`: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.

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.

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).

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 for details.

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`

. 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.
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.

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. Guard verification is also recommended.