`:meta`

rule (a hand-written simplifier)
Major Section: RULE-CLASSES

See rule-classes for a general discussion of rule classes and how they are
used to build rules from formulas. Meta rules extend the ACL2 simplifier
with hand-written code to transform certain terms to equivalent ones. To add
a meta rule, the `:`

`corollary`

formula must establish that the
hand-written ``metafunction'' preserves the meaning of the transformed term.

While our intention is that the set of ACL2 documentation topics is self-contained, readers might find it useful to see the following paper for an introduction to meta reasoning in ACL2.

W. A. Hunt, Jr., R. B. Krug, M. Kaufmann, J S. Moore and E. W. Smith, ``Meta Reasoning in ACL2.'' TPHOLs 2005, ed. J. Hurd and T. F. Melham, LNCS 3603, Springer-Verlag, Berlin, 2005, pp. 163-178.

Example `:`

`corollary`

formulas from which `:meta`

rules might be
built are:

Examples: (equal (evl x a) ; Modify the rewriter to use fn to (evl (fn x) a)) ; transform terms. The :trigger-fns ; of the :meta rule-class specify ; the top-most function symbols of ; those x that are candidates for ; this transformation.A non-(implies (and (pseudo-termp x) ; As above, but this illustrates (alistp a)) ; that without loss of generality we (equal (evl x a) ; may restrict x to be shaped like a (evl (fn x) a))) ; term and a to be an alist.

(implies (and (pseudo-termp x) ; As above (with or without the (alistp a) ; hypotheses on x and a) with the (evl (hyp-fn x) a)) ; additional restriction that the (equal (evl x a) ; meaning of (hyp-fn x) is true in (evl (fn x) a))) ; the current context. That is, the ; applicability of the transforma- ; tion may be dependent upon some ; computed hypotheses.

`nil`

list of function symbols must be supplied as the value
of the `:trigger-fns`

field in a `:meta`

rule class object.

### EVALUATOR-RESTRICTIONS -- some restrictions on the use of evaluators in meta-level rules

General Forms: (implies (and (pseudo-termp x) ; this hyp is optional (alistp a) ; this hyp is optional (ev (hyp-fn x ...) a)) ; this hyp is optional (equiv (ev x a) (ev (fn x ...) a)))where

`equiv`

is a known equivalence relation, `x`

and `a`

are
distinct variable names, and `ev`

is an evaluator function (see below), and
`fn`

is a function symbol, as is `hyp-fn`

when provided. The arguments
to `fn`

and `hyp-fn`

should be identical. In the most common case, both
take a single argument, `x`

, which denotes the term to be simplified. If
`fn`

and/or `hyp-fn`

are guarded, their guards should be
(implied by) `pseudo-termp`

. See pseudo-termp. We say the
theorem above is a ``metatheorem'' or ``metalemma'' and `fn`

is a
``metafunction'', and `hyp-fn`

is a ``hypothesis metafunction''.
If ```...`

'' is empty, i.e., the metafunctions take just one argument, we
say they are ``vanilla flavored.'' If ```...`

'' is non-empty, we say the
metafunctions are ``extended.'' Extended metafunctions can access
`state`

and context sensitive information to compute their results,
within certain limits. We discuss vanilla metafunctions here and recommend a
thorough understanding of them before proceeding (at which time
see extended-metafunctions).

Metafunctions and (if supplied) hypothesis metafunctions must be executable.
That is, they cannot be constrained (i.e., introduced in the signature
of `encapsulate`

events), nor can they be declared
`:`

`non-executable`

. After all, there is no point in installing a
simplifier that cannot be run!

We defer discussion of the case in which there is a hypothesis metafunction and for now address the case in which the other two hypotheses are present.

In the discussion below, we refer to the argument, `x`

, of `fn`

and
`hyp-fn`

as a ``term.'' When these metafunctions are executed by the
simplifier, they will be applied to (the quotations of) terms. But during
the proof of the metatheorem itself, `x`

may not be the quotation of a
term. If the `pseudo-termp`

hypothesis is omitted, `x`

may be any
object. Even with the `pseudo-termp`

hypothesis, `x`

may merely ``look
like a term'' but use non-function symbols or function symbols of incorrect
arity. In any case, the metatheorem is stronger than necessary to allow us
to apply the metafunctions to terms, as we do in the discussion below. We
return later to the question of proving the metatheorem.

Suppose the general form of the metatheorem above is proved with the
`pseudo-termp`

and `alistp`

hypotheses. Then when the simplifier
encounters a term, `(h t1 ... tn)`

, that begins with a function symbol,
`h`

, listed in `:trigger-fns`

, it applies the metafunction, `fn`

, to
the quotation of the term, i.e., it evaluates `(fn '(h t1 ... tn))`

to
obtain some result, which can be written as `'val`

. If `'val`

is
different from `'(h t1 ... tn)`

and `val`

is a term, then
`(h t1 ... tn)`

is replaced by `val`

, which is then passed along for
further rewriting. Because the metatheorem establishes the correctness of
`fn`

for all terms (even non-terms!), there is no restriction on which
function symbols are listed in the `:trigger-fns`

. Generally, of course,
they should be the symbols that head up the terms simplified by the
metafunction `fn`

. See term-table for how one obtains some assistance
towards guaranteeing that `val`

is indeed a term.

The ``evaluator'' function, `ev`

, is a function that can evaluate a certain
class of expressions, namely, all of those composed of variables, constants,
and applications of a fixed, finite set of function symbols, `g1`

, ...,
`gk`

. Generally speaking, the set of function symbols handled by `ev`

is
chosen to be exactly the function symbols recognized and manipulated by the
metafunctions being introduced. For example, if `fn`

manipulates
expressions in which `'`

`equal`

and `'`

`binary-append`

occur as
function symbols, then `ev`

is generally specified to handle `equal`

and `binary-append`

. The actual requirements on `ev`

become clear when
the metatheorem is proved. The standard way to introduce an evaluator is to
use the ACL2 macro `defevaluator`

, though this is not strictly necessary.
See defevaluator if you want details.

[Aside for the logic-minded.] Why are we justified in using metafunctions
this way? Suppose `(fn 'term1)`

is `'term2`

. What justifies replacing
`term1`

by `term2`

? The first step is to assert that `term1`

is
`(ev 'term1 a)`

, where `a`

is an alist that maps `'var`

to `var`

, for
each variable `var`

in `term1`

. This step is incorrect, because
`'term1`

may contain function symbols other than the ones, `g1`

, ...,
`gk`

, that `ev`

knows how to handle. But we can grow `ev`

to a
``larger'' evaluator, `ev*`

, an evaluator for all of the symbols that occur
in `term1`

or `term2`

. We can prove that `ev*`

satisfies the
constraints on `ev`

, provided no `defaxiom`

events are adding
constraints to `ev`

(or callers of `ev`

, and recursively); ACL2 checks
this additional property. Hence, the metatheorem holds for `ev*`

in place
of `ev`

, by functional instantiation. We can then carry out the proof of
the equivalence of `term1`

and `term2`

as follows: Fix `a`

to be
an alist that maps the quotations of the variables of `term1`

and `term2`

to themselves. Then,

term1 = (ev* 'term1 a) ; (1) by construction of ev* and a = (ev* (fn 'term1) a) ; (2) by the metatheorem for ev* = (ev* 'term2 a) ; (3) by evaluation of fn = term2 ; (4) by construction of ev* and aNote that in line (2) above, where we appeal to the (functional instantiation of the) metatheorem, we can relieve its (optional)

`pseudo-termp`

and `alistp`

hypotheses by appealing to the facts that
`term1`

is a term and `a`

is an alist by construction. [End of Aside for
the logic-minded.]
There are subtleties related to the notion of ``growing'' `ev`

to a
``larger'' evaluator, as mentioned in the paragraph just above. For
corresponding restrictions on `:meta`

rules, see evaluator-restrictions.

Finally, we turn to the second case, in which there is a hypothesis
metafunction. In that case, consider as before what happens when the
simplifier encounters a term, `(h t1 ... tn)`

, where `h`

is listed in
`:trigger-fns`

. This time, after it applies `fn`

to `'(h t1 ... tn)`

to obtain the quotation of some new term, `'val`

, it then applies the
hypothesis metafunction, `hyp-fn`

. That is, it evaluates
`(hyp-fn '(h t1 ... tn))`

to obtain some result, which can be written as
`'hyp-val`

. If `hyp-val`

is not in fact a term, the metafunction is not
used. Provided `hyp-val`

is a term, the simplifier attempts to establish
(by conventional backchaining) that this term is non-`nil`

in the current
context. If this attempt fails, then the meta rule is not applied.
Otherwise, `(h t1...tn)`

is replaced by `val`

as in the previous case
(where there was no hypothesis metafunction).

Why is it justified to make this extension to the case of hypothesis metafunctions? First, note that the rule

(implies (and (pseudo-termp x) (alistp a) (ev (hyp-fn x) a)) (equal (ev x a) (ev (fn x) a)))is logically equivalent to the rule

(implies (and (pseudo-termp x) (alistp a)) (equal (ev x a) (ev (new-fn x) a)))where

`(new-fn x)`

is defined to be `(list 'if (hyp-fn x) (fn x) x)`

.
(If we're careful, we realize that this argument depends on making an
extension of `ev`

to an evaluator `ev*`

that handles `if`

and the
functions manipulated by `hyp-fn`

.) If we write `'term`

for the
quotation of the present term, and if `(hyp-fn 'term)`

and `(fn 'term)`

are both terms, say `hyp1`

and `term1`

, then by the previous argument we
know it is sound to rewrite term to `(if hyp1 term1 term)`

. But since we
have established in the current context that `hyp1`

is non-`nil`

, we may
simplify `(if hyp1 term1 term)`

to `term1`

, as desired.
We now discuss the role of the `pseudo-termp`

hypothesis.
`(Pseudo-termp x)`

checks that `x`

has the shape of a term. Roughly
speaking, it ensures that `x`

is a symbol, a quoted constant, or a true
list consisting of a `lambda`

expression or symbol followed by some
pseudo-terms. Among the properties of terms not checked by
`pseudo-termp`

are that variable symbols never begin with ampersand,
`lambda`

expressions are closed, and function symbols are applied to the
correct number of arguments. See pseudo-termp.

There are two possible roles for `pseudo-termp`

in the development of a
metatheorem: it may be used as the guard of the metafunction and/or
hypothesis metafunction and it may be used as a hypothesis of the
metatheorem. Generally speaking, the `pseudo-termp`

hypothesis is
included in a metatheorem only if it makes it easier to prove. The choice is
yours. (An extreme example of this is when the metatheorem is invalid
without the hypothesis!) We therefore address ourselves the question: should
a metafunction have a `pseudo-termp`

guard? A `pseudo-termp`

guard for a metafunction, in connection with other considerations
described below, improves the efficiency with which the metafunction is used
by the simplifier.

To make a metafunction maximally efficient you should (a) provide it with a
`pseudo-termp`

guard and exploit the guard when possible in
coding the body of the function (see guards-and-evaluation, especially the
section on efficiency issues), (b) verify the guards of the metafunction
(see verify-guards), and (c) compile the metafunction (see comp). When
these three steps have been taken the simplifier can evaluate `(fn 'term1)`

by running the compiled ``primary code'' (see guards-and-evaluation) for
`fn`

directly in Common Lisp. (Note however that explicit compilation may
be suppressed; see compilation.)

Before discussing efficiency issues further, let us review for a moment the
general case in which we wish to evaluate `(fn 'obj)`

for some
`:`

`logic`

function. We must first ask whether the guards of
`fn`

have been verified. If not, we must evaluate `fn`

by executing its
logic definition. This effectively checks the guards of every
subroutine and so can be slow. If, on the other hand, the guards of
`fn`

have been verified, then we can run the primary code for `fn`

,
provided `'obj`

satisfies the guard of `fn`

. So we must next
evaluate the guard of `fn`

on `'obj`

. If the guard is met,
then we run the primary code for `fn`

, otherwise we run the logic code.

Now in the case of a metafunction for which the three steps above have been
followed, we know the guard is (implied by) `pseudo-termp`

and that
it has been verified. Furthermore, we know without checking that the
guard is met (because `term1`

is a term and hence `'term1`

is a
`pseudo-termp`

). Hence, we can use the compiled primary code directly.

We strongly recommend that you compile your metafunctions, as well as all their subroutines (unless explicit compilation is suppressed; see compilation). Guard verification is also recommended.

Finally, we present a very simple example of the use of `:meta`

rules,
based on one provided by Robert Krug. This example illustrates a trick for
avoiding undesired rewriting after applying a metafunction or any other form
of rewriting. To elaborate: in general, the term `t2`

obtained by applying
a metafunction to a term `t1`

is then handed immediately to the rewriter,
which descends recursively through the arguments of function calls to rewrite
`t2`

completely. But if `t2`

shares a lot of structure with `t1`

, then
it might not be worthwhile to rewrite `t2`

immediately. (A rewrite of
`t2`

will occur anyhow the next time a goal is generated.) The trick
involves avoiding this rewrite by wrapping `t2`

inside a call of
`hide`

, which in turn is inside a call of a user-defined ``unhiding''
function, `unhide`

.

(defun unhide (x) (declare (xargs :guard t)) x)(defthm unhide-hide (equal (unhide (hide x)) x) :hints (("Goal" :expand ((hide x)))))

(in-theory (disable unhide))

(defun my-plus (x y) (+ x y))

(in-theory (disable my-plus))

(defevaluator evl evl-list ((my-plus x y) (binary-+ x y) (unhide x) (hide x)))

(defun meta-fn (term) (declare (xargs :guard (pseudo-termp term))) (if (and (consp term) (equal (length term) 3) (equal (car term) 'my-plus)) `(UNHIDE (HIDE (BINARY-+ ,(cadr term) ,(caddr term)))) term))

(defthm my-meta-lemma (equal (evl term a) (evl (meta-fn term) a)) :hints (("Goal" :in-theory (enable my-plus))) :rule-classes ((:meta :trigger-fns (my-plus))))

Notice that in the following (silly) conjectre, ACL2 initially does only does
the simplification directed by the metafunction; a second goal is generated
before the commuativity of addition can be applied. If the above calls of
`UNHIDE`

and `HIDE`

had been stripped off, then `Goal'`

would have been
the term printed in `Goal''`

below.

ACL2 !>(thm (equal (my-plus b a) ccc))This simplifies, using the :meta rule MY-META-LEMMA and the :rewrite rule UNHIDE-HIDE, to

Goal' (EQUAL (+ B A) CCC).

This simplifies, using the :rewrite rule COMMUTATIVITY-OF-+, to

Goal'' (EQUAL (+ A B) CCC).

The discussion above probably suffices to make good use of this
`(UNHIDE (HIDE ...))`

trick. However, we invite the reader who wishes to
understand the trick in depth to evaluate the following form before
submitting the `thm`

form above.

(trace$ (rewrite :entry (list (take 2 arglist)) :exit (list (car values))) (rewrite-with-lemma :entry (list (take 2 arglist)) :exit (take 2 values)))The following annotated subset of the trace output (which may appear a bit different depending on the underlying Common Lisp implementation) explains how the trick works.

2> (REWRITE ((MY-PLUS B A) NIL))> 3> (REWRITE-WITH-LEMMA ((MY-PLUS B A) (REWRITE-RULE (:META MY-META-LEMMA) 1822 NIL EQUAL META-FN NIL META NIL NIL)))>We apply the meta rule, then recursively rewrite the result, which is the (UNHIDE (HIDE ...)) term shown just below.

4> (REWRITE ((UNHIDE (HIDE (BINARY-+ B A))) ((A . A) (B . B))))> 5> (REWRITE ((HIDE (BINARY-+ B A)) ((A . A) (B . B))))>

The HIDE protects its argument from being touched by the rewriter.

<5 (REWRITE (HIDE (BINARY-+ B A)))> 5> (REWRITE-WITH-LEMMA ((UNHIDE (HIDE (BINARY-+ B A))) (REWRITE-RULE (:REWRITE UNHIDE-HIDE) 1806 NIL EQUAL (UNHIDE (HIDE X)) X ABBREVIATION NIL NIL)))>

Now we apply UNHIDE-HIDE, then recursively rewrite its right-hand side in an environment where X is bound to (BINARY-+ B A).

6> (REWRITE (X ((X BINARY-+ B A))))>

Notice that at this point X is cached, so REWRITE just returns (BINARY-+ B A).

<6 (REWRITE (BINARY-+ B A))> <5 (REWRITE-WITH-LEMMA T (BINARY-+ B A))> <4 (REWRITE (BINARY-+ B A))> <3 (REWRITE-WITH-LEMMA T (BINARY-+ B A))> <2 (REWRITE (BINARY-+ B A))>