mirror of
https://github.com/achlipala/frap.git
synced 2024-12-01 00:26:18 +00:00
ModelChecking: an example of modularity
This commit is contained in:
Adam Chlipala
parent
e3bb90c4a1
commit
d04037a84f
1 changed files with 540 additions and 0 deletions
540
ModelChecking.v
540
ModelChecking.v
|
|
@ -794,3 +794,543 @@ Proof.
|
|||
|
||||
invert H2.
|
||||
Qed.
|
||||
|
||||
|
||||
(** * Modularity *)
|
||||
|
||||
Inductive stepWithInterference shared private (inv : shared -> Prop)
|
||||
(step : threaded_state shared private -> threaded_state shared private -> Prop)
|
||||
: threaded_state shared private -> threaded_state shared private -> Prop :=
|
||||
| StepSelf : forall st st',
|
||||
step st st'
|
||||
-> stepWithInterference inv step st st'
|
||||
| StepEnvironment : forall sh pr sh',
|
||||
inv sh'
|
||||
-> stepWithInterference inv step
|
||||
{| Shared := sh; Private := pr |}
|
||||
{| Shared := sh'; Private := pr |}.
|
||||
|
||||
Definition withInterference shared private (inv : shared -> Prop)
|
||||
(sys : trsys (threaded_state shared private))
|
||||
: trsys (threaded_state shared private) := {|
|
||||
Initial := sys.(Initial);
|
||||
Step := stepWithInterference inv sys.(Step)
|
||||
|}.
|
||||
|
||||
Inductive sharedInvariant shared private (inv : shared -> Prop)
|
||||
: threaded_state shared private -> Prop :=
|
||||
| SharedInvariant : forall sh pr,
|
||||
inv sh
|
||||
-> sharedInvariant inv {| Shared := sh; Private := pr |}.
|
||||
|
||||
Theorem withInterference_abstracts : forall shared private (inv : shared -> Prop)
|
||||
(sys : trsys (threaded_state shared private)),
|
||||
simulates (fun st st' => st = st') sys (withInterference inv sys).
|
||||
Proof.
|
||||
simplify.
|
||||
constructor; simplify.
|
||||
|
||||
exists st1; propositional.
|
||||
|
||||
exists st1'; propositional.
|
||||
constructor.
|
||||
equality.
|
||||
Qed.
|
||||
|
||||
Lemma withInterference_parallel1 : forall shared private1 private2
|
||||
(invs : shared -> Prop)
|
||||
(sys1 : trsys (threaded_state shared private1))
|
||||
(sys2 : trsys (threaded_state shared private2))
|
||||
st st',
|
||||
(withInterference invs (parallel sys1 sys2)).(Step)^* st st'
|
||||
-> forall st1 st2,
|
||||
(forall st1', (withInterference invs sys1).(Step)^* st1 st1' -> invs st1'.(Shared))
|
||||
-> (forall st2', (withInterference invs sys2).(Step)^* st2 st2' -> invs st2'.(Shared))
|
||||
-> (withInterference invs sys1).(Step)^* st1
|
||||
{| Shared := st.(Shared);
|
||||
Private := fst st.(Private) |}
|
||||
-> (withInterference invs sys2).(Step)^* st2
|
||||
{| Shared := st.(Shared);
|
||||
Private := snd st.(Private) |}
|
||||
-> (withInterference invs sys1).(Step)^* st1
|
||||
{| Shared := st'.(Shared);
|
||||
Private := fst st'.(Private) |}.
|
||||
Proof.
|
||||
induct 1; simplify.
|
||||
|
||||
assumption.
|
||||
|
||||
invert H; simplify.
|
||||
invert H5; simplify.
|
||||
|
||||
apply IHtrc with (st2 := {| Shared := sh'; Private := pr2 |}).
|
||||
simplify.
|
||||
apply H1.
|
||||
assumption.
|
||||
simplify.
|
||||
eapply H2.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
apply H1 with (st1' := {| Shared := sh'; Private := pr1' |}).
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
econstructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
assumption.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
econstructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
constructor.
|
||||
|
||||
apply IHtrc with (st2 := {| Shared := sh'; Private := pr2' |}).
|
||||
assumption.
|
||||
simplify.
|
||||
apply H2.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
constructor.
|
||||
eassumption.
|
||||
eassumption.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
apply H2 with (st2' := {| Shared := sh'; Private := pr2' |}).
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
econstructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
constructor.
|
||||
constructor.
|
||||
|
||||
apply IHtrc with (st2 := {| Shared := sh'; Private := snd pr |}).
|
||||
assumption.
|
||||
simplify.
|
||||
eapply H2.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
assumption.
|
||||
assumption.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
assumption.
|
||||
constructor.
|
||||
constructor.
|
||||
Qed.
|
||||
|
||||
Lemma withInterference_parallel2 : forall shared private1 private2
|
||||
(invs : shared -> Prop)
|
||||
(sys1 : trsys (threaded_state shared private1))
|
||||
(sys2 : trsys (threaded_state shared private2))
|
||||
st st',
|
||||
(withInterference invs (parallel sys1 sys2)).(Step)^* st st'
|
||||
-> forall st1 st2,
|
||||
(forall st1', (withInterference invs sys1).(Step)^* st1 st1' -> invs st1'.(Shared))
|
||||
-> (forall st2', (withInterference invs sys2).(Step)^* st2 st2' -> invs st2'.(Shared))
|
||||
-> (withInterference invs sys1).(Step)^* st1
|
||||
{| Shared := st.(Shared);
|
||||
Private := fst st.(Private) |}
|
||||
-> (withInterference invs sys2).(Step)^* st2
|
||||
{| Shared := st.(Shared);
|
||||
Private := snd st.(Private) |}
|
||||
-> (withInterference invs sys2).(Step)^* st2
|
||||
{| Shared := st'.(Shared);
|
||||
Private := snd st'.(Private) |}.
|
||||
Proof.
|
||||
induct 1; simplify.
|
||||
|
||||
assumption.
|
||||
|
||||
invert H; simplify.
|
||||
invert H5; simplify.
|
||||
|
||||
apply IHtrc with (st1 := {| Shared := sh'; Private := pr1' |}).
|
||||
simplify.
|
||||
apply H1.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
econstructor.
|
||||
eassumption.
|
||||
assumption.
|
||||
assumption.
|
||||
constructor.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
apply H1 with (st1' := {| Shared := sh'; Private := pr1' |}).
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
econstructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
constructor.
|
||||
|
||||
apply IHtrc with (st1 := {| Shared := sh'; Private := pr1 |}).
|
||||
simplify.
|
||||
apply H1.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
apply H2 with (st2' := {| Shared := sh'; Private := pr2' |}).
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
constructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
assumption.
|
||||
assumption.
|
||||
constructor.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
constructor.
|
||||
eassumption.
|
||||
constructor.
|
||||
|
||||
apply IHtrc with (st1 := {| Shared := sh'; Private := fst pr |}).
|
||||
simplify.
|
||||
eapply H1.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
assumption.
|
||||
assumption.
|
||||
assumption.
|
||||
constructor.
|
||||
eapply trc_trans.
|
||||
eassumption.
|
||||
eapply TrcFront.
|
||||
apply StepEnvironment with (sh' := sh').
|
||||
assumption.
|
||||
constructor.
|
||||
Qed.
|
||||
|
||||
Theorem withInterference_parallel : forall shared private1 private2
|
||||
(invs : shared -> Prop)
|
||||
(sys1 : trsys (threaded_state shared private1))
|
||||
(sys2 : trsys (threaded_state shared private2)),
|
||||
invariantFor (withInterference invs sys1)
|
||||
(fun st => invs st.(Shared))
|
||||
-> invariantFor (withInterference invs sys2)
|
||||
(fun st => invs st.(Shared))
|
||||
-> invariantFor (withInterference invs (parallel sys1 sys2))
|
||||
(fun st => invs st.(Shared)).
|
||||
Proof.
|
||||
unfold invariantFor.
|
||||
simplify.
|
||||
invert H1.
|
||||
|
||||
assert ((withInterference invs sys1).(Step)^*
|
||||
{| Shared := sh; Private := pr1 |}
|
||||
{| Shared := s'.(Shared); Private := fst s'.(Private) |}).
|
||||
apply withInterference_parallel1 with (sys2 := sys2)
|
||||
(st := {| Shared := sh; Private := (pr1, pr2) |})
|
||||
(st2 := {| Shared := sh; Private := pr2 |});
|
||||
simplify; propositional.
|
||||
apply H in H1; propositional.
|
||||
apply H0 in H1; propositional.
|
||||
constructor.
|
||||
constructor.
|
||||
|
||||
assert ((withInterference invs sys2).(Step)^*
|
||||
{| Shared := sh; Private := pr2 |}
|
||||
{| Shared := s'.(Shared); Private := snd s'.(Private) |}).
|
||||
apply withInterference_parallel2 with (sys1 := sys1)
|
||||
(st := {| Shared := sh; Private := (pr1, pr2) |})
|
||||
(st1 := {| Shared := sh; Private := pr1 |});
|
||||
simplify; propositional.
|
||||
apply H in H5; propositional.
|
||||
apply H0 in H5; propositional.
|
||||
constructor.
|
||||
constructor.
|
||||
|
||||
apply H in H1; try assumption.
|
||||
Qed.
|
||||
|
||||
(*
|
||||
|
||||
int global = 0;
|
||||
|
||||
f() {
|
||||
int local = 0;
|
||||
|
||||
while (true) {
|
||||
local = global;
|
||||
local = 3 + local;
|
||||
local = 7 + local;
|
||||
global = local;
|
||||
}
|
||||
}
|
||||
*)
|
||||
|
||||
Inductive twoadd_pc := ReadIt | Add3 | Add7 | WriteIt.
|
||||
|
||||
Definition twoadd_initial := { {| Shared := 0; Private := (ReadIt, 0) |} }.
|
||||
|
||||
Inductive twoadd_step : threaded_state nat (twoadd_pc * nat)
|
||||
-> threaded_state nat (twoadd_pc * nat) -> Prop :=
|
||||
| Step_ReadIt : forall g l,
|
||||
twoadd_step {| Shared := g; Private := (ReadIt, l) |}
|
||||
{| Shared := g; Private := (Add3, g) |}
|
||||
| Step_Add3 : forall g l,
|
||||
twoadd_step {| Shared := g; Private := (Add3, l) |}
|
||||
{| Shared := g; Private := (Add7, 3 + l) |}
|
||||
| Step_Add7 : forall g l,
|
||||
twoadd_step {| Shared := g; Private := (Add7, l) |}
|
||||
{| Shared := g; Private := (WriteIt, 7 + l) |}
|
||||
| Step_WriteIt : forall g l,
|
||||
twoadd_step {| Shared := g; Private := (WriteIt, l) |}
|
||||
{| Shared := l; Private := (ReadIt, l) |}.
|
||||
|
||||
Definition twoadd_sys := {|
|
||||
Initial := twoadd_initial;
|
||||
Step := twoadd_step
|
||||
|}.
|
||||
|
||||
Definition twoadd_correct private (st : threaded_state nat private) :=
|
||||
isEven st.(Shared).
|
||||
|
||||
Definition twoadd_ainitial := { {| Shared := true; Private := (ReadIt, true) |} }.
|
||||
|
||||
Inductive twoadd_astep : threaded_state bool (twoadd_pc * bool)
|
||||
-> threaded_state bool (twoadd_pc * bool) -> Prop :=
|
||||
| AStep_ReadIt : forall g l,
|
||||
twoadd_astep {| Shared := g; Private := (ReadIt, l) |}
|
||||
{| Shared := g; Private := (Add3, g) |}
|
||||
| AStep_Add3 : forall g l,
|
||||
twoadd_astep {| Shared := g; Private := (Add3, l) |}
|
||||
{| Shared := g; Private := (Add7, negb l) |}
|
||||
| AStep_Add7 : forall g l,
|
||||
twoadd_astep {| Shared := g; Private := (Add7, l) |}
|
||||
{| Shared := g; Private := (WriteIt, negb l) |}
|
||||
| AStep_WriteIt : forall g l,
|
||||
twoadd_astep {| Shared := g; Private := (WriteIt, l) |}
|
||||
{| Shared := l; Private := (ReadIt, l) |}
|
||||
| AStep_Someone_Made_It_Even : forall g pr,
|
||||
twoadd_astep {| Shared := g; Private := pr |}
|
||||
{| Shared := true; Private := pr |}.
|
||||
|
||||
Definition twoadd_asys := {|
|
||||
Initial := twoadd_ainitial;
|
||||
Step := twoadd_astep
|
||||
|}.
|
||||
|
||||
Definition even_R (n : nat) (b : bool) :=
|
||||
isEven n <-> b = true.
|
||||
|
||||
Lemma even_R_0 : even_R 0 true.
|
||||
Proof.
|
||||
unfold even_R; propositional.
|
||||
constructor.
|
||||
Qed.
|
||||
|
||||
Lemma even_R_forward : forall n, isEven n -> even_R n true.
|
||||
Proof.
|
||||
unfold even_R; propositional.
|
||||
Qed.
|
||||
|
||||
Lemma even_R_backward : forall n, even_R n true -> isEven n.
|
||||
Proof.
|
||||
unfold even_R; propositional.
|
||||
Qed.
|
||||
|
||||
Lemma even_R_add2 : forall n b,
|
||||
even_R n b
|
||||
-> even_R (S (S n)) b.
|
||||
Proof.
|
||||
unfold even_R; propositional.
|
||||
invert H; propositional.
|
||||
constructor; assumption.
|
||||
Qed.
|
||||
|
||||
Lemma isEven_decide : forall n,
|
||||
(isEven n /\ ~isEven (S n)) \/ (~isEven n /\ isEven (S n)).
|
||||
Proof.
|
||||
induct n; simplify; propositional.
|
||||
|
||||
left; propositional.
|
||||
constructor.
|
||||
invert H.
|
||||
|
||||
right; propositional.
|
||||
constructor; assumption.
|
||||
|
||||
left; propositional.
|
||||
invert H.
|
||||
propositional.
|
||||
Qed.
|
||||
|
||||
Lemma even_R_add1 : forall n b,
|
||||
even_R n b
|
||||
-> even_R (S n) (negb b).
|
||||
Proof.
|
||||
unfold even_R; simplify.
|
||||
assert ((isEven n /\ ~isEven (S n)) \/ (~isEven n /\ isEven (S n))).
|
||||
apply isEven_decide.
|
||||
cases b; simplify; propositional.
|
||||
equality.
|
||||
equality.
|
||||
Qed.
|
||||
|
||||
Inductive twoadd_R : threaded_state nat (twoadd_pc * nat)
|
||||
-> threaded_state bool (twoadd_pc * bool) -> Prop :=
|
||||
| Twoadd_R : forall pc gn ln gb lb,
|
||||
even_R gn gb
|
||||
-> even_R ln lb
|
||||
-> twoadd_R {| Shared := gn; Private := (pc, ln) |}
|
||||
{| Shared := gb; Private := (pc, lb) |}.
|
||||
|
||||
Lemma twoadd_ok :
|
||||
invariantFor (withInterference isEven twoadd_sys)
|
||||
(fun st => isEven (Shared st)).
|
||||
Proof.
|
||||
eapply invariant_weaken.
|
||||
apply invariant_simulates with (sys2 := twoadd_asys) (R := twoadd_R).
|
||||
|
||||
constructor; simplify.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := true; Private := (ReadIt, true) |}; propositional.
|
||||
constructor; propositional.
|
||||
apply even_R_0.
|
||||
apply even_R_0.
|
||||
constructor.
|
||||
equality.
|
||||
simplify.
|
||||
propositional.
|
||||
|
||||
invert H0.
|
||||
invert H1.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := gb; Private := (Add3, gb) |}; propositional.
|
||||
constructor; propositional.
|
||||
constructor.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := gb; Private := (Add7, negb lb) |}; propositional.
|
||||
constructor; propositional.
|
||||
apply even_R_add2.
|
||||
apply even_R_add1.
|
||||
assumption.
|
||||
constructor.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := gb; Private := (WriteIt, negb lb) |}; propositional.
|
||||
constructor; propositional.
|
||||
repeat apply even_R_add2.
|
||||
apply even_R_add1.
|
||||
assumption.
|
||||
constructor.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := lb; Private := (ReadIt, lb) |}; propositional.
|
||||
constructor; propositional.
|
||||
constructor.
|
||||
|
||||
invert H.
|
||||
exists {| Shared := true; Private := (pc0, lb) |}; propositional.
|
||||
constructor; propositional.
|
||||
apply even_R_forward.
|
||||
assumption.
|
||||
constructor.
|
||||
|
||||
model_check_infer.
|
||||
|
||||
invert 1.
|
||||
invert H0.
|
||||
simplify.
|
||||
propositional.
|
||||
|
||||
invert H0.
|
||||
apply even_R_backward.
|
||||
assumption.
|
||||
|
||||
invert H1.
|
||||
apply even_R_backward.
|
||||
assumption.
|
||||
|
||||
invert H0.
|
||||
apply even_R_backward.
|
||||
assumption.
|
||||
|
||||
invert H1.
|
||||
apply even_R_backward.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Theorem twoadd2_ok :
|
||||
invariantFor (parallel twoadd_sys twoadd_sys) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
eapply invariant_weaken.
|
||||
eapply invariant_simulates.
|
||||
apply withInterference_abstracts.
|
||||
apply withInterference_parallel.
|
||||
apply twoadd_ok.
|
||||
apply twoadd_ok.
|
||||
|
||||
unfold twoadd_correct.
|
||||
invert 1.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Ltac twoadd := eapply invariant_weaken; [ eapply invariant_simulates; [
|
||||
apply withInterference_abstracts
|
||||
| repeat (apply withInterference_parallel
|
||||
|| apply twoadd_ok) ]
|
||||
| unfold twoadd_correct; invert 1; assumption ].
|
||||
|
||||
Theorem twoadd3_ok :
|
||||
invariantFor (parallel twoadd_sys (parallel twoadd_sys twoadd_sys)) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
twoadd.
|
||||
Qed.
|
||||
|
||||
Fixpoint manyadds_state (n : nat) : Type :=
|
||||
match n with
|
||||
| O => twoadd_pc * nat
|
||||
| S n' => manyadds_state n' * manyadds_state n'
|
||||
end%type.
|
||||
|
||||
Fixpoint manyadds (n : nat) : trsys (threaded_state nat (manyadds_state n)) :=
|
||||
match n with
|
||||
| O => twoadd_sys
|
||||
| S n' => parallel (manyadds n') (manyadds n')
|
||||
end.
|
||||
|
||||
Eval simpl in manyadds 0.
|
||||
Eval simpl in manyadds 1.
|
||||
Eval simpl in manyadds 2.
|
||||
Eval simpl in manyadds 3.
|
||||
|
||||
Theorem twoadd4_ok :
|
||||
invariantFor (manyadds 4) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
twoadd.
|
||||
Qed.
|
||||
|
||||
Theorem twoadd6_ok :
|
||||
invariantFor (manyadds 6) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
twoadd.
|
||||
Qed.
|
||||
|
|
|
|||
Loading…
Reference in a new issue