The whole compiler and its proof of semantic preservation

Libraries.

Require Import Axioms.
Require Import Coqlib.
Require Import Maps.
Require Import Errors.
Require Import AST.
Require Import Values.
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 LTLin.
Require Linear.
Require Mach.
Require Machsem.
Require Asm.

Translation passes.

Require Initializers.
Require SimplExpr.
Require Cshmgen.
Require Cminorgen.
Require Selection.
Require RTLgen.
Require Tailcall.
Require Inlining.
Require Renumber.
Require Constprop.
Require CSE.
Require Allocation.
Require Tunneling.
Require Linearize.
Require CleanupLabels.
Require Reload.
Require RRE.
Require Stacking.
Require Asmgen.

Type systems.

Require RTLtyping.
Require LTLtyping.
Require LTLintyping.
Require Lineartyping.
Require Machtyping.

Proofs of semantic preservation and typing preservation.

Require SimplExprproof.
Require Cshmgenproof.
Require Cminorgenproof.
Require Selectionproof.
Require RTLgenproof.
Require Tailcallproof.
Require Inliningproof.
Require Renumberproof.
Require Constpropproof.
Require CSEproof.
Require Allocproof.
Require Alloctyping.
Require Tunnelingproof.
Require Tunnelingtyping.
Require Linearizeproof.
Require Linearizetyping.
Require CleanupLabelsproof.
Require CleanupLabelstyping.
Require Reloadproof.
Require Reloadtyping.
Require RREproof.
Require RREtyping.
Require Stackingproof.
Require Stackingtyping.
Require Asmgenproof.

Pretty-printers (defined in Caml).

Parameter print_Clight: Clight.program -> unit.
Parameter print_Cminor: Cminor.program -> unit.
Parameter print_RTL: RTL.program -> unit.
Parameter print_RTL_tailcall: RTL.program -> unit.
Parameter print_RTL_inline: RTL.program -> unit.
Parameter print_RTL_constprop: RTL.program -> unit.
Parameter print_RTL_cse: RTL.program -> unit.
Parameter print_LTLin: LTLin.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.

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.

For the loopbound analysis, we compile a C program to RTL and use it for the analysis

Definition transf_c_program_to_rtl (p: Csyntax.program) : res RTL.program :=
   OK p
  @@@ SimplExpr.transl_program
   @@ print print_Clight
  @@@ Cshmgen.transl_program
  @@@ Cminorgen.transl_program
   @@ print print_Cminor
   @@ Selection.sel_program
  @@@ RTLgen.transl_program
   @@ print print_RTL
   @@ Tailcall.transf_program
   @@ print print_RTL_tailcall
  @@@ Inlining.transf_program
   @@ Renumber.transf_program
   @@ print print_RTL_inline
   @@ Constprop.transf_program
   @@ Renumber.transf_program
   @@ print print_RTL_constprop
  @@@ CSE.transf_program
   @@ print print_RTL_cse.

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.

Proof.

intros; unfold print. destruct (printer prog); auto.
Qed.

Lemma map_partial_compose:
  forall (X A B C: Type)
         (ctx: X -> errmsg)
         (f1: A -> res B) (f2: B -> res C)
         (pa: list (X * A)) (pc: list (X * C)),
  map_partial ctx (fun f => f1 f @@@ f2) pa = OK pc ->
  { pb | map_partial ctx f1 pa = OK pb /\ map_partial ctx f2 pb = OK pc }.

Proof.

  induction pa; simpl.
  intros. inv H. econstructor; eauto.
  intro pc. destruct a as [x a].
  destruct (f1 a) as [] _eqn; simpl; try congruence.
  destruct (f2 b) as [] _eqn; simpl; try congruence.
  destruct (map_partial ctx (fun f => f1 f @@@ f2) pa) as [] _eqn; simpl; try congruence.
  intros. inv H.
  destruct (IHpa l) as [pb [P Q]]; auto.
  rewrite P; simpl.
  econstructor; split. eauto. simpl. rewrite Heqr0. rewrite Q. auto.
Qed.

Lemma transform_partial_program_compose:
  forall (A B C V: Type)
         (f1: A -> res B) (f2: B -> res C)
         (pa: program A V) (pc: program C V),
  transform_partial_program (fun f => f1 f @@@ f2) pa = OK pc ->
  { pb | transform_partial_program f1 pa = OK pb /\
         transform_partial_program f2 pb = OK pc }.

Proof.

  intros. unfold transform_partial_program in H.
  destruct (map_partial prefix_name (fun f : A => f1 f @@@ f2) (prog_funct pa)) as [] _eqn;
  simpl in H; inv H.
  destruct (map_partial_compose _ _ _ _ _ _ _ _ _ Heqr) as [xb [P Q]].
  exists (mkprogram xb (prog_main pa) (prog_vars pa)); split.
  unfold transform_partial_program. rewrite P; auto.
  unfold transform_partial_program. simpl. rewrite Q; auto.
Qed.

Lemma transform_program_partial_program:
  forall (A B V: Type) (f: A -> B) (p: program A V) (tp: program B V),
  transform_partial_program (fun x => OK (f x)) p = OK tp ->
  transform_program f p = tp.

Proof.

intros until tp. unfold transform_partial_program.
rewrite map_partial_total. simpl. intros. inv H. auto.
Qed.

Lemma transform_program_compose:
  forall (A B C V: Type)
         (f1: A -> res B) (f2: B -> C)
         (pa: program A V) (pc: program C V),
  transform_partial_program (fun f => f1 f @@ f2) pa = OK pc ->
  { pb | transform_partial_program f1 pa = OK pb /\
         transform_program f2 pb = pc }.

Proof.

  intros.
  replace (fun f : A => f1 f @@ f2)
     with (fun f : A => f1 f @@@ (fun x => OK (f2 x))) in H.
  destruct (transform_partial_program_compose _ _ _ _ _ _ _ _ H) as [pb [X Y]].
  exists pb; split. auto.
  apply transform_program_partial_program. auto.
  apply extensionality; intro. destruct(f1 x); auto.
Qed.

Lemma transform_partial_program_identity:
  forall (A V: Type) (pa pb: program A V),
  transform_partial_program (@OK A) pa = OK pb ->
  pa = pb.

Proof.

  intros until pb. unfold transform_partial_program.
  replace (@OK A) with (fun b => @OK A b).
  rewrite map_partial_identity. simpl. destruct pa; simpl; congruence.
  apply extensionality; auto.
Qed.

Lemma transform_program_print_identity:
forall (A V: Type) (p: program A V) (f: A -> unit),
transform_program (print f) p = p.

Proof.

  intros until f. unfold transform_program, transf_program.
  destruct p; simpl; f_equal.
  transitivity (map (fun x => x) prog_funct).
  apply list_map_exten; intros. destruct x; simpl. rewrite print_identity. auto.
  apply list_map_identity.
Qed.

Lemma compose_print_identity:
forall (A: Type) (x: res A) (f: A -> unit),
x @@ print f = x.

Proof.

intros. destruct x; simpl. rewrite print_identity. auto. auto.
Qed.

Semantic preservation

We prove that the transf_program translations preserve semantics by constructing the following simulations:

Forward simulations from Cstrategy / Cminor / RTL to Asm (composition of the forward simulations for each pass).
Backward simulations for the same languages (derived from the forward simulation, using receptiveness of the source language and determinacy of Asm).
Backward simulation from Csem to Asm (composition of two backward simulations).

These results establish the correctness of the whole compiler!

Ltac TransfProgInv :=
  match goal with
  | [ H: transform_partial_program (fun f => _ @@@ _) _ = OK _ |- _ ] =>
      let p := fresh "p" in let X := fresh "X" in let P := fresh "P" in
      destruct (transform_partial_program_compose _ _ _ _ _ _ _ _ H) as [p [X P]];
      clear H
  | [ H: transform_partial_program (fun f => _ @@ _) _ = OK _ |- _ ] =>
      let p := fresh "p" in let X := fresh "X" in let P := fresh "P" in
      destruct (transform_program_compose _ _ _ _ _ _ _ _ H) as [p [X P]];
      clear H
  end.

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).

Proof.

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).

Proof.

  intros.
  assert (F: forward_simulation (Cminor.semantics p) (Asm.semantics tp)).
  unfold transf_cminor_program in H.
  repeat rewrite compose_print_identity in H.
  simpl in H.
  set (p1 := Selection.sel_program p) in *.
  destruct (RTLgen.transl_program p1) as [p2|]_eqn; simpl in H; try discriminate.
  eapply compose_forward_simulation. apply Selectionproof.transf_program_correct.
  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.
Qed.

Theorem transf_clight_program_correct:
  forall p tp,
  transf_clight_program p = OK tp ->
  forward_simulation (Clight.semantics p) (Asm.semantics tp)
  * backward_simulation (Clight.semantics p) (Asm.semantics tp).

Proof.

  intros.
  assert (F: forward_simulation (Clight.semantics p) (Asm.semantics tp)).
  revert H; unfold transf_clight_program; simpl.
  rewrite print_identity.
  caseEq (Cshmgen.transl_program p); simpl; try congruence; intros p1 EQ1.
  caseEq (Cminorgen.transl_program p1); simpl; try congruence; intros p2 EQ2.
  intros EQ3.
  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.
Qed.

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).

Proof.

  intros.
  assert (F: forward_simulation (Cstrategy.semantics p) (Asm.semantics tp)).
  revert H; unfold transf_c_program; 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.
Qed.

Theorem transf_c_program_correct:
  forall p tp,
  transf_c_program p = OK tp ->
  backward_simulation (Csem.semantics p) (Asm.semantics tp).

Proof.

  intros.
  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)).
Qed.

Require GlobalBounds.
Definition bound := GlobalBounds.bound.
Definition headers := GlobalBounds.headers.
Require FullLoopbounds.
Definition full_bounds := FullLoopbounds.full_bounds.

Module Compiler

Composing the translation passes

Semantic preservation