TYPE-PRESCRIPTION

make a rule that specifies the type of a term 
Major Section:  RULE-CLASSES

See rule-classes for a general discussion of rule classes and how they are used to build rules from formulas. Some example :corollary formulas from which :type-prescription rules might be built are: 

Examples:
(implies                           (nth n lst) is of type characterp
 (and (character-listp lst)        provided the two hypotheses can
      (< n (length lst)))          be established by type reasoning
 (characterp (nth n lst))).

(implies                           (demodulize a lst 'value ans) is
 (and (atom a)                     either a nonnegative integer or
      (true-listp lst)             of the same type as ans, provided
      (member-equal a lst))        the hyps can be established by type
 (or (and                          reasoning
       (integerp
        (demodulize a lst 'value ans))
       (>= (demodulize a lst 'value ans) 0))
     (equal (demodulize a lst 'value ans) ans))).
To specify the term whose type (see type-set) is described by the rule, provide that term as the value of the :typed-term field of the rule class object. 

General Form (after preprocessing; see below):
(implies hyps
         (or type-restriction1-on-pat
             ...
             type-restrictionk-on-pat
             (equal pat var1)
             ...
             (equal pat varj)))
where pat is the application of some function symbol to some arguments, each type-restrictioni-on-pat is a term involving pat and containing no variables outside of the occurrences of pat, and each vari is one of the variables of pat. Generally speaking, the type-restriction terms ought to be terms that inform us as to the type of pat. Ideally, they should be ``primitive recognizing expressions'' about pat; see compound-recognizer. We describe preprocessing at the end of this topic.

If the :typed-term is not provided in the rule class object, it is computed heuristically by looking for a term in the conclusion whose type is being restricted. An error is caused if no such term is found.

Roughly speaking, the effect of adding such a rule is to inform the ACL2 typing mechanism that pat has the type described by the conclusion, when the hypotheses are true. In particular, the type of pat is within the union of the types described by the several disjuncts. The ``type described by'' (equal pat vari) is the type of vari.

More operationally, when asked to determine the type of a term that is an instance of pat, ACL2 will first attempt to establish the hypotheses. This is done by type reasoning alone, not rewriting! However, if some hypothesis is a call of force, then forcing may occur, which may ultimately invoke the rewriter; see force and see case-split. So-called free variables in hypotheses are treated specially; see free-variables. Provided the hypotheses are established by type reasoning, ACL2 then unions the types described by the type-restrictioni-on-pat terms together with the types of those subexpressions of pat identified by the vari. The final type computed for a term is the intersection of the types implied by each applicable rule. Type prescription rules may be disabled.

You can limit the recursive establishment of hypotheses of rules; see set-backchain-limit.

Because only type reasoning is used to establish the hypotheses of :type-prescription rules, some care must be taken with the hypotheses. Suppose, for example, that the non-recursive function my-statep is defined as 

  (defun my-statep (x)
    (and (true-listp x)
         (equal (length x) 2)))
and suppose (my-statep s) occurs as a hypothesis of a :type-prescription rule that is being considered for use in the proof attempt for a conjecture with the hypothesis (my-statep s). Since the hypothesis in the conjecture is rewritten, it will become the conjunction of (true-listp s) and (equal (length s) 2). Those two terms will be assumed to have type t in the context in which the :type-prescription rule is tried. But type reasoning will be unable to deduce that (my-statep s) has type t in this context. Thus, either my-statep should be disabled (see disable) during the proof attempt or else the occurrence of (my-statep s) in the :type-prescription rule should be replaced by the conjunction into which it rewrites.

While this example makes it clear how non-recursive predicates can cause problems, non-recursive functions in general can cause problems. For example, if (mitigate x) is defined to be (if (rationalp x) (1- x) x) then the hypothesis (pred (mitigate s)) in the conjecture will rewrite, opening mitigate and splitting the conjecture into two subgoals, one in which (rationalp s) and (pred (1- x)) are assumed and the other in which (not (rationalp s)) and (pred x) are assumed. But (pred (mitigate s)) will not be typed as t in either of these contexts. The moral is: beware of non-recursive functions occuring in the hypotheses of :type-prescription rules.

Because of the freedom one has in forming the conclusion of a type-prescription, we have to use heuristics to recover the pattern, pat, whose type is being specified. In some cases our heuristics may not identify the intended term and the :type-prescription rule will be rejected as illegal because the conclusion is not of the correct form. When this happens you may wish to specify the pat directly. This may be done by using a suitable rule class token. In particular, when the token :type-prescription is used it means ACL2 is to compute pat with its heuristics; otherwise the token should be of the form (:type-prescription :typed-term pat), where pat is the term whose type is being specified.

The defun event may generate a :type-prescription rule. Suppose fn is the name of the function concerned. Then (:type-prescription fn) is the rune given to the type-prescription, if any, generated for fn by defun. (The trivial rule, saying fn has unknown type, is not stored, but defun still allocates the rune and the corollary of this rune is known to be t.)

We close with a discussion of how, before a term is parsed into a :type-prescription rule, it is preprocessed. We describe this preprocessing in some detail below, but first consider the following (contrived) example. 

(defthm append-tp-example
  (let ((result (append x y)))
    (implies (nat-listp x)
             (implies (let ((second-hyp (integer-listp y)))
                        second-hyp)
                      (true-listp result))))
  :rule-classes :type-prescription)
This theorem is parsed into a type-prescription rule with the following hypotheses and conclusion. 
(nat-listp x) ; first hypothesis
((lambda (second-hyp) second-hyp) (integer-listp y)) ; second hypothesis
(true-listp (binary-append x y)) ; conclusion
Notice that the top-level LET was expanded, i.e., (append x y) was substituted for result -- more accurately, (binary-append x y) was substituted for result, since append is a macro that abbreviates binary-append. Also notice that the two hypotheses were ``flattened'' in the sense that they were gathered up into a list. Finally, notice that the LET in the second hypothesis was not expanded (it was merely translated to internal form, using LAMBDA). If you actually submit the theorem above, you will get warnings, which you may choose to ignore; the application of type-prescription rules is somewhat subtle, so if you use them then you may wish to experiment to see which forms work best for you.

Here is the detail promised above, for parsing a term into a :type-prescription rule. There are two steps. (1) ACL2 first translates the term, expanding all macros (see trans) and also expanding away calls of all so-called ``guard holders,'' mv-list and return-last (the latter resulting for example from calls of prog2$, mbe, or ec-call), as well as expansions of the macro `the'. (2) Then the the translated term is traversed top-down, expanding away lambdas (let, let*, and mv-let expressions) and flattening the IMPLIES structure, until the conclusion is exposed; then the conclusion's lambdas are also expanded away. The simplest way to understand (2) may be to look at the definition of ACL2 source function unprettyify-tp, which implements Step (2), say by evaluating :pe unprettyify-tp.