Up to now, we've used a quite restricted set of Coq's tactic facilities. In this chapter, we'll learn more about two very powerful features of Coq's tactic language: proof search via the auto and eauto tactics, and automated forward reasoning via the Ltac hypothesis matching machinery. Using these features together with Ltac's scripting facilities will enable us to make our proofs startlingly short! Used properly, they can also make proofs more maintainable and robust in the face of incremental changes to underlying definitions. This chapter introduces auto and eauto. A deeper treatment can be found in the UseAuto chapter. There's a third major category of automation we haven't fully studied yet, namely built-in decision procedures for specific kinds of problems: omega is one example, but there are others. This topic will be deferred for a while longer. Our motivating example will be this proof, repeated with just a few small changes from Imp. We will try to simplify this proof in several stages.

The auto and eauto Tactics

Thus far, we have generally written proof scripts that apply relevant hypotheses or lemmas by name. In particular, when a chain of hypothesis applications is needed, we have specified them explicitly. (The only exceptions we've seen to this are using assumption to find a matching unqualified hypothesis or (e)constructor to find a matching constructor.)

The auto tactic frees us from this drudgery by searching for a sequence of applications that will prove the goal

The auto tactic solves goals that are solvable by any combination of

• intros,
• apply (with a local hypothesis, by default).

The eauto tactic works just like auto, except that it uses eapply instead of apply. Using auto is always "safe" in the sense that it will never fail and will never change the proof state: either it completely solves the current goal, or it does nothing. A more complicated example:

Search can take an arbitrarily long time, so there are limits to how far auto will search by default.

When searching for potential proofs of the current goal, auto and eauto consider the hypotheses in the current context together with a hint database of other lemmas and constructors. Some of the lemmas and constructors we've already seen — e.g., eq_refl, conj, or_introl, and or_intror — are installed in this hint database by default.

If we want to see which facts auto is using, we can use info_auto instead.

We can extend the hint database just for the purposes of one application of auto or eauto by writing auto using ....

Of course, in any given development there will probably be some specific constructors and lemmas that are used very often in proofs. We can add these to the global hint database by writing at the top level, where T is a top-level theorem or a constructor of an inductively defined proposition (i.e., anything whose type is an implication). As a shorthand, we can write to tell Coq to do a Hint Resolve for all of the constructors from the inductive definition of c. It is also sometimes necessary to add where d is a defined symbol, so that auto knows to expand uses of d and enable further possibilities for applying lemmas that it knows about.

Now let's take a pass over ceval_deterministic using auto to simplify the proof script. We see that all simple sequences of hypothesis applications and all uses of reflexivity can be replaced by auto, which we add to the default tactic to be applied to each case.

When we are using a particular tactic many times in a proof, we can use a variant of the Proof command to make that tactic into a default within the proof. Saying Proof with t (where t is an arbitrary tactic) allows us to use t₁... as a shorthand for t₁;t within the proof. As an illustration, here is an alternate version of the previous proof, using Proof with auto.

Searching Hypotheses

The proof has become simpler, but there is still an annoying amount of repetition. Let's start by tackling the contradiction cases. Each of them occurs in a situation where we have both H₁: beval st b = false and H₂: beval st b = true as hypotheses. The contradiction is evident, but demonstrating it is a little complicated: we have to locate the two hypotheses H₁ and H₂ and do a rewrite following by an inversion. We'd like to automate this process. (Note: In fact, Coq has a built-in tactic congruence that will do the job. But we'll ignore the existence of this tactic for now, in order to demonstrate how to build forward search tactics by hand.) As a first step, we can abstract out the piece of script in question by writing a small amount of paramerized Ltac.

But this is not much better. We really want Coq to discover the relevant hypotheses for us. We can do this by using the match goal with ... end facility of Ltac.

In words, this match goal looks for two distinct hypotheses that have the form of equalities with the same arbitrary expression E on the left and conflicting boolean values on the right; if such hypotheses are found, it binds H₁ and H₂ to their names, and applies the rwinv tactic. Adding this tactic to our default string handles all the contradiction cases.

Let's see about the remaining cases. Each of them involves applying a conditional hypothesis to extract an equality. Currently we have phrased these as assertions, so that we have to predict what the resulting equality will be (although we can then use auto to prove it.) An alternative is to pick the relevant hypotheses to use, and then rewrite with them, as follows:

Now we can automate the task of finding the relevant hypotheses to rewrite with.

But there are several pairs of hypotheses that have the right general form, and it seems tricky to pick out the ones we actually need. A key trick is to realize that we can try them all! Here's how this works:

• rewrite will fail given a trivial equation of the form X = X;
• each execution of match goal will keep trying to find a valid pair of hypotheses until the tactic on the RHS of the match succeeds; if there are no such pairs, it fails;
• we can wrap the whole thing in a repeat, which will keep doing useful rewrites until only trivial ones are left.

The big payoff in this approach is that our proof script should be robust in the face of modest changes to our language. For example, we can add a REPEAT command to the language. (This was an exercise in Hoare.v.)

REPEAT behaves like WHILE, except that the loop guard is checked after each execution of the body, with the loop repeating as long as the guard stays false. Because of this, the body will always execute at least once.

These examples just give a flavor of what "hyper-automation" can do. The details of using match goal are a bit tricky, and debugging scripts using it is not very pleasant. But it is well worth adding at least simple uses to your proofs to avoid tedium and to "future proof" your scripts.