Major Section: ACL2-TUTORIAL

We present here some tips for using ACL2 effectively. Though this
collection is somewhat *ad hoc*, we try to provide some
organization, albeit somewhat artificial: for example, the sections
overlap, and no particular order is intended. This material has
been adapted by Bill Young from a very similar list for Nqthm that
appeared in the conclusion of: ``Interaction with the Boyer-Moore
Theorem Prover: A Tutorial Study Using the Arithmetic-Geometric Mean
Theorem,'' by Matt Kaufmann and Paolo Pecchiari, CLI Technical
Report 100, June, 1995. We also draw from a similar list in Chapter
13 of ``A Computational Logic Handbook'' by R.S. Boyer and J
S. Moore (Academic Press, 1988). We'll refer to this as ``ACLH''
below.

These tips are organized roughly as follows.

A. ACL2 Basics

B. Strategies for creating events

C. Dealing with failed proofs

D. Performance tips

E. Miscellaneous tips and knowledge

F. Some things you DON'T need to know

*ACL2 BASICS*

**A1. The ACL2 logic.**

This is a logic of total functions. For example, if `A`

and `B`

are less than or equal to each other, then we need to know something
more in order to conclude that they are equal (e.g., that they are
numbers). This kind of twist is important in writing definitions;
for example, if you expect a function to return a number, you may
want to apply the function `fix`

or some variant (e.g., `nfix`

or
`ifix`

) in case one of the formals is to be returned as the value.

ACL2's notion of ordinals is important on occasion in supplying ``measure hints'' for the acceptance of recursive definitions. Be sure that your measure is really an ordinal. Consider the following example, which ACL2 fails to admit (as explained below).

(defun cnt (name a i x) (declare (xargs :measure (+ 1 i))) (cond ((zp (+ 1 i)) 0) ((equal x (aref1 name a i)) (1+ (cnt name a (1- i) x))) (t (cnt name a (1- i) x))))One might think that

`(+ 1 i)`

is a reasonable measure, since we
know that `(+ 1 i)`

is a positive integer in any recursive call of
`cnt`

, and positive integers are ACL2 ordinals
(see o-p). However, the ACL2 logic requires that the
measure be an ordinal unconditionally, not just under the governing
assumptions that lead to recursive calls. An appropriate fix is to
apply `nfix`

to `(+ 1 i)`

, i.e., to use
(declare (xargs :measure (nfix (+ 1 i))))in order to guarantee that the measure will always be an ordinal (in fact, a positive integer).

For more about admissibility of recursive definitions, see defun, in particular the discussion of termination.

**A2. Simplification.**

The ACL2 simplifier is basically a rewriter, with some ``linear
arithmetic'' thrown in. One needs to understand the notion of
conditional rewriting. See rewrite.

**A3. Parsing of rewrite rules.**

ACL2 parses rewrite rules roughly as explained in ACLH, *except*
that it never creates ``unusual'' rule classes. In ACL2, if you
want a `:`

`linear`

rule, for example, you must specify `:`

`linear`

in
the `:`

`rule-classes`

. See rule-classes, and also
see rewrite and see linear.

**A4. Linear arithmetic.**

On this subject, it should suffice to know that the prover can
handle truths about `+`

and `-`

, and that linear rules (see above)
are somehow ``thrown in the pot'' when the prover is doing such
reasoning. Perhaps it's also useful to know that linear rules can
have hypotheses, and that conditional rewriting is used to relieve
those hypotheses.

**A5. Events.**

Over time, the expert ACL2 user will know some subtleties of its
events. For example, `in-theory`

events and hints are
important, and they distinguish between a function and its
executable counterpart.

*B. STRATEGIES FOR CREATING EVENTS*

In this section, we concentrate on the use of definitions and rewrite rules. There are quite a few kinds of rules allowed in ACL2 besides rewrite rules, though most beginning users probably won't usually need to be aware of them. See rule-classes for details. In particular, there is support for congruence rewriting. Also see rune (``RUle NamE'') for a description of the various kinds of rules in the system.

**B1. Use high-level strategy.**

Decompose theorems into ``manageable'' lemmas (admittedly,
experience helps here) that yield the main result ``easily.'' It's
important to be able to outline non-trivial proofs by hand (or in
your head). In particular, avoid submitting goals to the prover
when there's no reason to believe that the goal will be proved and
there's no ``sense'' of how an induction argument would apply. It
is often a good idea to avoid induction in complicated theorems
unless you have a reason to believe that it is appropriate.

**B2. Write elegant definitions.**

Try to write definitions in a reasonably modular style, especially
recursive ones. Think of ACL2 as a programming language whose
procedures are definitions and lemmas, hence we are really
suggesting that one follow good programming style (in order to avoid
duplication of ``code,'' for example).

When possible, complex functions are best written as compositions of simpler functions. The theorem prover generally performs better on primitive recursive functions than on more complicated recursions (such as those using accumulating parameters).

Avoid large non-recursive definitions which tend to lead to large case explosions. If such definitions are necessary, try to prove all relevant facts about the definitions and then disable them.

Whenever possible, avoid mutual recursion if you care to prove anything about your functions. The induction heuristics provide essentially no help with reasoning about mutually defined functions. Mutually recursive functions can usually be combined into a single function with a ``flag'' argument. (However, see mutual-recursion-proof-example for a small example of proof involving mutually recursive functions.)

**B3. Look for analogies.**

Sometimes you can easily edit sequences of lemmas into sequences of
lemmas about analogous functions.

**B4. Write useful rewrite rules.**

As explained in A3 above, every rewrite rule is a directive to the
theorem prover, usually to replace one term by another. The
directive generated is determined by the syntax of the `defthm`

submitted. Never submit a rewrite rule unless you have considered
its interpretation as a proof directive.

**B4a. Rewrite rules should simplify.**

Try to write rewrite rules whose right-hand sides are in some sense
``simpler than'' (or at worst, are variants of) the left-hand sides.
This will help to avoid infinite loops in the rewriter.

**B4b. Avoid needlessly expensive rules.**

Consider a rule whose conclusion's left-hand side (or, the entire
conclusion) is a term such as `(consp x)`

that matches many terms
encountered by the prover. If in addition the rule has complicated
hypotheses, this rule could slow down the prover greatly. Consider
switching the conclusion and a complicated hypothesis (negating
each) in that case.

**B4c. The ``Knuth-Bendix problem''.**

Be aware that left sides of rewrite rules should match the
``normalized forms'', where ``normalization'' (rewriting) is inside
out. Be sure to avoid the use of nonrecursive function symbols on
left sides of rewrite rules, except when those function symbols are
disabled, because they tend to be expanded away before the rewriter
would encounter an instance of the left side of the rule. Also
assure that subexpressions on the left hand side of a rule are in
simplified form.

**B4d. Avoid proving useless rules.**

Sometimes it's tempting to prove a rewrite rule even before you see
how it might find application. If the rule seems clean and
important, and not unduly expensive, that's probably fine,
especially if it's not too hard to prove. But unless it's either
part of the high-level strategy or, on the other hand, intended to
get the prover past a particular unproved goal, it may simply waste
your time to prove the rule, and then clutter the database of rules
if you are successful.

**B4e. State rules as strongly as possible, usually.**

It's usually a good idea to state a rule in the strongest way
possible, both by eliminating unnecessary hypotheses and by
generalizing subexpressions to variables.

Advanced users may choose to violate this policy on occasion, for example in order to avoid slowing down the prover by excessive attempted application of the rule. However, it's a good rule of thumb to make the strongest rule possible, not only because it will then apply more often, but also because the rule will often be easier to prove (see also B6 below). New users are sometimes tempted to put in extra hypotheses that have a ``type restriction'' appearance, without realizing that the way ACL2 handles (total) functions generally lets it handle trivial cases easily.

**B4f. Avoid circularity.**

A stack overflow in a proof attempt almost always results from
circular rewriting. Use `brr`

to investigate the stack;
see break-lemma. Because of the complex heuristics, it is not
always easy to define just when a rewrite will cause circularity.
See the very good discussion of this topic in ACLH.

See break-lemma for a trick involving use of the forms `brr t`

and `(cw-gstack)`

for inspecting loops in the rewriter.

**B4g. Remember restrictions on permutative rules.**

Any rule that permutes the variables in its left hand side could
cause circularity. For example, the following axiom is
automatically supplied by the system:

(defaxiom commutativity-of-+ (equal (+ x y) (+ y x))).This would obviously lead to dangerous circular rewriting if such ``permutative'' rules were not governed by a further restriction. The restriction is that such rules will not produce a term that is ``lexicographically larger than'' the original term (see loop-stopper). However, this sometimes prevents intended rewrites. See Chapter 13 of ACLH for a discussion of this problem.

**B5. Conditional vs. unconditional rewrite rules.**

It's generally preferable to form unconditional rewrite rules unless
there is a danger of case explosion. That is, rather than pairs of
rules such as

(implies p (equal term1 term2))and

(implies (not p) (equal term1 term3))consider:

(equal term1 (if p term2 term3))However, sometimes this strategy can lead to case explosions:

`IF`

terms introduce cases in ACL2. Use your judgment. (On the subject
of `IF`

: `COND`

, `CASE`

, `AND`

, and `OR`

are macros that
abbreviate `IF`

forms, and propositional functions such as
`IMPLIES`

quickly expand into `IF`

terms.)**B6. Create elegant theorems.**

Try to formulate lemmas that are as simple and general as possible.
For example, sometimes properties about several functions can be
``factored'' into lemmas about one function at a time. Sometimes
the elimination of unnecessary hypotheses makes the theorem easier
to prove, as does generalizing first by hand.

**B7. Use** `defaxiom`

s **temporarily to explore possibilities.**

When there is a difficult goal that seems to follow immediately (by
a `:use`

hint or by rewriting) from some other lemmas, you can
create those lemmas as `defaxiom`

events (or, the application of
`skip-proofs`

to `defthm`

events) and then double-check that the
difficult goal really does follow from them. Then you can go back
and try to turn each `defaxiom`

into a `defthm`

. When you do
that, it's often useful to disable any additional rewrite rules that
you prove in the process, so that the ``difficult goal'' will still
be proved from its lemmas when the process is complete.

Better yet, rather than disabling rewrite rules, use the `local`

mechanism offered by `encapsulate`

to make temporary rules
completely `local`

to the problem at hand. See encapsulate and
see local.

**B9. Use books.**

Consider using previously certified books, especially for arithmetic
reasoning. This cuts down the duplication of effort and starts your
specification and proof effort from a richer foundation.
See community-books.

*C. DEALING WITH FAILED PROOFS*

**C1. Look in proof output for goals that can't be further simplified.**

Use the ``proof-tree'' utility to explore the proof space.
However, you don't need to use that tool to use the ``checkpoint''
strategy. The idea is to think of ACL2 as a ``simplifier'' that
either proves the theorem or generates some goal to consider. That
goal is the first ``checkpoint,'' i.e., the first goal that does not
further simplify. Exception: it's also important to look at the
induction scheme in a proof by induction, and if induction seems
appropriate, then look at the first checkpoint *after* the
induction has begun.

Consider whether the goal on which you focus is even a theorem. Sometimes you can execute it for particular values to find a counterexample.

When looking at checkpoints, remember that you are looking for any reason at all to believe the goal is a theorem. So for example, sometimes there may be a contradiction in the hypotheses.

Don't be afraid to skip the first checkpoint if it doesn't seem very helpful. Also, be willing to look a few lines up or down from the checkpoint if you are stuck, bearing in mind however that this practice can be more distracting than helpful.

**C2. Use the ``break rewrite'' facility.**

`Brr`

and related utilities let you inspect the ``rewrite stack.''
These can be valuable tools in large proof efforts.
See break-lemma for an introduction to these tools, and
see break-rewrite for more complete information.

The break facility is especially helpful in showing you why a particular rewrite rule is not being applied.

**C3. Use induction hints when necessary.**
Of course, if you can define your functions so that they suggest the
correct inductions to ACL2, so much the better! But for complicated
inductions, induction hints are crucial. See hints for a
description of `:induct`

hints.

**C4. Use the ``Proof Checker'' to explore.**

The `verify`

command supplied by ACL2 allows one to explore problem
areas ``by hand.'' However, even if you succeed in proving a
conjecture with `verify`

, it is useful to prove it without using
it, an activity that will often require the discovery of rewrite
rules that will be useful in later proofs as well.

**C5. Don't have too much patience.**

Interrupt the prover fairly quickly when simplification isn't
succeeding.

**C6. Simplify rewrite rules.**

When it looks difficult to relieve the hypotheses of an existing
rewrite rule that ``should'' apply in a given setting, ask yourself
if you can eliminate a hypothesis from the existing rewrite rule.
If so, it may be easier to prove the new version from the old
version (and some additional lemmas), rather than to start from
scratch.

**C7. Deal with base cases first.**

Try getting past the base case(s) first in a difficult proof by
induction. Usually they're easier than the inductive step(s), and
rules developed in proving them can be useful in the inductive
step(s) too. Moreover, it's pretty common that mistakes in the
statement of a theorem show up in the base case(s) of its proof by
induction.

**C8. Use** `:expand`

**hints.**
Consider giving `:expand`

hints. These are especially useful when a
proof by induction is failing. It's almost always helpful to open
up a recursively defined function that is supplying the induction
scheme, but sometimes ACL2 is too timid to do so; or perhaps the
function in question is disabled.

*D. PERFORMANCE TIPS*

**D1. Disable rules.**

There are a number of instances when it is crucial to disable rules,
including (often) those named explicitly in `:use`

hints. Also,
disable recursively defined functions for which you can prove what
seem to be all the relevant properties. The prover can spend
significant time ``behind the scenes'' trying to open up recursively
defined functions, where the only visible effect is slowness.

**D2. Turn off the ``break rewrite'' facility.**
Remember to execute `:brr nil`

after you've finished with the
``break rewrite'' utility (see break-rewrite), in order to
bring the prover back up to full speed.

*E. MISCELLANEOUS TIPS AND KNOWLEDGE*

**E1. Order of application of rewrite rules.**

Keep in mind that the most recent rewrite rules in the history
are tried first.

**E2. Relieving hypotheses is not full-blown theorem proving.**

Relieving hypotheses on rewrite rules is done by rewriting and linear
arithmetic alone, not by case splitting or by other prover processes
``below'' simplification.

**E3. ``Free variables'' in rewrite rules.**

The set of ``free
variables'' of a rewrite rule is defined to contain those
variables occurring in the rule that do not occur in the left-hand
side of the rule. It's often a good idea to avoid rules containing
free variables because they are ``weak,'' in the sense that
hypotheses containing such variables can generally only be proved
when they are ``obviously'' present in the current context. This
weakness suggests that it's important to put the most
``interesting'' (specific) hypotheses about free variables first, so
that the right instances are considered. For example, suppose you
put a very general hypothesis such as `(consp x)`

first. If the
context has several terms around that are known to be
`consp`

s, then `x`

may be bound to the wrong one of them. For much
more information on free variables, see free-variables.

**E4. Obtaining information**

Use `:`

`pl`

`foo`

to inspect rewrite rules whose left hand sides are
applications of the function `foo`

. Another approach to seeing
which rewrite rules apply is to enter the proof-checker with
`verify`

, and use the `show-rewrites`

or `sr`

command.

**E5. Consider esoteric rules with care.**

If you care to see rule-classes and peruse the list of
subtopics (which will be listed right there in most versions of this
documentation), you'll see that ACL2 supports a wide variety of
rules in addition to `:`

rewrite rules. Should you use them?
This is a complex question that we are not ready to answer with any
generality. Our general advice is to avoid relying on such rules as
long as you doubt their utility. More specifically: be careful not
to use conditional type prescription rules, as these have been known
to bring ACL2 to its knees, unless you are conscious that you are
doing so and have reason to believe that they are working well.

*F. SOME THINGS YOU DON'T NEED TO KNOW*

Most generally: you shouldn't usually need to be able to predict too much about ACL2's behavior. You should mainly just need to be able to react to it.

**F1. Induction heuristics.**

Although it is often important to read the part of the prover's
output that gives the induction scheme chosen by the prover, it is
not necessary to understand how the prover made that choice.
(Granted, advanced users may occasionally gain minor insight from
such knowledge. But it's truly minor in many cases.) What *is*
important is to be able to tell it an appropriate induction when it
doesn't pick the right one (after noticing that it doesn't). See C3
above.

**F2. Heuristics for expanding calls of recursively defined functions.**

As with the previous topic, the important thing isn't to understand
these heuristics but, rather, to deal with cases where they don't
seem to be working. That amounts to supplying `:expand`

hints for
those calls that you want opened up, which aren't. See also C8
above.

**F3. The ``waterfall''.**

As discussed many times already, a good strategy for using ACL2 is
to look for checkpoints (goals stable under simplification) when a
proof fails, perhaps using the proof-tree facility. Thus, it
is reasonable to ignore almost all the prover output, and to avoid
pondering the meaning of the other ``processes'' that ACL2 uses
besides simplification (such as elimination, cross-fertilization,
generalization, and elimination of irrelevance). For example, you
don't need to worry about prover output that mentions ``type
reasoning'' or ``abbreviations,'' for example.