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SharedMemory: soundness of partial-order reduction (or one particular flavor thereof)
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Adam Chlipala
parent
50baaa91fe
commit
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1 changed files with 141 additions and 10 deletions
151
SharedMemory.v
151
SharedMemory.v
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@ -458,7 +458,7 @@ Definition trsys_ofC (h : heap) (l : locks) (cs : list (cmd * summary)) := {|
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|}.
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Lemma commutes_sound : forall h l c2 h' l' c2',
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Lemma commutes_sound' : forall h l c2 h' l' c2',
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step (h, l, c2) (h', l', c2')
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-> forall s c1 h'' l'' c1', step (h', l', c1) (h'', l'', c1')
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-> summarize c2 s
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@ -519,6 +519,94 @@ Proof.
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sets.
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Qed.
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Lemma step_nextAction_Return : forall r h l c h' l' c',
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step (h, l, c) (h', l', c')
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-> nextAction c (Return r)
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-> h' = h /\ l' = l /\ (forall h'' l'', step (h'', l'', c) (h'', l'', c')).
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Proof.
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induct 1; invert 1; propositional; eauto.
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Qed.
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Lemma step_nextAction_other : forall c0 h l c h' l' c',
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step (h, l, c) (h', l', c')
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-> nextAction c c0
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-> (forall r, c0 <> Return r)
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-> exists f c0', step (h, l, c0) (h', l', c0')
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/\ c = f c0
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/\ c' = f c0'
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/\ forall h1 l1 h2 l2 c0'', step (h1, l1, c0) (h2, l2, c0'')
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-> step (h1, l1, f c0) (h2, l2, f c0'').
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Proof.
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induct 1; invert 1; first_order; subst; eauto.
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exists (fun X => x <- x X; c2 x); eauto 10.
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invert H3.
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unfold not in *; exfalso; eauto.
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exists (fun X => X); eauto.
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exists (fun X => X); eauto.
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exists (fun X => X); eauto 10.
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exists (fun X => X); eauto 10.
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Qed.
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Lemma nextAction_couldBe : forall c c0,
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nextAction c c0
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-> match c0 with
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| Return _ => True
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| Fail => True
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| Read _ => True
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| Write _ _ => True
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| Lock _ => True
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| Unlock _ => True
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| _ => False
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end.
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Proof.
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induct 1; auto.
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Qed.
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Lemma commutes_sound : forall h l c2 h' l' c2',
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step (h, l, c2) (h', l', c2')
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-> forall s c1 c0 h'' l'' c1', step (h', l', c1) (h'', l'', c1')
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-> summarize c2 s
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-> nextAction c1 c0
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-> commutes c0 s
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-> exists h1 l1, step (h, l, c1) (h1, l1, c1')
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/\ step (h1, l1, c2) (h'', l'', c2').
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Proof.
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simplify.
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assert (Hc : commutes c0 s) by assumption.
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specialize (nextAction_couldBe H2).
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cases c0; propositional.
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assert (Hs : step (h', l', c1) (h'', l'', c1')) by assumption.
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eapply step_nextAction_Return in H0; eauto; propositional; subst.
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eauto.
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eapply step_nextAction_other in H0; eauto; first_order; subst; try equality.
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eapply commutes_sound' with (c2 := c2) (c1 := Read a) in H3; eauto.
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first_order; eauto.
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eapply step_nextAction_other in H0; eauto; first_order; subst; try equality.
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eapply commutes_sound' with (c2 := c2) (c1 := Write a v) in H3; eauto.
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first_order; eauto.
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eapply step_nextAction_other in H0; eauto; first_order; subst; try equality.
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invert H0.
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eapply step_nextAction_other in H0; eauto; first_order; subst; try equality.
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eapply commutes_sound' with (c2 := c2) (c1 := Lock a) in H3; eauto.
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first_order; eauto.
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eapply step_nextAction_other in H0; eauto; first_order; subst; try equality.
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eapply commutes_sound' with (c2 := c2) (c1 := Unlock a) in H3; eauto.
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first_order; eauto.
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Qed.
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Hint Constructors summarize.
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Lemma summarize_step : forall h l c h' l' c' s,
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@ -1137,11 +1225,21 @@ Lemma commutes_noblock : forall c c0,
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-> forall c1 s, summarize c1 s
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-> commutes c0 s
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-> forall h1' l1' c1', step (h, l, c1) (h1', l1', c1')
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-> exists h'' l'' c'', step (h1', l1', c) (h'', l'', c'').
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-> exists h'' l'', step (h1', l1', c) (h'', l'', c').
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Proof.
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induct 1; invert 1.
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induct 1; simplify; eauto.
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invert H0.
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invert H0.
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invert H3; eauto.
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invert H1; eauto.
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invert H1; eauto.
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assert (a0 <> a) by sets.
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replace (h' $! a) with (h' $+ (a0, v) $! a) by (simplify; equality).
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eauto.
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invert H1; eauto.
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invert H1; eauto.
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induct 1; simplify; eauto.
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@ -1152,11 +1250,11 @@ Proof.
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invert H2; eauto.
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invert H2; eauto.
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invert H2; eauto.
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do 3 eexists.
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do 2 eexists.
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constructor.
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sets.
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invert H2; eauto.
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do 3 eexists.
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do 2 eexists.
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constructor.
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sets.
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@ -1167,11 +1265,11 @@ Proof.
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invert H2; eauto.
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invert H2; eauto.
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invert H2; eauto.
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do 3 eexists.
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do 2 eexists.
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constructor.
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sets.
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invert H2; eauto.
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do 3 eexists.
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do 2 eexists.
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constructor.
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sets.
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@ -1181,11 +1279,11 @@ Proof.
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invert H5; eauto.
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invert H3; eauto.
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invert H3.
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do 3 eexists; eapply commute_writes in H2; eauto.
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do 2 eexists; eapply commute_writes in H2; eauto.
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invert H3.
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do 3 eexists; eapply commute_locks in H2; eauto.
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do 2 eexists; eapply commute_locks in H2; eauto.
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invert H3.
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do 3 eexists; eapply commute_unlocks in H2; eauto.
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do 2 eexists; eapply commute_unlocks in H2; eauto.
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eauto.
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Qed.
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@ -1217,6 +1315,35 @@ Qed.
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Hint Rewrite app_length.
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Lemma step_out_of_summarizeThreads : forall c cs1 c0 s cs2,
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summarizeThreads c (cs1 ++ (c0, s) :: cs2)
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-> forall h l c0' s' h' l', step (h, l, c0') (h', l', c0)
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-> summarize c0' s'
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-> exists c', summarizeThreads c' (cs1 ++ (c0', s') :: cs2)
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/\ step (h, l, c') (h', l', c).
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Proof.
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induct 1; simplify.
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apply split_center in x; first_order; subst.
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eapply IHsummarizeThreads1 in H1; try reflexivity; eauto.
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first_order.
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change (cs1 ++ (c0', s') :: x ++ ss2) with (cs1 ++ ((c0', s') :: x) ++ ss2).
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rewrite app_assoc.
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eauto.
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eapply IHsummarizeThreads2 in H1; try reflexivity; eauto.
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first_order.
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rewrite <- app_assoc.
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eauto.
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cases cs1; simplify.
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invert x.
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eauto using summarize_step.
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invert x.
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cases cs1; simplify; try equality.
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Qed.
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Lemma translate_trace_commute : forall i h l c h' l' c',
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stepsi (S i) (h, l, c) (h', l', c')
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-> (forall h'' l'' c'', step (h', l', c') (h'', l'', c'') -> False)
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@ -1277,7 +1404,11 @@ Proof.
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first_order.
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eapply IHstepsi in H3; clear IHstepsi; eauto using summarize_step.
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first_order.
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admit.
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eapply commutes_sound with (c1 := c0) (c2 := x4) (c0 := x) in H10; eauto.
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first_order.
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eapply step_out_of_summarizeThreads with (cs1 := (x2, s) :: x3) in H11; eauto.
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simplify; first_order.
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eauto 10.
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Qed.
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Lemma summarizeThreads_Forall : forall c cs,
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