## META

make a `: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.

• ### 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
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, in which case it is presumably unnecessary anyhow; 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))
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))>
```