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ModelChecking.v
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ModelChecking.v
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(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 7: Model Checking
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Set Warnings "-notation-overridden". (* <-- needed while we play with defining one
* of the book's notations ourselves locally *)
Require Import Frap TransitionSystems.
Set Implicit Arguments.
(* Coming up with invariants ourselves can be tedious! Let's investigate how we
* can automate the choice of invariants, for systems with only finitely many
* reachable states. This style is known as model checking. First, let's think
* more deliberately about how to grow a candidate invariant by adding new cases
* that we missed. Here's what it means for one invariant to retain all cases of
* another. *)
Definition oneStepClosure_current {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
forall st, invariant1 st
-> invariant2 st.
(* And here's what it means to add all new states reachable from the original
* set. *)
Definition oneStepClosure_new {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
forall st st', invariant1 st
-> sys.(Step) st st'
-> invariant2 st'.
(* Putting together the two conditions, we have a closure operator, for
* enriching a candidate invariant with all new states reachable from it in a
* single step. *)
Definition oneStepClosure {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
oneStepClosure_current sys invariant1 invariant2
/\ oneStepClosure_new sys invariant1 invariant2.
(* Here's a simple restatement of [oneStepClosure] as a theorem with two
* premises. *)
Theorem prove_oneStepClosure : forall state (sys : trsys state) (inv1 inv2 : state -> Prop),
(forall st, inv1 st -> inv2 st)
-> (forall st st', inv1 st -> sys.(Step) st st' -> inv2 st')
-> oneStepClosure sys inv1 inv2.
Proof.
unfold oneStepClosure.
propositional.
Qed.
(* Now imagine the following general procedure to find an invariant. Start with
* the initial states as the candidate invariant. Now take the one-step
* closure, adding all states reachable in one step. Then take it again, and
* again, until the invariant is "big enough." What is the formal meaning of
* this termination condition? We are done if one-step closure brings us back
* to the original set. (Of course, we must also retain all the initial
* states.) *)
Theorem oneStepClosure_done : forall state (sys : trsys state) (invariant : state -> Prop),
(forall st, sys.(Initial) st -> invariant st)
-> oneStepClosure sys invariant invariant
-> invariantFor sys invariant.
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new.
propositional.
apply invariant_induction.
assumption.
simplify.
eapply H2.
eassumption.
assumption.
Qed.
(* Now we define an inductive relation, capturing repeated closure until
* convergence. *)
Inductive multiStepClosure {state} (sys : trsys state)
: (state -> Prop) -> (state -> Prop) -> Prop :=
(* We might be done, if one-step closure has no effect. *)
| MscDone : forall inv,
oneStepClosure sys inv inv
-> multiStepClosure sys inv inv
(* Or we might need to run another one-step closure and recurse. *)
| MscStep : forall inv inv' inv'',
oneStepClosure sys inv inv'
-> multiStepClosure sys inv' inv''
-> multiStepClosure sys inv inv''.
(* Now, with the help of a lemma, we prove that multi-step closure is a sound
* way to find an invariant for any transition system. Note that we really do
* not have that silver bullet here, because, for many systems, multi-step
* closure does not terminate! However, if it does, we get a correct
* invariant. *)
Lemma multiStepClosure_ok' : forall state (sys : trsys state) (inv inv' : state -> Prop),
multiStepClosure sys inv inv'
-> (forall st, sys.(Initial) st -> inv st)
-> invariantFor sys inv'.
Proof.
induct 1; simplify.
apply oneStepClosure_done.
assumption.
assumption.
apply IHmultiStepClosure.
simplify.
unfold oneStepClosure, oneStepClosure_current in *.
propositional.
apply H3.
apply H1.
assumption.
Qed.
Theorem multiStepClosure_ok : forall state (sys : trsys state) (inv : state -> Prop),
multiStepClosure sys sys.(Initial) inv
-> invariantFor sys inv.
Proof.
simplify.
eapply multiStepClosure_ok'.
eassumption.
propositional.
Qed.
(* OK, great. We know how to find invariants if we can evaluate one-step
* closure efficiently. Here's one case that is particularly easy to evaluate,
* starting from the empty set as the invariant. We use a function [constant]
* from the FRAP library, for sets of finite size. In general, we write
* [constant [x1; ..., xN]] for the set [{x1, ..., xN}], and in fact the latter
* notation is available, too. *)
Theorem oneStepClosure_empty : forall state (sys : trsys state),
oneStepClosure sys (constant nil) (constant nil).
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; propositional.
Qed.
(* In general, for finite sets, we'll compute one-step closure by closing
* separately over each element of the set. This theorem implements one step of
* that process, where we learn that [inv1] accurately captures where we might
* get from state [st] in one step. States [sts] are those left over to
* consider. *)
Theorem oneStepClosure_split : forall state (sys : trsys state) st sts (inv1 inv2 : state -> Prop),
(forall st', sys.(Step) st st' -> inv1 st')
-> oneStepClosure sys (constant sts) inv2
-> oneStepClosure sys (constant (st :: sts)) ({st} \cup inv1 \cup inv2).
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; propositional.
invert H0.
left.
(* [left] and [right]: prove a disjunction by proving the left or right case,
* respectively. Note that here, we are using the fact that set union
* [\cup] is defined in terms of disjunction. *)
left.
simplify.
propositional.
right.
apply H1.
assumption.
simplify.
propositional.
right.
left.
apply H.
equality.
right.
right.
eapply H2.
eassumption.
assumption.
Qed.
(* A trivial fact about union and singleton sets.
* Note that we model sets as functions that are passed elements, deciding in
* each case whether that element belongs to the set. *)
Theorem singleton_in : forall {A} (x : A) rest,
({x} \cup rest) x.
Proof.
simplify.
left.
simplify.
equality.
Qed.
(* OK, back to our example from last chapter, of factorial as a transition
* system. Here's a good overall correctness condition, which we didn't bother
* to state before. *)
Definition fact_correct (original_input : nat) (st : fact_state) : Prop :=
match st with
| AnswerIs ans => fact original_input = ans
| WithAccumulator _ _ => True
end.
(* Let's also restate the initial-states set using a singleton set. *)
Theorem fact_init_is : forall original_input,
fact_init original_input = {WithAccumulator original_input 1}.
Proof.
simplify.
apply sets_equal; simplify.
(* Note the use of a theorem [sets_equal], saying that sets are equal if they
* have the same elements. *)
propositional.
invert H.
equality.
rewrite <- H0.
constructor.
Qed.
(* Now we will prove that factorial is correct, for the input 2, without needing
* to write out an inductive invariant ourselves. Note that it's important that
* we choose a small, constant input, so that the reachable state space is
* finite and tractable. *)
Theorem factorial_ok_2 :
invariantFor (factorial_sys 2) (fact_correct 2).
Proof.
simplify.
eapply invariant_weaken.
(* We begin like in last chapter, by strengthening to an inductive
* invariant. *)
apply multiStepClosure_ok.
(* The difference is that we will use multi-step closure to find the invariant
* automatically. Note that the invariant appears as an existential variable,
* whose name begins with a question mark. *)
simplify.
rewrite fact_init_is.
(* It's important to phrase the current candidate invariant explicitly as a
* finite set, before continuing. Otherwise, it won't be obvious how to take
* the one-step closure. *)
(* Compute which states are reachable after one step. *)
eapply MscStep.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
(* Compute which states are reachable after two steps. *)
eapply MscStep.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
(* Compute which states are reachable after three steps. *)
eapply MscStep.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_split; simplify.
invert H; simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
(* Now the candidate invariant is closed under single steps. Let's prove
* it. *)
apply MscDone.
apply prove_oneStepClosure; simplify.
propositional.
propositional; invert H0; try equality.
invert H; equality.
invert H1; equality.
(* Finally, we prove that our new invariant implies the simpler, noninductive
* one that we started with. *)
simplify.
propositional; subst; simplify; propositional.
(* [subst]: remove all hypotheses like [x = e] for variables [x], simply
* replacing all uses of [x] by [e]. *)
Qed.
(* That process was so rote that we can automate it all, in a generic way that
* will work for most transition systems that have finitely many reachable
* states. Here is a definition of some tactics to do the work.
* BEGIN CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
Local Hint Rewrite fact_init_is.
Ltac model_check_done :=
apply MscDone; apply prove_oneStepClosure; simplify; propositional; subst;
repeat match goal with
| [ H : _ |- _ ] => invert H
end; simplify; equality.
Theorem singleton_in_other : forall {A} (x : A) (s1 s2 : set A),
s2 x
-> (s1 \cup s2) x.
Proof.
simplify.
right.
right.
assumption.
Qed.
Ltac singletoner :=
repeat match goal with
| [ |- _ ?S ] => idtac S; apply singleton_in
| [ |- (_ \cup _) _ ] => apply singleton_in_other
end.
Ltac model_check_step0 :=
eapply MscStep; [
repeat ((apply oneStepClosure_empty; simplify)
|| (apply oneStepClosure_split; [ simplify;
repeat match goal with
| [ H : _ |- _ ] => invert H; try congruence
end; solve [ singletoner ] | ]))
| simplify ].
Ltac model_check_step :=
match goal with
| [ |- multiStepClosure _ ?inv1 _ ] =>
model_check_step0;
match goal with
| [ |- multiStepClosure _ ?inv2 _ ] =>
(assert (inv1 = inv2) by compare_sets; fail 3)
|| idtac
end
end.
Ltac model_check_steps1 := model_check_step || model_check_done.
Ltac model_check_steps := repeat model_check_steps1.
Ltac model_check_finish := simplify; propositional; subst; simplify; equality.
Ltac model_check_infer :=
apply multiStepClosure_ok; simplify; model_check_steps.
Ltac model_check_find_invariant :=
simplify; eapply invariant_weaken; [ model_check_infer | ]; cbv beta in *.
Ltac model_check := model_check_find_invariant; model_check_finish.
(* END CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
(* Now watch this. We can check various instances of factorial
* automatically. Notice that reachable states are printed as we encounter them
* in exploration, using [idtac] invocations above. This printing is for the
* user's understanding and has no logical meaning. *)
Theorem factorial_ok_2_snazzy :
invariantFor (factorial_sys 2) (fact_correct 2).
Proof.
model_check.
Qed.
Theorem factorial_ok_3 :
invariantFor (factorial_sys 3) (fact_correct 3).
Proof.
model_check.
Qed.
Theorem factorial_ok_5 :
invariantFor (factorial_sys 5) (fact_correct 5).
Proof.
model_check.
Qed.
(* Let's see that last one broken into two steps, so that we get a look at the
* inferred invariant. *)
Theorem factorial_ok_5_again :
invariantFor (factorial_sys 5) (fact_correct 5).
Proof.
model_check_find_invariant.
model_check_finish.
Qed.
(** * Abstraction *)
(* It's lovely when we happen to be analyzing a system with a finite state
* space, but usually we aren't that lucky. For instance, imagine that we are
* using a programming language with infinite-precision integers, and we want to
* check this program:
* <<
int global = 0;
thread() {
int local;
while (true) {
local = global;
global = local + 2;
}
}
>>
* The program loops indefinitely, adding 2 to a global variable. We want to
* prove that "global" always holds an even value. Here's how we can formalize
* evenness inductively. *)
Inductive isEven : nat -> Prop :=
| EvenO : isEven 0
| EvenSS : forall n, isEven n -> isEven (S (S n)).
(* And now we define a transition system for the program, in a process that
* should be routine by now. We use last chapter's concept of a multithreaded
* transition system. *)
Inductive add2_thread :=
| Read
| Write (local : nat).
Inductive add2_init : threaded_state nat add2_thread -> Prop :=
| Add2Init : add2_init {| Shared := 0; Private := Read |}.
Inductive add2_step : threaded_state nat add2_thread -> threaded_state nat add2_thread -> Prop :=
| StepRead : forall global,
add2_step {| Shared := global; Private := Read |}
{| Shared := global; Private := Write global |}
| StepWrite : forall global local,
add2_step {| Shared := global; Private := Write local |}
{| Shared := S (S local); Private := Read |}.
Definition add2_sys1 := {|
Initial := add2_init;
Step := add2_step
|}.
Definition add2_sys := parallel add2_sys1 add2_sys1.
(* Here is the invariant we want to prove. *)
Definition add2_correct (st : threaded_state nat (add2_thread * add2_thread)) :=
isEven st.(Shared).
(* We can't model-check [add2_sys] directly, because it can reach infinitely
* many states. Even if we worked with fixed-precision integers, say with 64
* bits, the state space would be impractically large to explore directly.
* Instead, we will *abstract* this system into another one that retains its
* essential properties. In particular, we want to find another transition
* system that *simulates* this one, in the sense made precise by this
* definition, where [sys1] will be [add2_sys] for this example. *)
Inductive simulates state1 state2 (R : state1 -> state2 -> Prop)
(* [R] is a relation connecting the states of the two systems. *)
(sys1 : trsys state1) (sys2 : trsys state2) : Prop :=
| Simulates :
(* Every initial state of [sys1] has some matching initial state of [sys2]. *)
(forall st1, sys1.(Initial) st1
-> exists st2, R st1 st2
/\ sys2.(Initial) st2)
(* Starting from a pair of related states, every step in [sys1] can be matched
* in [sys2], to destinations that are also related. *)
-> (forall st1 st2, R st1 st2
-> forall st1', sys1.(Step) st1 st1'
-> exists st2', R st1' st2'
/\ sys2.(Step) st2 st2')
-> simulates R sys1 sys2.
(* Given an invariant for [sys2], we now have a generic way of defining an
* invariant for [sys1], by composing with [R]. *)
Inductive invariantViaSimulation state1 state2 (R : state1 -> state2 -> Prop)
(inv2 : state2 -> Prop)
: state1 -> Prop :=
| InvariantViaSimulation : forall st1 st2, R st1 st2
-> inv2 st2
-> invariantViaSimulation R inv2 st1.
(* By way of a lemma, let's prove that, given a simulation, any
* invariant-via-simulation really is an invariant for the original system. *)
Lemma invariant_simulates' : forall state1 state2 (R : state1 -> state2 -> Prop)
(sys1 : trsys state1) (sys2 : trsys state2),
(forall st1 st2, R st1 st2
-> forall st1', sys1.(Step) st1 st1'
-> exists st2', R st1' st2'
/\ sys2.(Step) st2 st2')
-> forall st1 st1', sys1.(Step)^* st1 st1'
-> forall st2, R st1 st2
-> exists st2', R st1' st2'
/\ sys2.(Step)^* st2 st2'.
Proof.
induct 2.
simplify.
exists st2.
(* [exists E]: prove [exists x, P(x)] by proving [P(E)]. *)
propositional.
constructor.
simplify.
eapply H in H2.
first_order.
(* [first_order]: simplify first-order logic structure. Be forewarned: this
* one is especially likely to run forever! *)
apply IHtrc in H2.
first_order.
exists x1.
propositional.
econstructor.
eassumption.
assumption.
assumption.
Qed.
Theorem invariant_simulates : forall state1 state2 (R : state1 -> state2 -> Prop)
(sys1 : trsys state1) (sys2 : trsys state2) (inv2 : state2 -> Prop),
simulates R sys1 sys2
-> invariantFor sys2 inv2
-> invariantFor sys1 (invariantViaSimulation R inv2).
Proof.
simplify.
invert H.
unfold invariantFor; simplify.
apply H1 in H.
first_order.
apply invariant_simulates' with (sys2 := sys2) (R := R) (st2 := x) in H3; try assumption.
first_order.
unfold invariantFor in H0.
apply H0 with (s' := x0) in H4; try assumption.
econstructor.
eassumption.
assumption.
Qed.
(* OK, that's a general theory for abstracting a system with another one that
* simulates it. What abstraction will work for our example of the two threads
* and the counter? Here's another program that has replaced integers with
* Booleans, where the Boolean is true iff the matching integer is even.
* <<
bool global = true;
thread() {
bool local;
while (true) {
local = global;
global = local;
}
}
>>
* We can formalize this program as a transition system, too. *)
Inductive add2_bthread :=
| BRead
| BWrite (local : bool).
Inductive add2_binit : threaded_state bool add2_bthread -> Prop :=
| Add2BInit : add2_binit {| Shared := true; Private := BRead |}.
Inductive add2_bstep : threaded_state bool add2_bthread -> threaded_state bool add2_bthread -> Prop :=
| StepBRead : forall global,
add2_bstep {| Shared := global; Private := BRead |}
{| Shared := global; Private := BWrite global |}
| StepBWrite : forall global local,
add2_bstep {| Shared := global; Private := BWrite local |}
{| Shared := local; Private := BRead |}.
Definition add2_bsys1 := {|
Initial := add2_binit;
Step := add2_bstep
|}.
Definition add2_bsys := parallel add2_bsys1 add2_bsys1.
(* This invariant formalizes the connection between local states of threads, in
* the original and abstracted systems. *)
Inductive R_private1 : add2_thread -> add2_bthread -> Prop :=
| RpRead : R_private1 Read BRead
| RpWrite : forall n b, (b = true <-> isEven n)
-> R_private1 (Write n) (BWrite b).
(* We lift [R_private1] to a relation over whole states. *)
Inductive add2_R : threaded_state nat (add2_thread * add2_thread)
-> threaded_state bool (add2_bthread * add2_bthread)
-> Prop :=
| Add2_R : forall n b th1 th2 th1' th2',
(b = true <-> isEven n)
-> R_private1 th1 th1'
-> R_private1 th2 th2'
-> add2_R {| Shared := n; Private := (th1, th2) |}
{| Shared := b; Private := (th1', th2') |}.
(* Let's also recharacterize the initial states via a singleton set. *)
Theorem add2_init_is :
parallel_init add2_binit add2_binit = { {| Shared := true; Private := (BRead, BRead) |} }.
Proof.
simplify.
apply sets_equal; simplify.
propositional.
invert H.
invert H2.
invert H4.
equality.
invert H0.
constructor.
constructor.
constructor.
Qed.
(* We ask Coq to remember this lemma as a hint, which will be used by the
* model-checking tactics that we refrain from explaining in detail. *)
Local Hint Rewrite add2_init_is.
(* Now, let's verify the original system. *)
Theorem add2_ok :
invariantFor add2_sys add2_correct.
Proof.
(* First step: strengthen the invariant. We leave an underscore for the
* unknown invariant, to be found by model checking. *)
eapply invariant_weaken with (invariant1 := invariantViaSimulation add2_R _).
(* One way to find an invariant-by-simulation is to find an invariant for the
* abstracted system, as this step asks to do. *)
apply invariant_simulates with (sys2 := add2_bsys).
(* Now we must prove that the simulation via [add2_R] is valid, which is
* routine. *)
constructor; simplify.
invert H.
invert H0.
invert H1.
exists {| Shared := true; Private := (BRead, BRead) |}; simplify.
propositional.
constructor.
propositional.
constructor.
constructor.
constructor.
invert H.
invert H0; simplify.
invert H7.
invert H2.
exists {| Shared := b; Private := (BWrite b, th2') |}.
propositional.
constructor.
propositional.
constructor.
propositional.
assumption.
constructor.
constructor.
invert H2.
exists {| Shared := b0; Private := (BRead, th2') |}.
propositional.
constructor.
propositional.
constructor.
assumption.
invert H0.
propositional.
constructor.
assumption.
constructor.
constructor.
invert H7.
invert H3.
exists {| Shared := b; Private := (th1', BWrite b) |}.
propositional.
constructor.
propositional.
assumption.
constructor.
propositional.
constructor.
constructor.
invert H3.
exists {| Shared := b0; Private := (th1', BRead) |}.
propositional.
constructor.
propositional.
constructor.
assumption.
invert H0.
propositional.
assumption.
constructor.
constructor.
constructor.
(* OK, we're glad to have that over with! Such a process could also be
* automated, but we won't bother doing so here. However, we are now in a
* good state, where our model checker can find the invariant
* automatically. *)
model_check_infer.
(* It finds exactly four reachable states. We finish by showing that they all
* obey the original invariant. *)
invert 1. (* Note that this [1] means "first premise below the double
* line." *)
invert H0.
simplify.
unfold add2_correct.
simplify.
propositional; subst.
invert H.
propositional.
invert H1.
propositional.
invert H.
propositional.
invert H1.
propositional.
Qed.
(** * Another abstraction example *)
(* Let's try a fancier example of abstraction. Here's a simple integer
* function.
* <<
f(int n) {
int i, j;
i = 0;
j = 0;
while (n > 0) {
i = i + n;
j = j + n;
n = n - 1;
}
}
>>
* We might want to prove that "i" and "j" are always equal at the end.
* First, we formalize the transition system. *)
Inductive pc :=
| i_gets_0
| j_gets_0
| Loop
| i_add_n
| j_add_n
| n_sub_1
| Done.
Record vars := {
N : nat;
I : nat;
J : nat
}.
Record state := {
Pc : pc;
Vars : vars
}.
Inductive initial : state -> Prop :=
| Init : forall vs, initial {| Pc := i_gets_0; Vars := vs |}.
Inductive step : state -> state -> Prop :=
| Step_i_gets_0 : forall n i j,
step {| Pc := i_gets_0; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := j_gets_0; Vars := {| N := n;
I := 0;
J := j |} |}
| Step_j_gets_0 : forall n i j,
step {| Pc := j_gets_0; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := Loop; Vars := {| N := n;
I := i;
J := 0 |} |}
| Step_Loop_done : forall i j,
step {| Pc := Loop; Vars := {| N := 0;
I := i;
J := j |} |}
{| Pc := Done; Vars := {| N := 0;
I := i;
J := j |} |}
| Step_Loop_enter : forall n i j,
step {| Pc := Loop; Vars := {| N := S n;
I := i;
J := j |} |}
{| Pc := i_add_n; Vars := {| N := S n;
I := i;
J := j |} |}
| Step_i_add_n : forall n i j,
step {| Pc := i_add_n; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := j_add_n; Vars := {| N := n;
I := i + n;
J := j |} |}
| Step_j_add_n : forall n i j,
step {| Pc := j_add_n; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := n_sub_1; Vars := {| N := n;
I := i;
J := j + n |} |}
| Step_n_sub_1 : forall n i j,
step {| Pc := n_sub_1; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := Loop; Vars := {| N := n - 1;
I := i;
J := j |} |}.
Definition loopy_sys := {|
Initial := initial;
Step := step
|}.
Definition loopy_correct (st : state) :=
st.(Pc) = Done -> st.(Vars).(I) = st.(Vars).(J).
(* Which abstraction will give us a finite-state system? Unlike with factorial,
* here we are more ambitious, seeking an abstraction that will be finite-state
* even when considering all possible parameter values "n". Let's try this
* simple abstract version of variable state. *)
Inductive absvars :=
| Unknown
(* We don't know anything about the values of the variables. *)
| i_is_0
(* We know [i == 0]. *)
| i_eq_j
(* We know [i == j]. *)
| i_eq_j_plus_n.
(* We know [i == j + n]. *)
(* To get our abstract states, we keep the same program counters and just change
* out the variable state. *)
Record absstate := {
APc : pc;
AVars : absvars
}.
(* Here's the rather boring new abstract step relation. Note the clever state
* transformations, in terms of our new abstraction. *)
Inductive absstep : absstate -> absstate -> Prop :=
| AStep_i_gets_0 : forall vs,
absstep {| APc := i_gets_0; AVars := vs |}
{| APc := j_gets_0; AVars := i_is_0 |}
| AStep_j_gets_0_i_is_0 :
absstep {| APc := j_gets_0; AVars := i_is_0 |}
{| APc := Loop; AVars := i_eq_j |}
| AStep_j_gets_0_Other : forall vs,
vs <> i_is_0
-> absstep {| APc := j_gets_0; AVars := vs |}
{| APc := Loop; AVars := Unknown |}
| AStep_Loop_done : forall vs,
absstep {| APc := Loop; AVars := vs |}
{| APc := Done; AVars := vs |}
| AStep_Loop_enter : forall vs,
absstep {| APc := Loop; AVars := vs |}
{| APc := i_add_n; AVars := vs |}
| AStep_i_add_n_i_eq_j :
absstep {| APc := i_add_n; AVars := i_eq_j |}
{| APc := j_add_n; AVars := i_eq_j_plus_n |}
| AStep_i_add_n_Other : forall vs,
vs <> i_eq_j
-> absstep {| APc := i_add_n; AVars := vs |}
{| APc := j_add_n; AVars := Unknown |}
| AStep_j_add_n_i_eq_j_plus_n :
absstep {| APc := j_add_n; AVars := i_eq_j_plus_n |}
{| APc := n_sub_1; AVars := i_eq_j |}
| AStep_j_add_n_i_Other : forall vs,
vs <> i_eq_j_plus_n
-> absstep {| APc := j_add_n; AVars := vs |}
{| APc := n_sub_1; AVars := Unknown |}
| AStep_n_sub_1_bad :
absstep {| APc := n_sub_1; AVars := i_eq_j_plus_n |}
{| APc := Loop; AVars := Unknown |}
| AStep_n_sub_1_good : forall vs,
vs <> i_eq_j_plus_n
-> absstep {| APc := n_sub_1; AVars := vs |}
{| APc := Loop; AVars := vs |}.
Definition absloopy_sys := {|
Initial := { {| APc := i_gets_0; AVars := Unknown |} };
Step := absstep
|}.
(* Now we need our simulation relation. First, we define one just at the level
* of local-variable state. It formalizes our intuition about those values. *)
Inductive Rvars : vars -> absvars -> Prop :=
| Rv_Unknown : forall vs, Rvars vs Unknown
| Rv_i_is_0 : forall vs, vs.(I) = 0 -> Rvars vs i_is_0
| Rv_i_eq_j : forall vs, vs.(I) = vs.(J) -> Rvars vs i_eq_j
| Rv_i_eq_j_plus_n : forall vs, vs.(I) = vs.(J) + vs.(N) -> Rvars vs i_eq_j_plus_n.
(* We lift to full states in the obvious way. *)
Inductive R : state -> absstate -> Prop :=
| Rcon : forall pc vs avs, Rvars vs avs -> R {| Pc := pc; Vars := vs |}
{| APc := pc; AVars := avs |}.
(* Now we are ready to prove the original system correct. *)
Theorem loopy_ok :
invariantFor loopy_sys loopy_correct.
Proof.
eapply invariant_weaken with (invariant1 := invariantViaSimulation R _).
apply invariant_simulates with (sys2 := absloopy_sys).
(* Here comes another boring simulation proof. *)
constructor; simplify.
invert H.
exists {| APc := i_gets_0; AVars := Unknown |}.
propositional.
constructor.
constructor.
invert H0.
invert H.
exists {| APc := j_gets_0; AVars := i_is_0 |}.
propositional; repeat constructor.
invert H.
invert H3.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_eq_j |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Done; AVars := st2.(AVars) |}.
invert H; simplify; propositional; repeat constructor; equality.
exists {| APc := i_add_n; AVars := st2.(AVars) |}.
invert H; simplify; propositional; repeat constructor; equality.
invert H.
invert H3.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := j_add_n; AVars := i_eq_j_plus_n |}; repeat constructor; simplify; equality.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
invert H.
invert H3.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := i_eq_j |}; repeat constructor; simplify; equality.
invert H.
invert H3.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_is_0 |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_eq_j |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
(* Finally, we can call the model checker to find an invariant of the abstract
* system. *)
model_check_infer.
(* We get 7 neat little states, one per program counter. Next, we prove that
* each of them implies the original invariant. *)
invert 1.
invert H0.
unfold loopy_correct.
simplify.
propositional; subst.
(* Most of the hypotheses we invert are contradictory, implying that distinct
* program counters are equal. *)
invert H2.
invert H1.
invert H2.
invert H1.
invert H.
assumption.
invert H2.
invert H1.
invert H2.
Qed.
(** * Modularity *)
(* Throughout the book, we'll come again and again to our two main weapons in
* soundly modeling complex transition systems with simpler ones. We just