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:
(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.
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!
Of course, if some hypothesis is to be forced, it is so
treated; see force and see case-split. 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.
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.)
