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CompilerCorrectness: a new simulation condition to get trace equivalence for free
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Adam Chlipala
parent
2832696faa
commit
7c705bb2fb
1 changed files with 124 additions and 75 deletions
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@ -174,6 +174,74 @@ Fixpoint cfoldExprs (c : cmd) : cmd :=
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| Output e => Output (cfoldArith e)
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end.
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Theorem skip_or_step : forall v c,
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c = Skip
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\/ exists v' l c', cstep (v, c) l (v', c').
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Proof.
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induct c; simplify; first_order; subst;
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try match goal with
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| [ H : cstep _ _ _ |- _ ] => invert H
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end;
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try match goal with
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| [ |- context[cstep (?v, If ?e _ _)] ] => cases (interp e v ==n 0)
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| [ |- context[cstep (?v, While ?e _)] ] => cases (interp e v ==n 0)
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end; eauto 10.
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Qed.
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Lemma deterministic0 : forall vc l vc',
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step0 vc l vc'
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-> forall l' vc'', step0 vc l' vc''
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-> l = l' /\ vc'' = vc'.
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Proof.
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invert 1; invert 1; simplify; propositional.
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Qed.
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Theorem plug_function : forall C c1 c2, plug C c1 c2
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-> forall c2', plug C c1 c2'
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-> c2 = c2'.
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Proof.
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induct 1; invert 1; eauto.
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apply IHplug in H5.
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equality.
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Qed.
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Lemma peel_cseq : forall C1 C2 c (c1 c2 : cmd),
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C1 = C2 /\ c1 = c2
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-> CSeq C1 c = CSeq C2 c /\ c1 = c2.
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Proof.
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equality.
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Qed.
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Hint Resolve peel_cseq.
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Lemma plug_deterministic : forall v C c1 c2, plug C c1 c2
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-> forall l vc1, step0 (v, c1) l vc1
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-> forall C' c1', plug C' c1' c2
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-> forall l' vc1', step0 (v, c1') l' vc1'
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-> C' = C /\ c1' = c1.
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Proof.
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induct 1; invert 1; invert 1; invert 1; auto;
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try match goal with
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| [ H : plug _ _ _ |- _ ] => invert1 H
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end; eauto.
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Qed.
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Theorem deterministic : forall vc l vc',
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cstep vc l vc'
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-> forall l' vc'', cstep vc l' vc''
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-> l = l' /\ vc' = vc''.
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Proof.
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invert 1; invert 1; simplify.
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eapply plug_deterministic in H0; eauto.
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invert H0.
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eapply deterministic0 in H1; eauto.
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propositional; subst; auto.
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invert H0.
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auto.
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eapply plug_function in H2; eauto.
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equality.
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Qed.
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Section simulation.
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Variable R : valuation * cmd -> valuation * cmd -> Prop.
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@ -181,7 +249,10 @@ Section simulation.
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-> forall vc1' l, cstep vc1 l vc1'
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-> exists vc2', cstep vc2 l vc2' /\ R vc1' vc2'.
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Lemma simulation' : forall vc1 ns, generate vc1 ns
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Hypothesis agree_on_termination : forall v1 v2 c2, R (v1, Skip) (v2, c2)
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-> c2 = Skip.
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Lemma simulation_fwd' : forall vc1 ns, generate vc1 ns
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-> forall vc2, R vc1 vc2
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-> generate vc2 ns.
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Proof.
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@ -196,14 +267,59 @@ Section simulation.
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eauto.
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Qed.
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Theorem simulation : forall vc1 vc2, R vc1 vc2
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Theorem simulation_fwd : forall vc1 vc2, R vc1 vc2
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-> vc1 <| vc2.
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Proof.
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unfold traceInclusion; eauto using simulation'.
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unfold traceInclusion; eauto using simulation_fwd'.
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Qed.
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Lemma simulation_bwd' : forall vc2 ns, generate vc2 ns
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-> forall vc1, R vc1 vc2
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-> generate vc1 ns.
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Proof.
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induct 1; simplify; eauto.
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cases vc1; cases vc.
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assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
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first_order; subst.
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apply agree_on_termination in H1; subst.
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invert H.
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invert H3.
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invert H4.
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specialize (one_step H1 H2).
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first_order.
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eapply deterministic in H; eauto.
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propositional; subst.
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eauto.
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cases vc1; cases vc.
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assert (c = Skip \/ exists v' l c', cstep (v, c) l (v', c')) by apply skip_or_step.
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first_order; subst.
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apply agree_on_termination in H1; subst.
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invert H.
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invert H3.
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invert H4.
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specialize (one_step H1 H2).
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first_order.
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eapply deterministic in H; eauto.
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propositional; subst.
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eauto.
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Qed.
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Theorem simulation_bwd : forall vc1 vc2, R vc1 vc2
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-> vc2 <| vc1.
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Proof.
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unfold traceInclusion; eauto using simulation_bwd'.
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Qed.
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Theorem simulation : forall vc1 vc2, R vc1 vc2
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-> vc1 =| vc2.
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Proof.
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simplify; split; auto using simulation_fwd, simulation_bwd.
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Qed.
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End simulation.
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Lemma cfoldExprs_ok1' : forall v1 c1 l v2 c2,
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Lemma cfoldExprs_ok' : forall v1 c1 l v2 c2,
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step0 (v1, c1) l (v2, c2)
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-> step0 (v1, cfoldExprs c1) l (v2, cfoldExprs c2).
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Proof.
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@ -214,7 +330,7 @@ Proof.
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end; eauto.
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Qed.
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Hint Resolve cfoldExprs_ok1'.
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Hint Resolve cfoldExprs_ok'.
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Fixpoint cfoldExprsContext (C : context) : context :=
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match C with
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@ -230,8 +346,8 @@ Qed.
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Hint Resolve plug_cfoldExprs1.
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Lemma cfoldExprs_ok1 : forall v c,
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(v, c) <| (v, cfoldExprs c).
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Lemma cfoldExprs_ok : forall v c,
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(v, c) =| (v, cfoldExprs c).
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Proof.
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simplify.
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apply simulation with (R := fun vc1 vc2 => fst vc1 = fst vc2
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@ -239,74 +355,7 @@ Proof.
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simplify; propositional.
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invert H0; simplify; subst.
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apply cfoldExprs_ok1' in H3.
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apply cfoldExprs_ok' in H3.
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cases vc2; simplify; subst.
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eauto 7.
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Qed.
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Lemma cfoldExprs_ok2' : forall v1 c1 l v2 c2,
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step0 (v1, cfoldExprs c1) l (v2, c2)
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-> exists c2', step0 (v1, c1) l (v2, c2') /\ c2 = cfoldExprs c2'.
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Proof.
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invert 1;
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repeat match goal with
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| [ H : _ = cfoldExprs ?c |- _ ] => cases c; invert H
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end; repeat rewrite cfoldArith_ok in *; eauto.
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Qed.
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Hint Resolve cfoldExprs_ok2'.
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Lemma plug_cfoldExprs2 : forall C c1 c2, plug C c1 (cfoldExprs c2)
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-> exists C' c1', plug C' c1' c2
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/\ C = cfoldExprsContext C'
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/\ c1 = cfoldExprs c1'.
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Proof.
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induct 1; simplify; eauto.
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cases c2; simplify; invert x.
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specialize (IHplug _ eq_refl).
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first_order; subst.
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eauto 7.
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Qed.
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Lemma plug_cfoldExprs2' : forall C c1 c2, plug (cfoldExprsContext C) (cfoldExprs c1) c2
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-> exists c2', plug C c1 c2'
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/\ c2 = cfoldExprs c2'.
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Proof.
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induct 1; simplify; eauto.
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cases C; invert x.
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eauto.
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cases C; invert x.
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specialize (IHplug _ _ eq_refl eq_refl).
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first_order; subst.
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eauto.
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Qed.
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Lemma cfoldExprs_ok2 : forall v c,
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(v, cfoldExprs c) <| (v, c).
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Proof.
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simplify.
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apply simulation with (R := fun vc1 vc2 => fst vc1 = fst vc2
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/\ snd vc1 = cfoldExprs (snd vc2));
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simplify; propositional.
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invert H0; simplify; subst.
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eapply plug_cfoldExprs2 in H.
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first_order; subst.
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apply cfoldExprs_ok2' in H3.
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first_order; subst.
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cases vc2; simplify; subst.
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apply plug_cfoldExprs2' in H4.
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first_order; subst.
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eauto.
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Qed.
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Theorem cfoldExprs_ok : forall v c,
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(v, cfoldExprs c) =| (v, c).
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Proof.
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simplify.
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split.
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apply cfoldExprs_ok2.
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apply cfoldExprs_ok1.
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Qed.
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