: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.
(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.
A non-nil list of function symbols must be supplied as the value
of the :trigger-fns field in a :meta rule class object
(except that a macro alias can stand in for a function symbol;
see add-macro-alias).
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
; meta-extract hyps may also be included (see below)
)
(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
trivially implied by 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).
Additional hypotheses are supported, called ``meta-extract hypotheses''. These allow metafunctions to depend on the validity of certain terms extracted from the context or the logical world. These hypotheses provide a relatively advanced form of metatheorem so we explain them elsewhere; see meta-extract.
One might think that metafunctions and (if supplied) hypothesis metafunctions
must be executable: that is, not constrained (i.e., introduced in the
signature of encapsulate events), and not declared
:non-executable. After all, there is no point in installing a
simplifier that cannot be run! However, such a restriction is not enforced,
because one could introduce a metafunction using encapsulate and then
use defattach to attach it to an executable function; see defattach.
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 a
Note 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) conjecture, 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))>
