Module Compiler

The whole compiler and its proof of semantic preservation

Require Import String.
Require Import Coqlib.
Require Import Errors.
Require Import AST.
Require Import Smallstep.
Languages (syntax and semantics).
Require Csyntax.
Require Csem.
Require Cstrategy.
Require Cexec.
Require Clight.
Require Csharpminor.
Require Cminor.
Require CminorSel.
Require RTL.
Require LTL.
Require Linear.
Require Mach.
Require Asm.
Translation passes.
Require Initializers.
Require SimplExpr.
Require SimplLocals.
Require Cshmgen.
Require Cminorgen.
Require Selection.
Require RTLgen.
Require Tailcall.
Require Inlining.
Require Renumber.
Require Constprop.
Require CSE.
Require Deadcode.
Require Unusedglob.
Require Allocation.
Require Tunneling.
Require Linearize.
Require CleanupLabels.
Require Debugvar.
Require Stacking.
Require Asmgen.
Proofs of semantic preservation.
Require SimplExprproof.
Require SimplLocalsproof.
Require Cshmgenproof.
Require Cminorgenproof.
Require Selectionproof.
Require RTLgenproof.
Require Tailcallproof.
Require Inliningproof.
Require Renumberproof.
Require Constpropproof.
Require CSEproof.
Require Deadcodeproof.
Require Unusedglobproof.
Require Allocproof.
Require Tunnelingproof.
Require Linearizeproof.
Require CleanupLabelsproof.
Require Debugvarproof.
Require Stackingproof.
Require Asmgenproof.
Command-line flags.
Require Import Compopts.

Pretty-printers (defined in Caml).
Parameter print_Clight: Clight.program -> unit.
Parameter print_Cminor: Cminor.program -> unit.
Parameter print_RTL: Z -> RTL.program -> unit.
Parameter print_LTL: LTL.program -> unit.
Parameter print_Mach: Mach.program -> unit.

Open Local Scope string_scope.

Composing the translation passes

We first define useful monadic composition operators, along with funny (but convenient) notations.

Definition apply_total (A B: Type) (x: res A) (f: A -> B) : res B :=
  match x with Error msg => Error msg | OK x1 => OK (f x1) end.

Definition apply_partial (A B: Type)
                         (x: res A) (f: A -> res B) : res B :=
  match x with Error msg => Error msg | OK x1 => f x1 end.

Notation "a @@@ b" :=
   (apply_partial _ _ a b) (at level 50, left associativity).
Notation "a @@ b" :=
   (apply_total _ _ a b) (at level 50, left associativity).

Definition print {A: Type} (printer: A -> unit) (prog: A) : A :=
  let unused := printer prog in prog.

Definition time {A B: Type} (name: string) (f: A -> B) : A -> B := f.

Definition total_if {A: Type}
          (flag: unit -> bool) (f: A -> A) (prog: A) : A :=
  if flag tt then f prog else prog.

Definition partial_if {A: Type}
          (flag: unit -> bool) (f: A -> res A) (prog: A) : res A :=
  if flag tt then f prog else OK prog.

We define three translation functions for whole programs: one starting with a C program, one with a Cminor program, one with an RTL program. The three translations produce Asm programs ready for pretty-printing and assembling.

Definition transf_rtl_program (f: RTL.program) : res Asm.program :=
   OK f
   @@ print (print_RTL 0)
   @@ total_if Compopts.optim_tailcalls (time "Tail calls" Tailcall.transf_program)
   @@ print (print_RTL 1)
  @@@ time "Inlining" Inlining.transf_program
   @@ print (print_RTL 2)
   @@ time "Renumbering" Renumber.transf_program
   @@ print (print_RTL 3)
   @@ total_if Compopts.optim_constprop (time "Constant propagation" Constprop.transf_program)
   @@ print (print_RTL 4)
   @@ total_if Compopts.optim_constprop (time "Renumbering" Renumber.transf_program)
   @@ print (print_RTL 5)
  @@@ partial_if Compopts.optim_CSE (time "CSE" CSE.transf_program)
   @@ print (print_RTL 6)
  @@@ partial_if Compopts.optim_redundancy (time "Redundancy elimination" Deadcode.transf_program)
   @@ print (print_RTL 7)
  @@@ time "Unused globals" Unusedglob.transform_program
   @@ print (print_RTL 8)
  @@@ time "Register allocation" Allocation.transf_program
   @@ print print_LTL
   @@ time "Branch tunneling" Tunneling.tunnel_program
  @@@ time "CFG linearization" Linearize.transf_program
   @@ time "Label cleanup" CleanupLabels.transf_program
  @@@ partial_if Compopts.debug (time "Debugging info for local variables" Debugvar.transf_program)
  @@@ time "Mach generation" Stacking.transf_program
   @@ print print_Mach
  @@@ time "Asm generation" Asmgen.transf_program.

Definition transf_cminor_program (p: Cminor.program) : res Asm.program :=
   OK p
   @@ print print_Cminor
  @@@ time "Instruction selection" Selection.sel_program
  @@@ time "RTL generation" RTLgen.transl_program
  @@@ transf_rtl_program.

Definition transf_clight_program (p: Clight.program) : res Asm.program :=
  OK p
   @@ print print_Clight
  @@@ time "Simplification of locals" SimplLocals.transf_program
  @@@ time "C#minor generation" Cshmgen.transl_program
  @@@ time "Cminor generation" Cminorgen.transl_program
  @@@ transf_cminor_program.

Definition transf_c_program (p: Csyntax.program) : res Asm.program :=
  OK p
  @@@ time "Clight generation" SimplExpr.transl_program
  @@@ transf_clight_program.

Force Initializers and Cexec to be extracted as well.

Definition transl_init := Initializers.transl_init.
Definition cexec_do_step := Cexec.do_step.

The following lemmas help reason over compositions of passes.

Lemma print_identity:
  forall (A: Type) (printer: A -> unit) (prog: A),
  print printer prog = prog.
  intros; unfold print. destruct (printer prog); auto.

Lemma compose_print_identity:
  forall (A: Type) (x: res A) (f: A -> unit),
  x @@ print f = x.
  intros. destruct x; simpl. rewrite print_identity. auto. auto.

Remark forward_simulation_identity:
  forall sem, forward_simulation sem sem.
  intros. apply forward_simulation_step with (fun s1 s2 => s2 = s1); intros.
- auto.
- exists s1; auto.
- subst s2; auto.
- subst s2. exists s1'; auto.

Lemma total_if_simulation:
  forall (A: Type) (sem: A -> semantics) (flag: unit -> bool) (f: A -> A) (prog: A),
  (forall p, forward_simulation (sem p) (sem (f p))) ->
  forward_simulation (sem prog) (sem (total_if flag f prog)).
  intros. unfold total_if. destruct (flag tt). auto. apply forward_simulation_identity.

Lemma partial_if_simulation:
  forall (A: Type) (sem: A -> semantics) (flag: unit -> bool) (f: A -> res A) (prog tprog: A),
  partial_if flag f prog = OK tprog ->
  (forall p tp, f p = OK tp -> forward_simulation (sem p) (sem tp)) ->
  forward_simulation (sem prog) (sem tprog).
  intros. unfold partial_if in *. destruct (flag tt). eauto. inv H. apply forward_simulation_identity.

Semantic preservation

We prove that the transf_program translations preserve semantics by constructing the following simulations: These results establish the correctness of the whole compiler!

Theorem transf_rtl_program_correct:
  forall p tp,
  transf_rtl_program p = OK tp ->
  forward_simulation (RTL.semantics p) (Asm.semantics tp)
  * backward_simulation (RTL.semantics p) (Asm.semantics tp).
  assert (F: forward_simulation (RTL.semantics p) (Asm.semantics tp)).
  unfold transf_rtl_program, time in H.
  repeat rewrite compose_print_identity in H.
  simpl in H.
  set (p1 := total_if optim_tailcalls Tailcall.transf_program p) in *.
  destruct (Inlining.transf_program p1) as [p11|] eqn:?; simpl in H; try discriminate.
  set (p12 := Renumber.transf_program p11) in *.
  set (p2 := total_if optim_constprop Constprop.transf_program p12) in *.
  set (p21 := total_if optim_constprop Renumber.transf_program p2) in *.
  destruct (partial_if optim_CSE CSE.transf_program p21) as [p3|] eqn:?; simpl in H; try discriminate.
  destruct (partial_if optim_redundancy Deadcode.transf_program p3) as [p31|] eqn:?; simpl in H; try discriminate.
  destruct (Unusedglob.transform_program p31) as [p32|] eqn:?; simpl in H; try discriminate.
  destruct (Allocation.transf_program p32) as [p4|] eqn:?; simpl in H; try discriminate.
  set (p5 := Tunneling.tunnel_program p4) in *.
  destruct (Linearize.transf_program p5) as [p6|] eqn:?; simpl in H; try discriminate.
  set (p7 := CleanupLabels.transf_program p6) in *.
  destruct (partial_if debug Debugvar.transf_program p7) as [p71|] eqn:?; simpl in H; try discriminate.
  destruct (Stacking.transf_program p71) as [p8|] eqn:?; simpl in H; try discriminate.
  apply compose_forward_simulation with (RTL.semantics p1).
    apply total_if_simulation. apply Tailcallproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p11).
    apply Inliningproof.transf_program_correct; auto.
  apply compose_forward_simulation with (RTL.semantics p12).
    apply Renumberproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p2).
    apply total_if_simulation. apply Constpropproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p21).
    apply total_if_simulation. apply Renumberproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p3).
    eapply partial_if_simulation; eauto. apply CSEproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p31).
    eapply partial_if_simulation; eauto. apply Deadcodeproof.transf_program_correct.
  apply compose_forward_simulation with (RTL.semantics p32).
    apply Unusedglobproof.transf_program_correct; auto.
  apply compose_forward_simulation with (LTL.semantics p4).
    apply Allocproof.transf_program_correct; auto.
  apply compose_forward_simulation with (LTL.semantics p5).
    apply Tunnelingproof.transf_program_correct.
  apply compose_forward_simulation with (Linear.semantics p6).
    apply Linearizeproof.transf_program_correct; auto.
  apply compose_forward_simulation with (Linear.semantics p7).
    apply CleanupLabelsproof.transf_program_correct.
  apply compose_forward_simulation with (Linear.semantics p71).
    eapply partial_if_simulation; eauto. apply Debugvarproof.transf_program_correct.
  apply compose_forward_simulation with (Mach.semantics Asmgenproof0.return_address_offset p8).
    apply Stackingproof.transf_program_correct.
    exact Asmgenproof.return_address_exists.
  apply Asmgenproof.transf_program_correct; eauto.
  split. auto.
  apply forward_to_backward_simulation. auto.
  apply RTL.semantics_receptive.
  apply Asm.semantics_determinate.

Theorem transf_cminor_program_correct:
  forall p tp,
  transf_cminor_program p = OK tp ->
  forward_simulation (Cminor.semantics p) (Asm.semantics tp)
  * backward_simulation (Cminor.semantics p) (Asm.semantics tp).
  assert (F: forward_simulation (Cminor.semantics p) (Asm.semantics tp)).
  unfold transf_cminor_program, time in H.
  repeat rewrite compose_print_identity in H.
  simpl in H.
  destruct (Selection.sel_program p) as [p1|] eqn:?; simpl in H; try discriminate.
  destruct (RTLgen.transl_program p1) as [p2|] eqn:?; simpl in H; try discriminate.
  eapply compose_forward_simulation. apply Selectionproof.transf_program_correct. eauto.
  eapply compose_forward_simulation. apply RTLgenproof.transf_program_correct. eassumption.
  exact (fst (transf_rtl_program_correct _ _ H)).

  split. auto.
  apply forward_to_backward_simulation. auto.
  apply Cminor.semantics_receptive.
  apply Asm.semantics_determinate.

Theorem transf_clight_program_correct:
  forall p tp,
  transf_clight_program p = OK tp ->
  forward_simulation (Clight.semantics1 p) (Asm.semantics tp)
  * backward_simulation (Clight.semantics1 p) (Asm.semantics tp).
  assert (F: forward_simulation (Clight.semantics1 p) (Asm.semantics tp)).
  revert H; unfold transf_clight_program, time; simpl.
  rewrite print_identity.
  caseEq (SimplLocals.transf_program p); simpl; try congruence; intros p0 EQ0.
  caseEq (Cshmgen.transl_program p0); simpl; try congruence; intros p1 EQ1.
  caseEq (Cminorgen.transl_program p1); simpl; try congruence; intros p2 EQ2.
  intros EQ3.
  eapply compose_forward_simulation. apply SimplLocalsproof.transf_program_correct. eauto.
  eapply compose_forward_simulation. apply Cshmgenproof.transl_program_correct. eauto.
  eapply compose_forward_simulation. apply Cminorgenproof.transl_program_correct. eauto.
  exact (fst (transf_cminor_program_correct _ _ EQ3)).

  split. auto.
  apply forward_to_backward_simulation. auto.
  apply Clight.semantics_receptive.
  apply Asm.semantics_determinate.

Theorem transf_cstrategy_program_correct:
  forall p tp,
  transf_c_program p = OK tp ->
  forward_simulation (Cstrategy.semantics p) (Asm.semantics tp)
  * backward_simulation (atomic (Cstrategy.semantics p)) (Asm.semantics tp).
  assert (F: forward_simulation (Cstrategy.semantics p) (Asm.semantics tp)).
  revert H; unfold transf_c_program, time; simpl.
  caseEq (SimplExpr.transl_program p); simpl; try congruence; intros p0 EQ0.
  intros EQ1.
  eapply compose_forward_simulation. apply SimplExprproof.transl_program_correct. eauto.
  exact (fst (transf_clight_program_correct _ _ EQ1)).

  split. auto.
  apply forward_to_backward_simulation.
  apply factor_forward_simulation. auto. eapply sd_traces. eapply Asm.semantics_determinate.
  apply atomic_receptive. apply Cstrategy.semantics_strongly_receptive.
  apply Asm.semantics_determinate.

Theorem transf_c_program_correct:
  forall p tp,
  transf_c_program p = OK tp ->
  backward_simulation (Csem.semantics p) (Asm.semantics tp).
  apply compose_backward_simulation with (atomic (Cstrategy.semantics p)).
  eapply sd_traces; eapply Asm.semantics_determinate.
  apply factor_backward_simulation.
  apply Cstrategy.strategy_simulation.
  apply Csem.semantics_single_events.
  eapply ssr_well_behaved; eapply Cstrategy.semantics_strongly_receptive.
  exact (snd (transf_cstrategy_program_correct _ _ H)).