Major Section: MISCELLANEOUS

General Form of an Extended :Meta theorem: (implies (and (pseudo-termp x) ; this hyp is optional (alistp a) ; this hyp is optional (ev (hyp-fn x mfc state) a)) ; this hyp is optional (equiv (ev x a) (ev (fn x mfc state) a)))where the restrictions are as described in the documentation for

`meta`

and, in addition, `mfc`

and `state`

are distinct variable
symbols (different also from `x`

and `a`

) and `state`

is literally the
symbol `STATE`

. A `:meta`

theorem of the above form installs `fn`

as a
metatheoretic simplifier with hypothesis function `hyp-fn`

, exactly as for
vanilla metafunctions. The only difference is that when the metafunctions
are applied, some contextual information is passed in via the `mfc`

argument and the ACL2 `state`

is made available.
See meta for a discussion of vanilla flavored metafunctions. This
documentation assumes you are familiar with the simpler situation, in
particular, how to define a vanilla flavored metafunction, `fn`

, and its
associated hypothesis metafunction, `hyp-fn`

, and how to state and prove
metatheorems installing such functions. Defining extended metafunctions
requires that you also be familiar with many ACL2 implementation details.
This documentation is sketchy on these details; see the ACL2 source code or
email the acl2-help list if you need more help.

The metafunction context, `mfc`

, is a list containing many different data
structures used by various internal ACL2 functions. We do not document the
form of `mfc`

. Your extended metafunction should just take `mfc`

as its
second formal and pass it into the functions mentioned below. The ACL2
`state`

is well-documented (see state). The following expressions may be
useful in defining extended metafunctions.

`(mfc-clause mfc)`

: returns the current goal, in clausal form. A clause is
a list of ACL2 terms, implicitly denoting the disjunction of the listed
terms. The clause returned by `mfc-clause`

is the clausal form of the
translation (see trans) of the goal or subgoal on which the rewriter is
working. When a metafunction calls `mfc-clause`

, the term being rewritten
by the metafunction either occurs somewhere in this clause or, perhaps more
commonly, is being rewritten on behalf of some lemma to which the rewriter
has backchained while trying to rewrite a term in the clause.

`(mfc-ancestors mfc)`

: returns the current list of the negations of the
backchaining hypotheses being pursued. In particular,
`(null (mfc-ancestors mfc))`

will be true if and only if the term being
rewritten is part of the current goal; otherwise, that term is part of a
hypothesis from a rule currently being considered for use. Exception: An
ancestor of the form `(:binding-hyp hyp unify-subst)`

indicates that
`hyp`

has been encountered as a hypothesis of the form `(equal var term)`

or `(equiv var (double-rewrite-term))`

that binds variable `var`

to the
result of rewriting `term`

under `unify-subst`

.

`(mfc-type-alist mfc)`

: returns the type-alist governing the occurrence of
the term, `x`

, being rewritten by the metafunction. A type-alist is an
association list, each element of which is of the form `(term ts . ttree)`

.
Such an element means that the term `term`

has the type-set `ts`

.
The `ttree`

component is probably irrelevant here. All the terms in the
type-alist are in translated form (see trans). The `ts`

are numbers
denoting finite Boolean combinations of ACL2's primitive types
(see type-set). The type-alist includes not only information gleaned from the
conditions governing the term being rewritten but also that gleaned from
forward chaining from the (negations of the) other literals in the clause
returned by `mfc-clause`

.

`(w state)`

: returns the ACL2 logical `world`

.

`(mfc-ts term mfc state)`

: returns the `type-set`

of `term`

in
the current context; see type-set.

`(mfc-rw term obj equiv-info mfc state)`

: returns the result of rewriting
`term`

in the current context, `mfc`

, with objective `obj`

and the
equivalence relation described by `equiv-info`

. `Obj`

should be `t`

,
`nil`

, or `?`

, and describes your objective: try to show that `term`

is
true, false, or anything. `Equiv-info`

is either `nil`

, `t`

, a
function symbol `fn`

naming a known equivalence relation, or a list of
congruence rules. `Nil`

means return a term that is `equal`

to `term`

.
`T`

means return a term that is propositionally equivalent (i.e., in the
`iff`

sense) to `term`

, while `fn`

means return a term
`fn`

-equivalent to `term`

. The final case, which is intended only for
advanced users, allows the specification of generated equivalence relations,
as supplied to the `geneqv`

argument of `rewrite`

. Generally, if you
wish to establish that `term`

is true in the current context, use the idiom

(equal (mfc-rw term t t mfc state) *t*)The constant

`*t*`

is set to the internal form of the constant term `t`

,
i.e., `'t`

.
`(mfc-rw+ term alist obj equiv-info mfc state)`

: if `alist`

is `nil`

then this is equivalent to `(mfc-rw term obj equiv-info mfc state)`

.
However, the former takes an argument, `alist`

, that binds variables to
terms, and returns the result of rewriting `term`

under that `alist`

,
where this rewriting is as described for `mfc-rw`

above. The function
`mfc-rw+`

can be more efficient than `mfc-rw|`

, because the terms in the
binding alist have generally already been rewritten, and it can be
inefficient to rewrite them again. For example, consider a rewrite rule of
the following form.

(implies (and ... (syntaxp (... (mfc-rw `(bar ,x) ...) ...)) ...) (equal (... x ...) ...))Here,

`x`

is bound in the conclusion to the result of rewriting some term,
say, `tm`

. Then the above call of `mfc-rw`

will rewrite `tm`

, which is
probably unnecessary. So a preferable form of the rule above may be as
follows, so that `tm`

is not further rewritten by `mfc-rw+`

.
(implies (and ... (syntaxp (... (mfc-rw+ '(bar v) `((v . ,x)) ...) ...)) ...) (equal (... x ...) ...))However, you may find that the additional rewriting done by

`mfc-rw`

is
useful in some cases.
`(mfc-ap term mfc state)`

: returns `t`

or `nil`

according to whether
linear arithmetic can determine that `term`

is false. To the cognoscenti:
returns the contradiction flag produced by linearizing `term`

and adding it
to the `linear-pot-lst`

.

During the execution of a metafunction by the theorem prover, the expressions
above compute the results specified. However, there are no axioms about the
`mfc-`

function symbols: they are uninterpreted function symbols. Thus, in
the proof of the correctness of a metafunction, no information is available
about the results of these functions. Thus,
*these functions can be used for heuristic purposes only.*

For example, your metafunction may use these functions to decide whether to
perform a given transformation, but the transformation must be sound
regardless of the value that your metafunction returns. We illustrate this
below. It is sometimes possible to use the hypothesis metafunction,
`hyp-fn`

, to generate a sufficient hypothesis to justify the
transformation. The generated hypothesis might have already been ``proved''
internally by your use of `mfc-ts`

or `mfc-rw`

, but the system will have
to prove it ``officially'' by relieving it. We illustrate this below also.

We conclude with a script that defines, verifies, and uses several extended metafunctions. This script is based on one provided by Robert Krug, who was instrumental in the development of this style of metafunction and whose help we gratefully acknowledge.

; Here is an example. I will define (foo i j) simply to be (+ i j). ; But I will keep its definition disabled so the theorem prover ; doesn't know that. Then I will define an extended metafunction ; that reduces (foo i (- i)) to 0 provided i has integer type and the ; expression (< 10 i) occurs as a hypothesis in the clause.

; Note that (foo i (- i)) is 0 in any case.

(defun foo (i j) (+ i j))

(defevaluator eva eva-lst ((foo i j) (unary-- i) ; I won't need these two cases until the last example below, but I ; include them now. (if x y z) (integerp x)))

(set-state-ok t)

(defun metafn (x mfc state) (cond ((and (consp x) (equal (car x) 'foo) (equal (caddr x) (list 'unary-- (cadr x))))

; So x is of the form (foo i (- i)). Now I want to check some other ; conditions.

(cond ((and (ts-subsetp (mfc-ts (cadr x) mfc state) *ts-integer*) (member-equal `(NOT (< '10 ,(cadr x))) (mfc-clause mfc))) (quote (quote 0))) (t x))) (t x)))

(defthm metafn-correct (equal (eva x a) (eva (metafn x mfc state) a)) :rule-classes ((:meta :trigger-fns (foo))))

(in-theory (disable foo))

; The following will fail because the metafunction won't fire. ; We don't know enough about i.

(thm (equal (foo i (- i)) 0))

; Same here.

(thm (implies (and (integerp i) (< 0 i)) (equal (foo i (- i)) 0)))

; But this will work.

(thm (implies (and (integerp i) (< 10 i)) (equal (foo i (- i)) 0)))

; This won't, because the metafunction looks for (< 10 i) literally, ; not just something that implies it. (thm (implies (and (integerp i) (< 11 i)) (equal (foo i (- i)) 0)))

; Now I will undo the defun of metafn and repeat the exercise, but ; this time check the weaker condition that (< 10 i) is provable ; (by rewriting it) rather than explicitly present.

(ubt 'metafn) (defun metafn (x mfc state) (cond ((and (consp x) (equal (car x) 'foo) (equal (caddr x) (list 'unary-- (cadr x)))) (cond ((and (ts-subsetp (mfc-ts (cadr x) mfc state) *ts-integer*) (equal (mfc-rw `(< '10 ,(cadr x)) t t mfc state) *t*)) ; The mfc-rw above rewrites (< 10 i) with objective t and iffp t. The ; objective means the theorem prover will try to establish it. The ; iffp means the theorem prover can rewrite maintaining propositional ; equivalence instead of strict equality.

(quote (quote 0))) (t x))) (t x)))

(defthm metafn-correct (equal (eva x a) (eva (metafn x mfc state) a)) :rule-classes ((:meta :trigger-fns (foo))))

(in-theory (disable foo))

; Now it will prove both:

(thm (implies (and (integerp i) (< 10 i)) (equal (foo i (- i)) 0)))

(thm (implies (and (integerp i) (< 11 i)) (equal (foo i (- i)) 0)))

; Now I undo the defun of metafn and change the problem entirely. ; This time I will rewrite (integerp (foo i j)) to t. Note that ; this is true if i and j are integers. I can check this ; internally, but have to generate a hyp-fn to make it official.

(ubt 'metafn) (defun metafn (x mfc state) (cond ((and (consp x) (equal (car x) 'integerp) (consp (cadr x)) (equal (car (cadr x)) 'foo)) ; So x is (integerp (foo i j)). Now check that i and j are ; ``probably'' integers.

(cond ((and (ts-subsetp (mfc-ts (cadr (cadr x)) mfc state) *ts-integer*) (ts-subsetp (mfc-ts (caddr (cadr x)) mfc state) *ts-integer*)) *t*) (t x))) (t x)))

; To justify this transformation, I generate the appropriate hyps.

(defun hyp-fn (x mfc state)

(declare (ignore mfc state))

; The hyp-fn is run only if the metafn produces an answer different ; from its input. Thus, we know at execution time that x is of the ; form (integerp (foo i j)) and we know that metafn rewrote ; (integerp i) and (integerp j) both to t. So we just produce their ; conjunction. Note that we must produce a translated term; we ; cannot use the macro AND and must quote constants! Sometimes you ; must do tests in the hyp-fn to figure out which case the metafn ; produced, but not in this example. `(if (integerp ,(cadr (cadr x))) (integerp ,(caddr (cadr x))) 'nil)) (defthm metafn-correct (implies (eva (hyp-fn x mfc state) a) (equal (eva x a) (eva (metafn x mfc state) a))) :rule-classes ((:meta :trigger-fns (integerp))))

(in-theory (disable foo))

; This will not be proved.

(thm (implies (and (rationalp x) (integerp i)) (integerp (foo i j))))

; But this will be.

(thm (implies (and (rationalp x) (integerp i) (integerp j)) (integerp (foo i j))))