state and context sensitive metafunctions

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.

(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. Iffp 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-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))))