Module Tunnelingproof


Correctness proof for the branch tunneling optimization.

Require Import Coqlib Maps UnionFind.
Require Import AST Linking.
Require Import Values Memory Events Globalenvs Smallstep.
Require Import Op Locations LTL.
Require Import Tunneling.

Definition match_prog (p tp: program) :=
  match_program (fun ctx f tf => tf = tunnel_fundef f) eq p tp.

Lemma transf_program_match:
  forall p, match_prog p (tunnel_program p).
Proof.
  intros. eapply match_transform_program; eauto.
Qed.

Properties of the branch map computed using union-find.


A variant of record_goto that also incrementally computes a measure f: node -> nat counting the number of Lnop instructions starting at a given pc that were eliminated.

Definition measure_edge (u: U.t) (pc s: node) (f: node -> nat) : node -> nat :=
  fun x => if peq (U.repr u s) pc then f x
           else if peq (U.repr u x) pc then (f x + f s + 1)%nat
           else f x.

Definition record_goto' (uf: U.t * (node -> nat)) (pc: node) (b: bblock) : U.t * (node -> nat) :=
  match b with
  | Lbranch s :: b' => let (u, f) := uf in (U.union u pc s, measure_edge u pc s f)
  | _ => uf
  end.

Definition branch_map_correct (c: code) (uf: U.t * (node -> nat)): Prop :=
  forall pc,
  match c!pc with
  | Some(Lbranch s :: b) =>
      U.repr (fst uf) pc = pc \/ (U.repr (fst uf) pc = U.repr (fst uf) s /\ snd uf s < snd uf pc)%nat
  | _ =>
      U.repr (fst uf) pc = pc
  end.

Lemma record_gotos'_correct:
  forall c,
  branch_map_correct c (PTree.fold record_goto' c (U.empty, fun (x: node) => O)).
Proof.
  intros.
  apply PTree_Properties.fold_rec with (P := fun c uf => branch_map_correct c uf).

- (* extensionality *)
  intros. red; intros. rewrite <- H. apply H0.

- (* base case *)
  red; intros; simpl. rewrite PTree.gempty. apply U.repr_empty.

- (* inductive case *)
  intros m uf pc bb; intros. destruct uf as [u f].
  assert (PC: U.repr u pc = pc).
    generalize (H1 pc). rewrite H. auto.
  assert (record_goto' (u, f) pc bb = (u, f)
          \/ exists s, exists bb', bb = Lbranch s :: bb' /\ record_goto' (u, f) pc bb = (U.union u pc s, measure_edge u pc s f)).
    unfold record_goto'; simpl. destruct bb; auto. destruct i; auto. right. exists s; exists bb; auto.
  destruct H2 as [B | [s [bb' [EQ B]]]].

+ (* u and f are unchanged *)
  rewrite B.
  red. intro pc'. simpl. rewrite PTree.gsspec. destruct (peq pc' pc). subst pc'.
  destruct bb; auto. destruct i; auto.
  apply H1.

+ (* b is Lbranch s, u becomes union u pc s, f becomes measure_edge u pc s f *)
  rewrite B.
  red. intro pc'. simpl. rewrite PTree.gsspec. destruct (peq pc' pc). subst pc'. rewrite EQ.

* (* The new instruction *)
  rewrite (U.repr_union_2 u pc s); auto. rewrite U.repr_union_3.
  unfold measure_edge. destruct (peq (U.repr u s) pc). auto. right. split. auto.
  rewrite PC. rewrite peq_true. omega.

* (* An old instruction *)
  assert (U.repr u pc' = pc' -> U.repr (U.union u pc s) pc' = pc').
  { intro. rewrite <- H2 at 2. apply U.repr_union_1. congruence. }
  generalize (H1 pc'). simpl. destruct (m!pc'); auto. destruct b; auto. destruct i; auto.
  intros [P | [P Q]]. left; auto. right.
  split. apply U.sameclass_union_2. auto.
  unfold measure_edge. destruct (peq (U.repr u s) pc). auto.
  rewrite P. destruct (peq (U.repr u s0) pc). omega. auto.
Qed.

Definition record_gotos' (f: function) :=
  PTree.fold record_goto' f.(fn_code) (U.empty, fun (x: node) => O).

Lemma record_gotos_gotos':
  forall f, fst (record_gotos' f) = record_gotos f.
Proof.
  intros. unfold record_gotos', record_gotos.
  repeat rewrite PTree.fold_spec.
  generalize (PTree.elements (fn_code f)) (U.empty) (fun _ : node => O).
  induction l; intros; simpl.
  auto.
  unfold record_goto' at 2. unfold record_goto at 2.
  destruct (snd a). apply IHl. destruct i; apply IHl.
Qed.

Definition branch_target (f: function) (pc: node) : node :=
  U.repr (record_gotos f) pc.

Definition count_gotos (f: function) (pc: node) : nat :=
  snd (record_gotos' f) pc.

Theorem record_gotos_correct:
  forall f pc,
  match f.(fn_code)!pc with
  | Some(Lbranch s :: b) =>
       branch_target f pc = pc \/
       (branch_target f pc = branch_target f s /\ count_gotos f s < count_gotos f pc)%nat
  | _ => branch_target f pc = pc
  end.
Proof.
  intros.
  generalize (record_gotos'_correct f.(fn_code) pc). simpl.
  fold (record_gotos' f). unfold branch_map_correct, branch_target, count_gotos.
  rewrite record_gotos_gotos'. auto.
Qed.

Preservation of semantics


Section PRESERVATION.

Variables prog tprog: program.
Hypothesis TRANSL: match_prog prog tprog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.

Lemma functions_translated:
  forall v f,
  Genv.find_funct ge v = Some f ->
  Genv.find_funct tge v = Some (tunnel_fundef f).
Proof (Genv.find_funct_transf TRANSL).

Lemma function_ptr_translated:
  forall v f,
  Genv.find_funct_ptr ge v = Some f ->
  Genv.find_funct_ptr tge v = Some (tunnel_fundef f).
Proof (Genv.find_funct_ptr_transf TRANSL).

Lemma symbols_preserved:
  forall id,
  Genv.find_symbol tge id = Genv.find_symbol ge id.
Proof (Genv.find_symbol_transf TRANSL).

Lemma senv_preserved:
  Senv.equiv ge tge.
Proof (Genv.senv_transf TRANSL).

Lemma sig_preserved:
  forall f, funsig (tunnel_fundef f) = funsig f.
Proof.
  destruct f; reflexivity.
Qed.

The proof of semantic preservation is a simulation argument based on diagrams of the following form:
           st1 --------------- st2
            |                   |
           t|                  ?|t
            |                   |
            v                   v
           st1'--------------- st2'
The match_states predicate, defined below, captures the precondition between states st1 and st2, as well as the postcondition between st1' and st2'. One transition in the source code (left) can correspond to zero or one transition in the transformed code (right). The "zero transition" case occurs when executing a Lgoto instruction in the source code that has been removed by tunneling. In the definition of match_states, what changes between the original and transformed codes is mainly the control-flow (in particular, the current program point pc), but also some values and memory states, since some Vundef values can become more defined as a consequence of eliminating useless Lcond instructions.

Definition tunneled_block (f: function) (b: bblock) :=
  tunnel_block (record_gotos f) b.

Definition tunneled_code (f: function) :=
  PTree.map1 (tunneled_block f) (fn_code f).

Definition locmap_lessdef (ls1 ls2: locset) : Prop :=
  forall l, Val.lessdef (ls1 l) (ls2 l).

Inductive match_stackframes: stackframe -> stackframe -> Prop :=
  | match_stackframes_intro:
      forall f sp ls0 bb tls0,
      locmap_lessdef ls0 tls0 ->
      match_stackframes
         (Stackframe f sp ls0 bb)
         (Stackframe (tunnel_function f) sp tls0 (tunneled_block f bb)).

Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp pc ls m ts tls tm
        (STK: list_forall2 match_stackframes s ts)
        (LS: locmap_lessdef ls tls)
        (MEM: Mem.extends m tm),
      match_states (State s f sp pc ls m)
                   (State ts (tunnel_function f) sp (branch_target f pc) tls tm)
  | match_states_block:
      forall s f sp bb ls m ts tls tm
        (STK: list_forall2 match_stackframes s ts)
        (LS: locmap_lessdef ls tls)
        (MEM: Mem.extends m tm),
      match_states (Block s f sp bb ls m)
                   (Block ts (tunnel_function f) sp (tunneled_block f bb) tls tm)
  | match_states_interm:
      forall s f sp pc bb ls m ts tls tm
        (STK: list_forall2 match_stackframes s ts)
        (LS: locmap_lessdef ls tls)
        (MEM: Mem.extends m tm),
      match_states (Block s f sp (Lbranch pc :: bb) ls m)
                   (State ts (tunnel_function f) sp (branch_target f pc) tls tm)
  | match_states_call:
      forall s f ls m ts tls tm
        (STK: list_forall2 match_stackframes s ts)
        (LS: locmap_lessdef ls tls)
        (MEM: Mem.extends m tm),
      match_states (Callstate s f ls m)
                   (Callstate ts (tunnel_fundef f) tls tm)
  | match_states_return:
      forall s ls m ts tls tm
        (STK: list_forall2 match_stackframes s ts)
        (LS: locmap_lessdef ls tls)
        (MEM: Mem.extends m tm),
      match_states (Returnstate s ls m)
                   (Returnstate ts tls tm).

Properties of locmap_lessdef

Lemma reglist_lessdef:
  forall rl ls1 ls2,
  locmap_lessdef ls1 ls2 -> Val.lessdef_list (reglist ls1 rl) (reglist ls2 rl).
Proof.
  induction rl; simpl; intros; auto.
Qed.

Lemma locmap_set_lessdef:
  forall ls1 ls2 v1 v2 l,
  locmap_lessdef ls1 ls2 -> Val.lessdef v1 v2 -> locmap_lessdef (Locmap.set l v1 ls1) (Locmap.set l v2 ls2).
Proof.
  intros; red; intros l'. unfold Locmap.set. destruct (Loc.eq l l').
- destruct l; auto using Val.load_result_lessdef.
- destruct (Loc.diff_dec l l'); auto.
Qed.

Lemma locmap_set_undef_lessdef:
  forall ls1 ls2 l,
  locmap_lessdef ls1 ls2 -> locmap_lessdef (Locmap.set l Vundef ls1) ls2.
Proof.
  intros; red; intros l'. unfold Locmap.set. destruct (Loc.eq l l').
- destruct l; auto. destruct ty; auto.
- destruct (Loc.diff_dec l l'); auto.
Qed.

Lemma locmap_undef_regs_lessdef:
  forall rl ls1 ls2,
  locmap_lessdef ls1 ls2 -> locmap_lessdef (undef_regs rl ls1) (undef_regs rl ls2).
Proof.
  induction rl as [ | r rl]; intros; simpl. auto. apply locmap_set_lessdef; auto.
Qed.

Lemma locmap_undef_regs_lessdef_1:
  forall rl ls1 ls2,
  locmap_lessdef ls1 ls2 -> locmap_lessdef (undef_regs rl ls1) ls2.
Proof.
  induction rl as [ | r rl]; intros; simpl. auto. apply locmap_set_undef_lessdef; auto.
Qed.


Lemma locmap_getpair_lessdef:
  forall p ls1 ls2,
  locmap_lessdef ls1 ls2 -> Val.lessdef (Locmap.getpair p ls1) (Locmap.getpair p ls2).
Proof.
  intros; destruct p; simpl; auto using Val.longofwords_lessdef.
Qed.

Lemma locmap_getpairs_lessdef:
  forall pl ls1 ls2,
  locmap_lessdef ls1 ls2 ->
  Val.lessdef_list (map (fun p => Locmap.getpair p ls1) pl) (map (fun p => Locmap.getpair p ls2) pl).
Proof.
  intros. induction pl; simpl; auto using locmap_getpair_lessdef.
Qed.

Lemma locmap_setpair_lessdef:
  forall p ls1 ls2 v1 v2,
  locmap_lessdef ls1 ls2 -> Val.lessdef v1 v2 -> locmap_lessdef (Locmap.setpair p v1 ls1) (Locmap.setpair p v2 ls2).
Proof.
  intros; destruct p; simpl; auto using locmap_set_lessdef, Val.loword_lessdef, Val.hiword_lessdef.
Qed.

Lemma locmap_setres_lessdef:
  forall res ls1 ls2 v1 v2,
  locmap_lessdef ls1 ls2 -> Val.lessdef v1 v2 -> locmap_lessdef (Locmap.setres res v1 ls1) (Locmap.setres res v2 ls2).
Proof.
  induction res; intros; simpl; auto using locmap_set_lessdef, Val.loword_lessdef, Val.hiword_lessdef.
Qed.

Lemma find_function_translated:
  forall ros ls tls fd,
  locmap_lessdef ls tls ->
  find_function ge ros ls = Some fd ->
  find_function tge ros tls = Some (tunnel_fundef fd).
Proof.
  intros. destruct ros; simpl in *.
- assert (E: tls (R m) = ls (R m)).
  { exploit Genv.find_funct_inv; eauto. intros (b & EQ).
    generalize (H (R m)). rewrite EQ. intros LD; inv LD. auto. }
  rewrite E. apply functions_translated; auto.
- rewrite symbols_preserved. destruct (Genv.find_symbol ge i); inv H0.
  apply function_ptr_translated; auto.
Qed.

Lemma call_regs_lessdef:
  forall ls1 ls2, locmap_lessdef ls1 ls2 -> locmap_lessdef (call_regs ls1) (call_regs ls2).
Proof.
  intros; red; intros. destruct l as [r | [] ofs ty]; simpl; auto.
Qed.

Lemma return_regs_lessdef:
  forall caller1 callee1 caller2 callee2,
  locmap_lessdef caller1 caller2 ->
  locmap_lessdef callee1 callee2 ->
  locmap_lessdef (return_regs caller1 callee1) (return_regs caller2 callee2).
Proof.
  intros; red; intros. destruct l; simpl.
- destruct (Conventions1.is_callee_save r); auto.
- auto.
Qed.

To preserve non-terminating behaviours, we show that the transformed code cannot take an infinity of "zero transition" cases. We use the following measure function over source states, which decreases strictly in the "zero transition" case.

Definition measure (st: state) : nat :=
  match st with
  | State s f sp pc ls m => (count_gotos f pc * 2)%nat
  | Block s f sp (Lbranch pc :: _) ls m => (count_gotos f pc * 2 + 1)%nat
  | Block s f sp bb ls m => 0%nat
  | Callstate s f ls m => 0%nat
  | Returnstate s ls m => 0%nat
  end.

Lemma match_parent_locset:
  forall s ts,
  list_forall2 match_stackframes s ts ->
  locmap_lessdef (parent_locset s) (parent_locset ts).
Proof.
  induction 1; simpl.
- red; auto.
- inv H; auto.
Qed.

Lemma tunnel_step_correct:
  forall st1 t st2, step ge st1 t st2 ->
  forall st1' (MS: match_states st1 st1'),
  (exists st2', step tge st1' t st2' /\ match_states st2 st2')
  \/ (measure st2 < measure st1 /\ t = E0 /\ match_states st2 st1')%nat.
Proof.
  induction 1; intros; try inv MS.

- (* entering a block *)
  assert (DEFAULT: branch_target f pc = pc ->
    (exists st2' : state,
     step tge (State ts (tunnel_function f) sp (branch_target f pc) tls tm) E0 st2'
     /\ match_states (Block s f sp bb rs m) st2')).
  { intros. rewrite H0. econstructor; split.
    econstructor. simpl. rewrite PTree.gmap1. rewrite H. simpl. eauto.
    econstructor; eauto. }

  generalize (record_gotos_correct f pc). rewrite H.
  destruct bb; auto. destruct i; auto.
  intros [A | [B C]]. auto.
  right. split. simpl. omega.
  split. auto.
  rewrite B. econstructor; eauto.

- (* Lop *)
  exploit eval_operation_lessdef. apply reglist_lessdef; eauto. eauto. eauto.
  intros (tv & EV & LD).
  left; simpl; econstructor; split.
  eapply exec_Lop with (v := tv); eauto.
  rewrite <- EV. apply eval_operation_preserved. exact symbols_preserved.
  econstructor; eauto using locmap_set_lessdef, locmap_undef_regs_lessdef.
- (* Lload *)
  exploit eval_addressing_lessdef. apply reglist_lessdef; eauto. eauto.
  intros (ta & EV & LD).
  exploit Mem.loadv_extends. eauto. eauto. eexact LD.
  intros (tv & LOAD & LD').
  left; simpl; econstructor; split.
  eapply exec_Lload with (a := ta).
  rewrite <- EV. apply eval_addressing_preserved. exact symbols_preserved.
  eauto. eauto.
  econstructor; eauto using locmap_set_lessdef, locmap_undef_regs_lessdef.
- (* Lgetstack *)
  left; simpl; econstructor; split.
  econstructor; eauto.
  econstructor; eauto using locmap_set_lessdef, locmap_undef_regs_lessdef.
- (* Lsetstack *)
  left; simpl; econstructor; split.
  econstructor; eauto.
  econstructor; eauto using locmap_set_lessdef, locmap_undef_regs_lessdef.
- (* Lstore *)
  exploit eval_addressing_lessdef. apply reglist_lessdef; eauto. eauto.
  intros (ta & EV & LD).
  exploit Mem.storev_extends. eauto. eauto. eexact LD. apply LS.
  intros (tm' & STORE & MEM').
  left; simpl; econstructor; split.
  eapply exec_Lstore with (a := ta).
  rewrite <- EV. apply eval_addressing_preserved. exact symbols_preserved.
  eauto. eauto.
  econstructor; eauto using locmap_undef_regs_lessdef.
- (* Lcall *)
  left; simpl; econstructor; split.
  eapply exec_Lcall with (fd := tunnel_fundef fd); eauto.
  eapply find_function_translated; eauto.
  rewrite sig_preserved. auto.
  econstructor; eauto.
  constructor; auto.
  constructor; auto.
- (* Ltailcall *)
  exploit Mem.free_parallel_extends. eauto. eauto. intros (tm' & FREE & MEM').
  left; simpl; econstructor; split.
  eapply exec_Ltailcall with (fd := tunnel_fundef fd); eauto.
  eapply find_function_translated; eauto using return_regs_lessdef, match_parent_locset.
  apply sig_preserved.
  econstructor; eauto using return_regs_lessdef, match_parent_locset.
- (* Lbuiltin *)
  exploit eval_builtin_args_lessdef. eexact LS. eauto. eauto. intros (tvargs & EVA & LDA).
  exploit external_call_mem_extends; eauto. intros (tvres & tm' & A & B & C & D).
  left; simpl; econstructor; split.
  eapply exec_Lbuiltin; eauto.
  eapply eval_builtin_args_preserved with (ge1 := ge); eauto. exact symbols_preserved.
  eapply external_call_symbols_preserved. apply senv_preserved. eauto.
  econstructor; eauto using locmap_setres_lessdef, locmap_undef_regs_lessdef.
- (* Lbranch (preserved) *)
  left; simpl; econstructor; split.
  eapply exec_Lbranch; eauto.
  fold (branch_target f pc). econstructor; eauto.
- (* Lbranch (eliminated) *)
  right; split. simpl. omega. split. auto. constructor; auto.

- (* Lcond *)
  simpl tunneled_block.
  set (s1 := U.repr (record_gotos f) pc1). set (s2 := U.repr (record_gotos f) pc2).
  destruct (peq s1 s2).
+ left; econstructor; split.
  eapply exec_Lbranch.
  destruct b.
* constructor; eauto using locmap_undef_regs_lessdef_1.
* rewrite e. constructor; eauto using locmap_undef_regs_lessdef_1.
+ left; econstructor; split.
  eapply exec_Lcond; eauto. eapply eval_condition_lessdef; eauto using reglist_lessdef.
  destruct b; econstructor; eauto using locmap_undef_regs_lessdef.

- (* Ljumptable *)
  assert (tls (R arg) = Vint n).
  { generalize (LS (R arg)); rewrite H; intros LD; inv LD; auto. }
  left; simpl; econstructor; split.
  eapply exec_Ljumptable.
  eauto. rewrite list_nth_z_map. change U.elt with node. rewrite H0. reflexivity. eauto.
  econstructor; eauto using locmap_undef_regs_lessdef.
- (* Lreturn *)
  exploit Mem.free_parallel_extends. eauto. eauto. intros (tm' & FREE & MEM').
  left; simpl; econstructor; split.
  eapply exec_Lreturn; eauto.
  constructor; eauto using return_regs_lessdef, match_parent_locset.
- (* internal function *)
  exploit Mem.alloc_extends. eauto. eauto. apply Z.le_refl. apply Z.le_refl.
  intros (tm' & ALLOC & MEM').
  left; simpl; econstructor; split.
  eapply exec_function_internal; eauto.
  simpl. econstructor; eauto using locmap_undef_regs_lessdef, call_regs_lessdef.
- (* external function *)
  exploit external_call_mem_extends; eauto using locmap_getpairs_lessdef.
  intros (tvres & tm' & A & B & C & D).
  left; simpl; econstructor; split.
  eapply exec_function_external; eauto.
  eapply external_call_symbols_preserved; eauto. apply senv_preserved.
  simpl. econstructor; eauto using locmap_setpair_lessdef.
- (* return *)
  inv STK. inv H1.
  left; econstructor; split.
  eapply exec_return; eauto.
  constructor; auto.
Qed.

Lemma transf_initial_states:
  forall st1, initial_state prog st1 ->
  exists st2, initial_state tprog st2 /\ match_states st1 st2.
Proof.
  intros. inversion H.
  exists (Callstate nil (tunnel_fundef f) (Locmap.init Vundef) m0); split.
  econstructor; eauto.
  apply (Genv.init_mem_transf TRANSL); auto.
  rewrite (match_program_main TRANSL).
  rewrite symbols_preserved. eauto.
  apply function_ptr_translated; auto.
  rewrite <- H3. apply sig_preserved.
  constructor. constructor. red; simpl; auto. apply Mem.extends_refl.
Qed.

Lemma transf_final_states:
  forall st1 st2 r,
  match_states st1 st2 -> final_state st1 r -> final_state st2 r.
Proof.
  intros. inv H0. inv H. inv STK.
  set (p := map_rpair R (Conventions1.loc_result signature_main)) in *.
  generalize (locmap_getpair_lessdef p _ _ LS). rewrite H1; intros LD; inv LD.
  econstructor; eauto.
Qed.

Theorem transf_program_correct:
  forward_simulation (LTL.semantics prog) (LTL.semantics tprog).
Proof.
  eapply forward_simulation_opt.
  apply senv_preserved.
  eexact transf_initial_states.
  eexact transf_final_states.
  eexact tunnel_step_correct.
Qed.

End PRESERVATION.