_regs_same; auto.
  rewrite Mach.undef_regs_other; auto. rewrite H0; auto.
  apply preg_of_data.
  rewrite preg_notin_charact. intros; red; intros. elim n.
  exploit preg_of_injective; eauto. congruence.
Qed.

Lemma agree_set_undef_mreg:
  forall ms sp rs r v rl rs',
  agree ms sp rs ->
  Val.lessdef v (rs'#(preg_of r)) ->
  (forall r', data_preg r' = true -> r' <> preg_of r -> preg_notin r' rl -> rs'#r' = rs#r') ->
  agree (Regmap.set r v (Mach.undef_regs rl ms)) sp rs'.

Proof.

  intros. apply agree_set_mreg with (rs'#(preg_of r) <- (rs#(preg_of r))); auto.
  apply agree_undef_regs with rs; auto.
  intros. unfold Pregmap.set. destruct (PregEq.eq r' (preg_of r)).
  congruence. auto.
  intros. rewrite Pregmap.gso; auto.
Qed.

Lemma agree_change_sp:
  forall ms sp rs sp',
  agree ms sp rs -> sp' <> Vundef ->
  agree ms sp' (rs#SP <- sp').

Proof.

intros. inv H. split; auto.
intros. rewrite Pregmap.gso; auto with asmgen.
Qed.

Connection between Mach and Asm calling conventions for external functions.

Lemma extcall_arg_match:
  forall ms sp rs m m' l v,
  agree ms sp rs ->
  Mem.extends m m' ->
  Mach.extcall_arg ms m sp l v ->
  exists v', Asm.extcall_arg rs m' l v' /\ Val.lessdef v v'.

Proof.

  intros. inv H1.
  exists (rs#(preg_of r)); split. constructor. eapply preg_val; eauto.
  unfold load_stack in H2.
  exploit Mem.loadv_extends; eauto. intros [v' [A B]].
  rewrite (sp_val _ _ _ H) in A.
  exists v'; split; auto.
  econstructor. eauto. assumption.
Qed.

Lemma extcall_args_match:
  forall ms sp rs m m', agree ms sp rs -> Mem.extends m m' ->
  forall ll vl,
  list_forall2 (Mach.extcall_arg ms m sp) ll vl ->
  exists vl', list_forall2 (Asm.extcall_arg rs m') ll vl' /\ Val.lessdef_list vl vl'.

Proof.

  induction 3; intros.
  exists (@nil val); split. constructor. constructor.
  exploit extcall_arg_match; eauto. intros [v1' [A B]].
  destruct IHlist_forall2 as [vl' [C D]].
  exists (v1' :: vl'); split; constructor; auto.
Qed.

Lemma extcall_arguments_match:
  forall ms m m' sp rs sg args,
  agree ms sp rs -> Mem.extends m m' ->
  Mach.extcall_arguments ms m sp sg args ->
  exists args', Asm.extcall_arguments rs m' sg args' /\ Val.lessdef_list args args'.

Proof.

unfold Mach.extcall_arguments, Asm.extcall_arguments; intros.
eapply extcall_args_match; eauto.
Qed.

Translation of arguments and results to builtins.

Remark builtin_arg_match:
  forall ge (rs: regset) sp m a v,
  eval_builtin_arg ge (fun r => rs (preg_of r)) sp m a v ->
  eval_builtin_arg ge rs sp m (map_builtin_arg preg_of a) v.

Proof.

induction 1; simpl; eauto with barg.
Qed.

Lemma builtin_args_match:
  forall ge ms sp rs m m', agree ms sp rs -> Mem.extends m m' ->
  forall al vl, eval_builtin_args ge ms sp m al vl ->
  exists vl', eval_builtin_args ge rs sp m' (map (map_builtin_arg preg_of) al) vl'
           /\ Val.lessdef_list vl vl'.

Proof.

  induction 3; intros; simpl.
  exists (@nil val); split; constructor.
  exploit (@eval_builtin_arg_lessdef _ ge ms (fun r => rs (preg_of r))); eauto.
  intros; eapply preg_val; eauto.
  intros (v1' & A & B).
  destruct IHlist_forall2 as [vl' [C D]].
  exists (v1' :: vl'); split; constructor; auto. apply builtin_arg_match; auto.
Qed.

Lemma agree_set_res:
  forall res ms sp rs v v',
  agree ms sp rs ->
  Val.lessdef v v' ->
  agree (Mach.set_res res v ms) sp (Asm.set_res (map_builtin_res preg_of res) v' rs).

Proof.

  induction res; simpl; intros.
- eapply agree_set_mreg; eauto. rewrite Pregmap.gss. auto.
  intros. apply Pregmap.gso; auto.
- auto.
- apply IHres2. apply IHres1. auto.
  apply Val.hiword_lessdef; auto.
  apply Val.loword_lessdef; auto.
Qed.

Lemma set_res_other:
  forall r res v rs,
  data_preg r = false ->
  set_res (map_builtin_res preg_of res) v rs r = rs r.

Proof.

induction res; simpl; intros.
- apply Pregmap.gso. red; intros; subst r. rewrite preg_of_data in H; discriminate.
- auto.
- rewrite IHres2, IHres1; auto.
Qed.

Correspondence between Mach code and Asm code

Lemma find_instr_in:
forall c pos i,
find_instr pos c = Some i -> In i c.

Proof.

  induction c; simpl. intros; discriminate.
  intros until i. case (zeq pos 0); intros.
  left; congruence. right; eauto.
Qed.

The ``code tail'' of an instruction list c is the list of instructions starting at PC pos.

Inductive code_tail: Z -> code -> code -> Prop :=
  | code_tail_0: forall c,
      code_tail 0 c c
  | code_tail_S: forall pos i c1 c2,
      code_tail pos c1 c2 ->
      code_tail (pos + 1) (i :: c1) c2.

Lemma code_tail_pos:
  forall pos c1 c2, code_tail pos c1 c2 -> pos >= 0.

Proof.

induction 1. omega. omega.
Qed.

Lemma find_instr_tail:
  forall c1 i c2 pos,
  code_tail pos c1 (i :: c2) ->
  find_instr pos c1 = Some i.

Proof.

  induction c1; simpl; intros.
  inv H.
  destruct (zeq pos 0). subst pos.
  inv H. auto. generalize (code_tail_pos _ _ _ H4). intro. omegaContradiction.
  inv H. congruence. replace (pos0 + 1 - 1) with pos0 by omega.
  eauto.
Qed.

Remark code_tail_bounds_1:
forall fn ofs c,
code_tail ofs fn c -> 0 <= ofs <= list_length_z fn.

Proof.

  induction 1; intros; simpl.
  generalize (list_length_z_pos c). omega.
  rewrite list_length_z_cons. omega.
Qed.

Remark code_tail_bounds_2:
forall fn ofs i c,
code_tail ofs fn (i :: c) -> 0 <= ofs < list_length_z fn.

Proof.

  assert (forall ofs fn c, code_tail ofs fn c ->
          forall i c', c = i :: c' -> 0 <= ofs < list_length_z fn).
  induction 1; intros; simpl.
  rewrite H. rewrite list_length_z_cons. generalize (list_length_z_pos c'). omega.
  rewrite list_length_z_cons. generalize (IHcode_tail _ _ H0). omega.
  eauto.
Qed.

Lemma code_tail_next:
  forall fn ofs i c,
  code_tail ofs fn (i :: c) ->
  code_tail (ofs + 1) fn c.

Proof.

  assert (forall ofs fn c, code_tail ofs fn c ->
          forall i c', c = i :: c' -> code_tail (ofs + 1) fn c').
  induction 1; intros.
  subst c. constructor. constructor.
  constructor. eauto.
  eauto.
Qed.

Lemma code_tail_next_int:
  forall fn ofs i c,
  list_length_z fn <= Int.max_unsigned ->
  code_tail (Int.unsigned ofs) fn (i :: c) ->
  code_tail (Int.unsigned (Int.add ofs Int.one)) fn c.

Proof.

  intros. rewrite Int.add_unsigned.
  change (Int.unsigned Int.one) with 1.
  rewrite Int.unsigned_repr. apply code_tail_next with i; auto.
  generalize (code_tail_bounds_2 _ _ _ _ H0). omega.
Qed.

transl_code_at_pc pc fb f c ep tf tc holds if the code pointer pc points within the Asm code generated by translating Mach function f, and tc is the tail of the generated code at the position corresponding to the code pointer pc.

Inductive transl_code_at_pc (ge: Mach.genv):
    val -> block -> Mach.function -> Mach.code -> bool -> Asm.function -> Asm.code -> Prop :=
  transl_code_at_pc_intro:
    forall b ofs f c ep tf tc,
    Genv.find_funct_ptr ge b = Some(Internal f) ->
    transf_function f = Errors.OK tf ->
    transl_code f c ep = OK tc ->
    code_tail (Int.unsigned ofs) (fn_code tf) tc ->
    transl_code_at_pc ge (Vptr b ofs) b f c ep tf tc.

Equivalence between transl_code and transl_code'.

Local Open Scope error_monad_scope.

Lemma transl_code_rec_transl_code:
forall f il ep k,
transl_code_rec f il ep k = (do c <- transl_code f il ep; k c).

Proof.

  induction il; simpl; intros.
  auto.
  rewrite IHil.
  destruct (transl_code f il (it1_is_parent ep a)); simpl; auto.
Qed.

Lemma transl_code'_transl_code:
forall f il ep,
transl_code' f il ep = transl_code f il ep.

Proof.

intros. unfold transl_code'. rewrite transl_code_rec_transl_code.
destruct (transl_code f il ep); auto.
Qed.

Predictor for return addresses in generated Asm code. The return_address_offset predicate defined here is used in the semantics for Mach to determine the return addresses that are stored in activation records.

Consider a Mach function f and a sequence c of Mach instructions representing the Mach code that remains to be executed after a function call returns. The predicate return_address_offset f c ofs holds if ofs is the integer offset of the PPC instruction following the call in the Asm code obtained by translating the code of f. Graphically:

     Mach function f    |--------- Mcall ---------|
         Mach code c    |                |--------|
                        |                 \        \
                        |                  \        \
                        |                   \        \
     Asm code           |                    |--------|
     Asm function       |------------- Pcall ---------|

                        <-------- ofs ------->

Definition return_address_offset (f: Mach.function) (c: Mach.code) (ofs: int) : Prop :=
  forall tf tc,
  transf_function f = OK tf ->
  transl_code f c false = OK tc ->
  code_tail (Int.unsigned ofs) (fn_code tf) tc.

We now show that such an offset always exists if the Mach code c is a suffix of f.(fn_code). This holds because the translation from Mach to PPC is compositional: each Mach instruction becomes zero, one or several PPC instructions, but the order of instructions is preserved.

Lemma is_tail_code_tail:
forall c1 c2, is_tail c1 c2 -> exists ofs, code_tail ofs c2 c1.

Proof.

induction 1. exists 0; constructor.
destruct IHis_tail as [ofs CT]. exists (ofs + 1); constructor; auto.
Qed.

Section RETADDR_EXISTS.

Hypothesis transl_instr_tail:
  forall f i ep k c, transl_instr f i ep k = OK c -> is_tail k c.
Hypothesis transf_function_inv:
  forall f tf, transf_function f = OK tf ->
  exists tc, exists ep, transl_code f (Mach.fn_code f) ep = OK tc /\ is_tail tc (fn_code tf).
Hypothesis transf_function_len:
  forall f tf, transf_function f = OK tf -> list_length_z (fn_code tf) <= Int.max_unsigned.

Lemma transl_code_tail:
  forall f c1 c2, is_tail c1 c2 ->
  forall tc2 ep2, transl_code f c2 ep2 = OK tc2 ->
  exists tc1, exists ep1, transl_code f c1 ep1 = OK tc1 /\ is_tail tc1 tc2.

Proof.

  induction 1; simpl; intros.
  exists tc2; exists ep2; split; auto with coqlib.
  monadInv H0. exploit IHis_tail; eauto. intros [tc1 [ep1 [A B]]].
  exists tc1; exists ep1; split. auto.
  apply is_tail_trans with x. auto. eapply transl_instr_tail; eauto.
Qed.

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. destruct (transf_function f) as [tf|] eqn:TF.
+ exploit transf_function_inv; eauto. intros (tc1 & ep1 & TR1 & TL1).
  exploit transl_code_tail; eauto. intros (tc2 & ep2 & TR2 & TL2).
Opaque transl_instr.
  monadInv TR2.
  assert (TL3: is_tail x (fn_code tf)).
  { apply is_tail_trans with tc1; auto.
    apply is_tail_trans with tc2; auto.
    eapply transl_instr_tail; eauto. }
  exploit is_tail_code_tail. eexact TL3. intros [ofs CT].
  exists (Int.repr ofs). red; intros.
  rewrite Int.unsigned_repr. congruence.
  exploit code_tail_bounds_1; eauto.
  apply transf_function_len in TF. omega.
+ exists Int.zero; red; intros. congruence.
Qed.

End RETADDR_EXISTS.

Remark code_tail_no_bigger:
forall pos c1 c2, code_tail pos c1 c2 -> (length c2 <= length c1)%nat.

Proof.

induction 1; simpl; omega.
Qed.

Remark code_tail_unique:
forall fn c pos pos',
code_tail pos fn c -> code_tail pos' fn c -> pos = pos'.

Proof.

  induction fn; intros until pos'; intros ITA CT; inv ITA; inv CT; auto.
  generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; omega.
  generalize (code_tail_no_bigger _ _ _ H3); simpl; intro; omega.
  f_equal. eauto.
Qed.

Lemma return_address_offset_correct:
  forall ge b ofs fb f c tf tc ofs',
  transl_code_at_pc ge (Vptr b ofs) fb f c false tf tc ->
  return_address_offset f c ofs' ->
  ofs' = ofs.

Proof.

  intros. inv H. red in H0.
  exploit code_tail_unique. eexact H12. eapply H0; eauto. intro.
  rewrite <- (Int.repr_unsigned ofs).
  rewrite <- (Int.repr_unsigned ofs').
  congruence.
Qed.

The find_label function returns the code tail starting at the given label. A connection with code_tail is then established.

Fixpoint find_label (lbl: label) (c: code) {struct c} : option code :=
  match c with
  | nil => None
  | instr :: c' =>
      if is_label lbl instr then Some c' else find_label lbl c'
  end.

Lemma label_pos_code_tail:
  forall lbl c pos c',
  find_label lbl c = Some c' ->
  exists pos',
  label_pos lbl pos c = Some pos'
  /\ code_tail (pos' - pos) c c'
  /\ pos < pos' <= pos + list_length_z c.

Proof.

  induction c.
  simpl; intros. discriminate.
  simpl; intros until c'.
  case (is_label lbl a).
  intro EQ; injection EQ; intro; subst c'.
  exists (pos + 1). split. auto. split.
  replace (pos + 1 - pos) with (0 + 1) by omega. constructor. constructor.
  rewrite list_length_z_cons. generalize (list_length_z_pos c). omega.
  intros. generalize (IHc (pos + 1) c' H). intros [pos' [A [B C]]].
  exists pos'. split. auto. split.
  replace (pos' - pos) with ((pos' - (pos + 1)) + 1) by omega.
  constructor. auto.
  rewrite list_length_z_cons. omega.
Qed.

Helper lemmas to reason about

the "code is tail of" property
correct translation of labels.

Definition tail_nolabel (k c: code) : Prop :=
is_tail k c /\ forall lbl, find_label lbl c = find_label lbl k.

Lemma tail_nolabel_refl:
forall c, tail_nolabel c c.

Proof.

intros; split. apply is_tail_refl. auto.
Qed.

Lemma tail_nolabel_trans:
forall c1 c2 c3, tail_nolabel c2 c3 -> tail_nolabel c1 c2 -> tail_nolabel c1 c3.

Proof.

  intros. destruct H; destruct H0; split.
  eapply is_tail_trans; eauto.
  intros. rewrite H1; auto.
Qed.

Definition nolabel (i: instruction) :=
  match i with Plabel _ => False | _ => True end.

Hint Extern 1 (nolabel _) => exact I : labels.

Lemma tail_nolabel_cons:
  forall i c k,
  nolabel i -> tail_nolabel k c -> tail_nolabel k (i :: c).

Proof.

  intros. destruct H0. split.
  constructor; auto.
  intros. simpl. rewrite <- H1. destruct i; reflexivity || contradiction.
Qed.

Execution of straight-line code

Section STRAIGHTLINE.

Variable ge: genv.
Variable fn: function.

Straight-line code is composed of processor instructions that execute in sequence (no branches, no function calls and returns). The following inductive predicate relates the machine states before and after executing a straight-line sequence of instructions. Instructions are taken from the first list instead of being fetched from memory.

Inductive exec_straight: code -> regset -> mem ->
                         code -> regset -> mem -> Prop :=
  | exec_straight_one:
      forall i1 c rs1 m1 rs2 m2,
      exec_instr ge fn i1 rs1 m1 = Next rs2 m2 ->
      rs2#PC = Val.add rs1#PC Vone ->
      exec_straight (i1 :: c) rs1 m1 c rs2 m2
  | exec_straight_step:
      forall i c rs1 m1 rs2 m2 c' rs3 m3,
      exec_instr ge fn i rs1 m1 = Next rs2 m2 ->
      rs2#PC = Val.add rs1#PC Vone ->
      exec_straight c rs2 m2 c' rs3 m3 ->
      exec_straight (i :: c) rs1 m1 c' rs3 m3.

Lemma exec_straight_trans:
  forall c1 rs1 m1 c2 rs2 m2 c3 rs3 m3,
  exec_straight c1 rs1 m1 c2 rs2 m2 ->
  exec_straight c2 rs2 m2 c3 rs3 m3 ->
  exec_straight c1 rs1 m1 c3 rs3 m3.

Proof.

  induction 1; intros.
  apply exec_straight_step with rs2 m2; auto.
  apply exec_straight_step with rs2 m2; auto.
Qed.

Lemma exec_straight_two:
  forall i1 i2 c rs1 m1 rs2 m2 rs3 m3,
  exec_instr ge fn i1 rs1 m1 = Next rs2 m2 ->
  exec_instr ge fn i2 rs2 m2 = Next rs3 m3 ->
  rs2#PC = Val.add rs1#PC Vone ->
  rs3#PC = Val.add rs2#PC Vone ->
  exec_straight (i1 :: i2 :: c) rs1 m1 c rs3 m3.

Proof.

intros. apply exec_straight_step with rs2 m2; auto.
apply exec_straight_one; auto.
Qed.

Lemma exec_straight_three:
  forall i1 i2 i3 c rs1 m1 rs2 m2 rs3 m3 rs4 m4,
  exec_instr ge fn i1 rs1 m1 = Next rs2 m2 ->
  exec_instr ge fn i2 rs2 m2 = Next rs3 m3 ->
  exec_instr ge fn i3 rs3 m3 = Next rs4 m4 ->
  rs2#PC = Val.add rs1#PC Vone ->
  rs3#PC = Val.add rs2#PC Vone ->
  rs4#PC = Val.add rs3#PC Vone ->
  exec_straight (i1 :: i2 :: i3 :: c) rs1 m1 c rs4 m4.

Proof.

intros. apply exec_straight_step with rs2 m2; auto.
eapply exec_straight_two; eauto.
Qed.

The following lemmas show that straight-line executions (predicate exec_straight) correspond to correct Asm executions.

Lemma exec_straight_steps_1:
  forall c rs m c' rs' m',
  exec_straight c rs m c' rs' m' ->
  list_length_z (fn_code fn) <= Int.max_unsigned ->
  forall b ofs,
  rs#PC = Vptr b ofs ->
  Genv.find_funct_ptr ge b = Some (Internal fn) ->
  code_tail (Int.unsigned ofs) (fn_code fn) c ->
  plus step ge (State rs m) E0 (State rs' m').

Proof.

  induction 1; intros.
  apply plus_one.
  econstructor; eauto.
  eapply find_instr_tail. eauto.
  eapply plus_left'.
  econstructor; eauto.
  eapply find_instr_tail. eauto.
  apply IHexec_straight with b (Int.add ofs Int.one).
  auto. rewrite H0. rewrite H3. reflexivity.
  auto.
  apply code_tail_next_int with i; auto.
  traceEq.
Qed.

Lemma exec_straight_steps_2:
  forall c rs m c' rs' m',
  exec_straight c rs m c' rs' m' ->
  list_length_z (fn_code fn) <= Int.max_unsigned ->
  forall b ofs,
  rs#PC = Vptr b ofs ->
  Genv.find_funct_ptr ge b = Some (Internal fn) ->
  code_tail (Int.unsigned ofs) (fn_code fn) c ->
  exists ofs',
     rs'#PC = Vptr b ofs'
  /\ code_tail (Int.unsigned ofs') (fn_code fn) c'.

Proof.

  induction 1; intros.
  exists (Int.add ofs Int.one). split.
  rewrite H0. rewrite H2. auto.
  apply code_tail_next_int with i1; auto.
  apply IHexec_straight with (Int.add ofs Int.one).
  auto. rewrite H0. rewrite H3. reflexivity. auto.
  apply code_tail_next_int with i; auto.
Qed.

End STRAIGHTLINE.