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ModelChecking.v
486
ModelChecking.v
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@ -8,22 +8,35 @@ Require Import Frap TransitionSystems.
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Set Implicit Arguments.
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(* Coming up with invariants ourselves can be tedious! Let's investigate how we
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* can automate the choice of invariants, for systems with only finitely many
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* reachable states. This style is known as model checking. First, let's think
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* more deliberately about how to grow a candidate invariant by adding new cases
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* that we missed. Here's what it means for one invariant to retain all cases of
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* another. *)
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Definition oneStepClosure_current {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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forall st, invariant1 st
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-> invariant2 st.
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(* And here's what it means to add all new states reachable from the original
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* set. *)
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Definition oneStepClosure_new {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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forall st st', invariant1 st
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-> sys.(Step) st st'
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-> invariant2 st'.
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(* Putting together the two conditions, we have a closure operator, for
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* enriching a candidate invariant with all new states reachable from it in a
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* single step. *)
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Definition oneStepClosure {state} (sys : trsys state)
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(invariant1 invariant2 : state -> Prop) :=
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oneStepClosure_current sys invariant1 invariant2
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/\ oneStepClosure_new sys invariant1 invariant2.
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(* Here's a simple restatement of [oneStepClosure] as a theorem with two
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* premises. *)
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Theorem prove_oneStepClosure : forall state (sys : trsys state) (inv1 inv2 : state -> Prop),
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(forall st, inv1 st -> inv2 st)
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-> (forall st st', inv1 st -> sys.(Step) st st' -> inv2 st')
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@ -33,6 +46,13 @@ Proof.
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propositional.
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Qed.
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(* Now imagine the following general procedure to find an invariant. Start with
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* the initial states as the candidate invariant. Now take the one-step
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* closure, adding all states reachable in one step. Then take it again, and
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* again, until the invariant is "big enough." What is the formal meaning of
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* this termination condition? We are done if one-step closure brings us back
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* to the original set. (Of course, we must also retain all the initial
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* states.) *)
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Theorem oneStepClosure_done : forall state (sys : trsys state) (invariant : state -> Prop),
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(forall st, sys.(Initial) st -> invariant st)
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-> oneStepClosure sys invariant invariant
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@ -48,16 +68,28 @@ Proof.
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assumption.
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Qed.
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(* Now we define an inductive relation, capturing repeated closure until
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* convergence. *)
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Inductive multiStepClosure {state} (sys : trsys state)
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: (state -> Prop) -> (state -> Prop) -> Prop :=
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(* We might be done, if one-step closure has no effect. *)
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| MscDone : forall inv,
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oneStepClosure sys inv inv
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-> multiStepClosure sys inv inv
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(* Or we might need to run another one-step closure and recurse. *)
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| MscStep : forall inv inv' inv'',
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oneStepClosure sys inv inv'
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-> multiStepClosure sys inv' inv''
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-> multiStepClosure sys inv inv''.
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(* Now, with the help of a lemma, we prove that multi-step closure is a sound
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* way to find an invariant for any transition system. Note that we really do
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* not have that silver bullet here, because, for many systems, multi-step
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* closure does not terminate! However, if it does, we get a correct
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* invariant. *)
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Lemma multiStepClosure_ok' : forall state (sys : trsys state) (inv inv' : state -> Prop),
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multiStepClosure sys inv inv'
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-> (forall st, sys.(Initial) st -> inv st)
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@ -88,12 +120,23 @@ Proof.
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propositional.
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Qed.
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(* OK, great. We know how to find invariants if we can evaluate one-step
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* closure efficiently. Here's one case that is particularly easy to evaluate,
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* starting from the empty set as the invariant. We use a function [constant]
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* from the FRAP library, for sets of finite size. In general, we write
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* [constant [x1; ..., xN]] for the set [{x1, ..., xN}], and in fact the latter
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* notation is available, too. *)
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Theorem oneStepClosure_empty : forall state (sys : trsys state),
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oneStepClosure sys (constant nil) (constant nil).
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Proof.
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unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; propositional.
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Qed.
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(* In general, for finite sets, we'll compute one-step closure by closing
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* separately over each element of the set. This theorem implements one step of
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* that process, where we learn that [inv1] accurately captures where we might
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* get from state [st] in one step. States [sts] are those left over to
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* consider. *)
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Theorem oneStepClosure_split : forall state (sys : trsys state) st sts (inv1 inv2 : state -> Prop),
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(forall st', sys.(Step) st st' -> inv1 st')
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-> oneStepClosure sys (constant sts) inv2
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@ -104,6 +147,9 @@ Proof.
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invert H0.
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left.
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(* [left] and [right]: prove a disjunction by proving the left or right case,
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* respectively. Note that here, we are using the fact that set union
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* [\cup] is defined in terms of disjunction. *)
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left.
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simplify.
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propositional.
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@ -126,12 +172,26 @@ Proof.
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assumption.
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Qed.
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(* A trivial fact about union and singleton sets. *)
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Theorem singleton_in : forall {A} (x : A) rest,
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({x} \cup rest) x.
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Proof.
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simplify.
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left.
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simplify.
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equality.
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Qed.
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(* OK, back to our example from last chapter, of factorial as a transition
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* system. Here's a good overall correctness condition, which we didn't bother
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* to state before. *)
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Definition fact_correct (original_input : nat) (st : fact_state) : Prop :=
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match st with
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| AnswerIs ans => fact original_input = ans
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| WithAccumulator _ _ => True
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end.
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(* Let's also restate the initial-states set using a singleton set. *)
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Theorem fact_init_is : forall original_input,
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fact_init original_input = {WithAccumulator original_input 1}.
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Proof.
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@ -146,15 +206,91 @@ Proof.
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constructor.
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Qed.
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Theorem singleton_in : forall {A} (x : A) rest,
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({x} \cup rest) x.
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(* Now we will prove that factorial is correct, for the input 2, without needing
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* to write out an inductive invariant ourselves. Note that it's important that
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* we choose a small, constant input, so that the reachable state space is
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* finite. *)
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Theorem factorial_ok_2 :
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invariantFor (factorial_sys 2) (fact_correct 2).
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Proof.
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simplify.
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left.
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eapply invariant_weaken.
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(* We begin like in last chapter, by strengthening to an inductive
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* invariant. *)
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apply multiStepClosure_ok.
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(* The difference is that we will use multi-step closure to find the invariant
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* automatically. Note that the invariant appears as an existential variable,
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* whose name begins with a question mark. *)
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simplify.
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equality.
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rewrite fact_init_is.
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(* It's important to phrase the current candidate invariant explicitly as a
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* finite set, before continuing. Otherwise, it won't be obvious how to take
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* the one-step closure. *)
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(* Compute which states are reachable after one step. *)
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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(* Compute which states are reachable after two steps. *)
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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(* Compute which states are reachable after three steps. *)
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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(* Now the candidate invariatn is closed under single steps. Let's prove
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* it. *)
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apply MscDone.
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apply prove_oneStepClosure; simplify.
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propositional.
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propositional; invert H0; try equality.
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invert H; equality.
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invert H1; equality.
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(* Finally, we prove that our new invariant implies the simpler, noninductive
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* one that we started with. *)
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simplify.
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propositional; subst; simplify; propositional.
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(* [subst]: remove all hypotheses like [x = e] for variables [x], simply
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* replacing all uses of [x] by [e]. *)
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Qed.
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(* That process was so rote that we can automate it all, in a generic way that
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* will work for most transition systems that have finitely many reachable
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* states. Here is a definition of some tactics to do the work.
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* BEGIN CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
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Hint Rewrite fact_init_is.
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Ltac model_check_done :=
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apply MscDone; apply prove_oneStepClosure; simplify; propositional; subst;
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repeat match goal with
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| [ H : _ |- _ ] => invert H
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end; simplify; equality.
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Theorem singleton_in_other : forall {A} (x : A) (s1 s2 : set A),
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s2 x
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-> (s1 \cup s2) x.
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assumption.
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Qed.
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Theorem factorial_ok_2 :
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invariantFor (factorial_sys 2) (fact_correct 2).
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Proof.
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simplify.
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eapply invariant_weaken.
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apply multiStepClosure_ok.
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simplify.
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rewrite fact_init_is.
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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eapply MscStep.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_split; simplify.
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invert H; simplify.
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apply singleton_in.
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apply oneStepClosure_empty.
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simplify.
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apply MscDone.
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apply prove_oneStepClosure; simplify.
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propositional.
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propositional; invert H0; try equality.
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invert H; equality.
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invert H1; equality.
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simplify.
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propositional; subst; simplify; propositional.
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(* ^-- *)
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Qed.
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Hint Rewrite fact_init_is.
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Ltac model_check_done :=
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apply MscDone; apply prove_oneStepClosure; simplify; propositional; subst;
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repeat match goal with
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| [ H : _ |- _ ] => invert H
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end; simplify; equality.
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Ltac singletoner :=
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repeat match goal with
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| _ => apply singleton_in
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@ -252,6 +328,11 @@ Ltac model_check_find_invariant :=
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Ltac model_check := model_check_find_invariant; model_check_finish.
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(* END CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
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(* Now watch this. We can check various instances of factorial
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* automatically. *)
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Theorem factorial_ok_2_snazzy :
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invariantFor (factorial_sys 2) (fact_correct 2).
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Proof.
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@ -270,27 +351,46 @@ Proof.
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model_check.
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Qed.
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(* Let's see that last one broken into two steps, so that we get a look at the
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* inferred invariant. *)
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Theorem factorial_ok_5_again :
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invariantFor (factorial_sys 5) (fact_correct 5).
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Proof.
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model_check_find_invariant.
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model_check_finish.
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Qed.
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(** * Abstraction *)
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(*
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(* It's lovely when we happen to be analyzing a system with a finite state
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* space, but usually we aren't that lucky. For instance, imagine that we are
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* using a programming language with infinite-precision integers, and we want to
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* check this program:
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* <<
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int global = 0;
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thread() {
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int local;
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int global = 0;
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thread() {
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int local;
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while (true) {
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local = global;
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global = local + 2;
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}
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}
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*)
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while (true) {
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local = global;
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global = local + 2;
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}
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}
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>>
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* The program loops indefinitely, adding 2 to a global variable. We want to
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* prove that "global" always holds an even value. Here's how we can formalize
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* evenness inductively. *)
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Inductive isEven : nat -> Prop :=
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| EvenO : isEven 0
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| EvenSS : forall n, isEven n -> isEven (S (S n)).
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(* And now we define a transition system for the program, in a process that
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* should be routine by now. We use last chapter's concept of a multithreaded
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* transition system. *)
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Inductive add2_thread :=
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| Read
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| Write (local : nat).
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@ -313,18 +413,37 @@ Definition add2_sys1 := {|
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Definition add2_sys := parallel add2_sys1 add2_sys1.
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(* Here is the invariant we want to prove. *)
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Definition add2_correct (st : threaded_state nat (add2_thread * add2_thread)) :=
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isEven st.(Shared).
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(* We can't model-check [add2_sys] directly, because it can reach infinitely
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* many states. Even if we worked with fixed-precision integers, say with 64
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* bits, the state space would be impractically large to explore directly.
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* Instead, we will *abstract* this system into another one that retains its
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* essential properties. In particular, we want to find another transition
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* system that *simulates* this one, in the sense made precise by this
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* definition, where [sys1] will be [add2_sys] for this example. *)
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Inductive simulates state1 state2 (R : state1 -> state2 -> Prop)
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(* [R] is a relation connecting the states of the two systems. *)
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(sys1 : trsys state1) (sys2 : trsys state2) : Prop :=
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| Simulates :
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(* Every initial state of [sys1] has some matching initial state of [sys2]. *)
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(forall st1, sys1.(Initial) st1
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-> exists st2, R st1 st2
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/\ sys2.(Initial) st2)
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(* Starting from a pair of related states, every step in [sys1] can be matched
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* in [sys2], to destinations that are also related. *)
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-> (forall st1 st2, R st1 st2
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-> forall st1', sys1.(Step) st1 st1'
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-> exists st2', R st1' st2'
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/\ sys2.(Step) st2 st2')
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-> simulates R sys1 sys2.
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(* Given an invariant for [sys2], we now have a generic way of defining an
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* invariant for [sys1], by composing with [R]. *)
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Inductive invariantViaSimulation state1 state2 (R : state1 -> state2 -> Prop)
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(inv2 : state2 -> Prop)
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: state1 -> Prop :=
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@ -332,6 +451,8 @@ Inductive invariantViaSimulation state1 state2 (R : state1 -> state2 -> Prop)
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-> inv2 st2
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-> invariantViaSimulation R inv2 st1.
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(* By way of a lemma, let's prove that, given a simulation, any
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* invariant-via-simulation really is an invariant for the original system. *)
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Lemma invariant_simulates' : forall state1 state2 (R : state1 -> state2 -> Prop)
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(sys1 : trsys state1) (sys2 : trsys state2),
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(forall st1 st2, R st1 st2
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@ -347,12 +468,15 @@ Proof.
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simplify.
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exists st2.
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(* [exists E]: prove [exists x, P(x)] by proving [P(E)]. *)
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propositional.
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constructor.
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simplify.
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eapply H in H2.
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first_order.
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(* [first_order]: simplify first-order logic structure. Be forewarned: this
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* one is especially likely to run forever! *)
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apply IHtrc in H2.
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first_order.
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exists x1.
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@ -383,19 +507,23 @@ Proof.
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assumption.
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Qed.
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(* Abstracted program:
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(* OK, that's a general theory for abstracting a system with another one that
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* simulates it. What abstraction will work for our example of the two threads
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* and the counter? Here's another program that has replaced integers with
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* Booleans, where the Boolean is true iff the matching integer is even.
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* <<
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bool global = true;
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bool global = true;
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thread() {
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bool local;
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thread() {
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bool local;
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while (true) {
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local = global;
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global = local;
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}
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}
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*)
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while (true) {
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local = global;
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global = local;
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}
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}
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>>
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* We can formalize this program as a transition system, too. *)
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Inductive add2_bthread :=
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| BRead
|
||||
|
@ -419,14 +547,14 @@ Definition add2_bsys1 := {|
|
|||
|
||||
Definition add2_bsys := parallel add2_bsys1 add2_bsys1.
|
||||
|
||||
Definition add2_correct (st : threaded_state nat (add2_thread * add2_thread)) :=
|
||||
isEven st.(Shared).
|
||||
|
||||
(* 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 :=
|
||||
|
@ -437,6 +565,7 @@ Inductive add2_R : threaded_state nat (add2_thread * add2_thread)
|
|||
-> 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 :
|
||||
parallel1 add2_binit add2_binit = { {| Shared := true; Private := (BRead, BRead) |} }.
|
||||
Proof.
|
||||
|
@ -455,14 +584,24 @@ Proof.
|
|||
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. *)
|
||||
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.
|
||||
|
@ -533,7 +672,13 @@ Proof.
|
|||
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.
|
||||
invert H0.
|
||||
|
@ -558,20 +703,23 @@ Qed.
|
|||
|
||||
(** * Another abstraction example *)
|
||||
|
||||
(*
|
||||
(* Let's try a fancier example of abstraction. Here's a simple integer
|
||||
* function.
|
||||
* <<
|
||||
f(int n) {
|
||||
int i, j;
|
||||
|
||||
f(int n) {
|
||||
int i, j;
|
||||
|
||||
i = 0;
|
||||
j = 0;
|
||||
while (n > 0) {
|
||||
i = i + n;
|
||||
j = j + n;
|
||||
n = n - 1;
|
||||
}
|
||||
}
|
||||
*)
|
||||
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
|
||||
|
@ -652,13 +800,32 @@ Definition loopy_sys := {|
|
|||
Step := step
|
||||
|}.
|
||||
|
||||
Inductive absvars := Unknown | i_is_0 | i_eq_j | i_eq_j_plus_n.
|
||||
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 |}
|
||||
|
@ -703,25 +870,27 @@ Definition absloopy_sys := {|
|
|||
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 |}.
|
||||
|
||||
Definition loopy_correct (st : state) :=
|
||||
st.(Pc) = Done -> st.(Vars).(I) = st.(Vars).(J).
|
||||
|
||||
(* 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.
|
||||
|
@ -770,14 +939,22 @@ Proof.
|
|||
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 1. (* Note that this [1] means "first premise below the double
|
||||
* line." *)
|
||||
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.
|
||||
|
@ -798,18 +975,40 @@ 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
|
||||
* learned about *abstraction*, to replace a full system with a simpler one.
|
||||
* The other key one is *modularity*, to replace a system with several others.
|
||||
* Let's study one example that helps with model checking, allowing us to check
|
||||
* programs with arbitrarily many threads running the same code, while still
|
||||
* finding an invariant automatically by brute-force enumeration. *)
|
||||
|
||||
(* The key to this particular technique is instrumenting a step relation to
|
||||
* consider *interference*, or the actions that other threads might take, in
|
||||
* between steps of the thread that we focus on. This relation is parameterized
|
||||
* on an invariant [inv] that the other threads guarantee to preserve on the
|
||||
* shared state. That is, the other threads may mess with the shared state
|
||||
* arbitrarily between our own steps, *but* they guarantee that every value they
|
||||
* set for it satisfies [inv]. *)
|
||||
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 :=
|
||||
|
||||
(* First kind of step: this thread runs in the normal way. *)
|
||||
| StepSelf : forall st st',
|
||||
step st st'
|
||||
-> stepWithInterference inv step st st'
|
||||
|
||||
(* Second kind of step: other threads change shared state to some new value
|
||||
* satisfying [inv]. *)
|
||||
| StepEnvironment : forall sh pr sh',
|
||||
inv sh'
|
||||
-> stepWithInterference inv step
|
||||
{| Shared := sh; Private := pr |}
|
||||
{| Shared := sh'; Private := pr |}.
|
||||
|
||||
(* Via this relation, we have an operator to build a new transition system from
|
||||
* an old one, given [inv]. *)
|
||||
Definition withInterference shared private (inv : shared -> Prop)
|
||||
(sys : trsys (threaded_state shared private))
|
||||
: trsys (threaded_state shared private) := {|
|
||||
|
@ -817,12 +1016,17 @@ Definition withInterference shared private (inv : shared -> Prop)
|
|||
Step := stepWithInterference inv sys.(Step)
|
||||
|}.
|
||||
|
||||
(* We also have a ready-made invariant we could try to prove for any such
|
||||
* system, asserting that [inv] always holds of the shared state. *)
|
||||
Inductive sharedInvariant shared private (inv : shared -> Prop)
|
||||
: threaded_state shared private -> Prop :=
|
||||
| SharedInvariant : forall sh pr,
|
||||
inv sh
|
||||
-> sharedInvariant inv {| Shared := sh; Private := pr |}.
|
||||
|
||||
(* Tired of simulation proofs yet? Then you'll love this theorem, which shows
|
||||
* a free simulation for any use of [withInterference]! We even get to pick the
|
||||
* trivial simulation relation, state equality. *)
|
||||
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).
|
||||
|
@ -837,6 +1041,12 @@ Proof.
|
|||
equality.
|
||||
Qed.
|
||||
|
||||
(* That proof was pretty straightforward, because we could construct the
|
||||
* simulation using only the first rule of [stepWithInterference], ignoring the
|
||||
* possibility for steps by other threads. We go next for a theorem with an
|
||||
* intimidating statement and a much more interesting proof, whose details we
|
||||
* nonetheless won't comment on in text. It may make sense to skip past the
|
||||
* next two lemma statements to the main theorem [withInterference_parallel]. *)
|
||||
Lemma withInterference_parallel1 : forall shared private1 private2
|
||||
(invs : shared -> Prop)
|
||||
(sys1 : trsys (threaded_state shared private1))
|
||||
|
@ -1025,6 +1235,12 @@ Proof.
|
|||
constructor.
|
||||
Qed.
|
||||
|
||||
(* OK, we made it to the main theorem. It helps us find an invariant for a
|
||||
* [parallel] system to which we have applied the [withInterference]
|
||||
* construction. Crucially, we may check the invariant for each constituent
|
||||
* thread *separately*, avoiding the combinatorial state-space explosion that
|
||||
* would come from analyzing the combined system directly. This is the essence
|
||||
* of modularity! *)
|
||||
Theorem withInterference_parallel : forall shared private1 private2
|
||||
(invs : shared -> Prop)
|
||||
(sys1 : trsys (threaded_state shared private1))
|
||||
|
@ -1040,6 +1256,8 @@ Proof.
|
|||
simplify.
|
||||
invert H1.
|
||||
|
||||
(* [assert P]: first prove proposition [P], then continue with it as a new
|
||||
* hypothesis. *)
|
||||
assert ((withInterference invs sys1).(Step)^*
|
||||
{| Shared := sh; Private := pr1 |}
|
||||
{| Shared := s'.(Shared); Private := fst s'.(Private) |}).
|
||||
|
@ -1067,21 +1285,23 @@ Proof.
|
|||
apply H in H1; try assumption.
|
||||
Qed.
|
||||
|
||||
(*
|
||||
(* Let's apply this principle on a concrete example. Consider a program with
|
||||
* many threads running calls to this function.
|
||||
* <<
|
||||
int global = 0;
|
||||
|
||||
int global = 0;
|
||||
f() {
|
||||
int local = 0;
|
||||
|
||||
f() {
|
||||
int local = 0;
|
||||
|
||||
while (true) {
|
||||
local = global;
|
||||
local = 3 + local;
|
||||
local = 7 + local;
|
||||
global = local;
|
||||
while (true) {
|
||||
local = global;
|
||||
local = 3 + local;
|
||||
local = 7 + local;
|
||||
global = local;
|
||||
}
|
||||
}
|
||||
}
|
||||
*)
|
||||
>>
|
||||
* Here's the usual formalization as a transition system. *)
|
||||
|
||||
Inductive twoadd_pc := ReadIt | Add3 | Add7 | WriteIt.
|
||||
|
||||
|
@ -1107,9 +1327,13 @@ Definition twoadd_sys := {|
|
|||
Step := twoadd_step
|
||||
|}.
|
||||
|
||||
(* Invariant to prove: the global variable is always even, again. *)
|
||||
Definition twoadd_correct private (st : threaded_state nat private) :=
|
||||
isEven st.(Shared).
|
||||
|
||||
(* Here's an abstract version of the system where, much like before, we model
|
||||
* integers as Booleans, recording whether they are even or not. *)
|
||||
|
||||
Definition twoadd_ainitial := { {| Shared := true; Private := (ReadIt, true) |} }.
|
||||
|
||||
Inductive twoadd_astep : threaded_state bool (twoadd_pc * bool)
|
||||
|
@ -1135,9 +1359,13 @@ Definition twoadd_asys := {|
|
|||
Step := twoadd_astep
|
||||
|}.
|
||||
|
||||
(* Here's a simulation relation at the level of integers and their Boolean
|
||||
* counterparts. *)
|
||||
Definition even_R (n : nat) (b : bool) :=
|
||||
isEven n <-> b = true.
|
||||
|
||||
(* A few unsurprising properties hold of [even_R]. *)
|
||||
|
||||
Lemma even_R_0 : even_R 0 true.
|
||||
Proof.
|
||||
unfold even_R; propositional.
|
||||
|
@ -1163,6 +1391,7 @@ Proof.
|
|||
constructor; assumption.
|
||||
Qed.
|
||||
|
||||
(* The cases for evenness of an integer and its successor *)
|
||||
Lemma isEven_decide : forall n,
|
||||
(isEven n /\ ~isEven (S n)) \/ (~isEven n /\ isEven (S n)).
|
||||
Proof.
|
||||
|
@ -1192,6 +1421,7 @@ Proof.
|
|||
equality.
|
||||
Qed.
|
||||
|
||||
(* Here's the top-level simulation relation for our choice of abstraction. *)
|
||||
Inductive twoadd_R : threaded_state nat (twoadd_pc * nat)
|
||||
-> threaded_state bool (twoadd_pc * bool) -> Prop :=
|
||||
| Twoadd_R : forall pc gn ln gb lb,
|
||||
|
@ -1200,6 +1430,7 @@ Inductive twoadd_R : threaded_state nat (twoadd_pc * nat)
|
|||
-> twoadd_R {| Shared := gn; Private := (pc, ln) |}
|
||||
{| Shared := gb; Private := (pc, lb) |}.
|
||||
|
||||
(* Step 1 of main proof: model-check an individual thread. *)
|
||||
Lemma twoadd_ok :
|
||||
invariantFor (withInterference isEven twoadd_sys)
|
||||
(fun st => isEven (Shared st)).
|
||||
|
@ -1207,6 +1438,7 @@ Proof.
|
|||
eapply invariant_weaken.
|
||||
apply invariant_simulates with (sys2 := twoadd_asys) (R := twoadd_R).
|
||||
|
||||
(* Boring simulation proof begins here. *)
|
||||
constructor; simplify.
|
||||
|
||||
invert H.
|
||||
|
@ -1255,8 +1487,10 @@ Proof.
|
|||
assumption.
|
||||
constructor.
|
||||
|
||||
(* Now find an invariant automatically. *)
|
||||
model_check_infer.
|
||||
|
||||
(* Now prove that the invariant implies the correctness condition. *)
|
||||
invert 1.
|
||||
invert H0.
|
||||
simplify.
|
||||
|
@ -1279,6 +1513,8 @@ Proof.
|
|||
assumption.
|
||||
Qed.
|
||||
|
||||
(* Step 2: lift that result to the two-thread system, with no new model
|
||||
* checking. *)
|
||||
Theorem twoadd2_ok :
|
||||
invariantFor (parallel twoadd_sys twoadd_sys) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
|
@ -1294,18 +1530,27 @@ Proof.
|
|||
assumption.
|
||||
Qed.
|
||||
|
||||
(* In fact, this modularity technique is so powerful that we now get correctness
|
||||
* for any number of threads, "for free"! Here's a tactic definition, which we
|
||||
* won't explain, but which is able to derive correctness for any number of
|
||||
* threads, just by repeating use of [withInterference_parallel] and
|
||||
* [twoadd_ok]. *)
|
||||
Ltac twoadd := eapply invariant_weaken; [ eapply invariant_simulates; [
|
||||
apply withInterference_abstracts
|
||||
| repeat (apply withInterference_parallel
|
||||
|| apply twoadd_ok) ]
|
||||
| unfold twoadd_correct; invert 1; assumption ].
|
||||
|
||||
(* For instance, let's verify the three-thread version. *)
|
||||
Theorem twoadd3_ok :
|
||||
invariantFor (parallel twoadd_sys (parallel twoadd_sys twoadd_sys)) (twoadd_correct (private := _)).
|
||||
Proof.
|
||||
twoadd.
|
||||
Qed.
|
||||
|
||||
(* To save us time defining versions with many threads, here's a recursive
|
||||
* function, creating exponentially many threads with respect to its
|
||||
* parameter. *)
|
||||
Fixpoint manyadds_state (n : nat) : Type :=
|
||||
match n with
|
||||
| O => twoadd_pc * nat
|
||||
|
@ -1318,6 +1563,7 @@ Fixpoint manyadds (n : nat) : trsys (threaded_state nat (manyadds_state n)) :=
|
|||
| S n' => parallel (manyadds n') (manyadds n')
|
||||
end.
|
||||
|
||||
(* Here are some examples of the systems we produce. *)
|
||||
Eval simpl in manyadds 0.
|
||||
Eval simpl in manyadds 1.
|
||||
Eval simpl in manyadds 2.
|
||||
|
|
|
@ -11,3 +11,4 @@ Interpreters_template.v
|
|||
Interpreters.v
|
||||
TransitionSystems_template.v
|
||||
TransitionSystems.v
|
||||
ModelChecking.v
|
||||
|
|
Loading…
Reference in a new issue