/usr/share/acl2-8.0dfsg/books/tools/defthmg.lisp is in acl2-books-source 8.0dfsg-1.
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; Written by Matt Kaufmann
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
(in-package "ACL2")
(include-book "xdoc/top" :dir :system)
; IMPLIEZ is defined in std/basic/defs.lisp:
(include-book "std/basic/defs" :dir :system)
(program)
(set-state-ok t)
(defun impliez-ify (form ctx state)
; We return a special value, (value nil), if IMPLIES does not occur in the
; translation of form. The caller should handle that appropriately.
(let ((wrld (w state)))
(er-let*
((term1 (translate form t t t ctx wrld state)))
(cond
((ffnnamep 'implies term1)
(mv-let (vars term2)
; This is an odd use of sublis-fn, since it replaces a function with a macro.
; But it seems to be OK, and since defthmg generates event forms that are
; checked, there is no soundness issue.
(sublis-fn '((implies . impliez)) term1 nil)
(declare (ignore vars))
(value (untranslate term2 nil wrld))))
(t (value nil))))))
(defun defthmg-rule-classes (rule-classes thm)
; We leave some of the error-checking to the generated defthm.
(cond ((atom rule-classes) rule-classes) ; should be nil
(t (cons (let ((x (car rule-classes)))
; The following code is adapted from source function translate-rule-class.
(cond
((keywordp x)
(list x
:COROLLARY
thm
:hints
'(("Goal" :in-theory (theory 'minimal-theory)))))
((and (consp x)
(keywordp (car x))
(keyword-value-listp (cdr x)))
(cond
((assoc-keyword :COROLLARY (cdr x)) x)
(t `(,(car x)
:COROLLARY ,thm
,@(if (assoc-keyword :hints (cdr x))
nil
`(:hints
'(("Goal"
:in-theory (theory 'minimal-theory)))))
,@(cdr x)))))
(t ; bad rule-class; let defthm catch this
x)))
(defthmg-rule-classes (cdr rule-classes) thm)))))
(defun defthmg-fn (event-form name term rule-classes verify-guards
verify-guards-p guard-hints state)
; Note that rule-classes is (:REWRITE) if no rule-classes were supplied with
; the original defthm.
(let ((ctx (cons 'defthmg name)))
(er-let* ((term-z (impliez-ify term ctx state)))
(mv-let (implies-p term-z)
(cond ((null term-z)
(mv nil term))
(t
(mv t term-z)))
(let ((rule-classes
(if implies-p
(defthmg-rule-classes
; Borrowing code from source function translate-rule-classes:
(cond ((null rule-classes) nil)
((atom rule-classes) (list rule-classes))
(t rule-classes))
term)
rule-classes)))
(cond
((eq rule-classes :error)
; This error message is less informative than one gets with defthm. But it
; doesn't seem worth the effort to improve it, since it already points to
; suitable :doc.
(er soft ctx
"Illegal :RULE-CLASSES argument. See :DOC rule-classes."))
(t
(let ((defthm-form
`(defthm ,name
,term-z
:rule-classes ,rule-classes
,@(remove-keyword
:verify-guards
(remove-keyword
:guard-hints
(remove-keyword
:rule-classes
(cdddr event-form))))))
(vg (if verify-guards-p
verify-guards
(not (eql (default-verify-guards-eagerness (w state))
0)))))
(value (cond
(vg `(progn ,defthm-form
(verify-guards ,name
,@(and guard-hints
`(:hints ,guard-hints)))))
(t defthm-form)))))))))))
(defmacro defthmg (&whole event-form
name term
&key (rule-classes '(:REWRITE))
(verify-guards 'nil ; irrelevant value
verify-guards-p)
guard-hints
&allow-other-keys) ; other defthm keyword args
(declare (xargs :guard (booleanp verify-guards)))
`(make-event (defthmg-fn ',event-form ',name ',term ',rule-classes
',verify-guards ',verify-guards-p ',guard-hints
state)))
(defxdoc defthmg
; Technical point: because defthmg's implementation using make-event, one might
; think that the make-event expansion during the second pass of encapsulate or
; certify-book could differ from that of the first pass, because of the
; dependence on the world. However, that dependence is only on
; verify-guards-eagerness, which is carried in the acl2-defaults-table and
; hence must actually be the same at expansion time on both passes. This seems
; like too subtle a point to make in user-level documentation, so we only make
; it here in a Lisp comment.
:parents (events guard)
:short "Variant of @(tsee defthm) supporting @(see guard) verification"
:long "<p>After a @(tsee defthm) event introduces a name, @(tsee
verify-guards) can be called on that theorem name, just as it can be called on
a function symbol. However, the proof obligation for verifying @(see guard)s
is often not a theorem. After presenting the general form for @('defthmg'),
we give a running example, which illustrates a problem with @('implies') for
guard verification and how @('defthmg') solves that problem.</p>
@({
Example Form:
(defthmg true-listp-member-equal
(implies (true-listp x)
(true-listp (member-equal a x))))
General Form:
(defthmg name
body
;;; defthm arguments:
:rule-classes rule-classes
:instructions instructions
:hints hints
:otf-flg otf-flg
;;; new arguments:
:verify-guards verify-guards
:guard-hints guard-hints)
})
<p>where all but the last two keyword arguments are exactly as for @(tsee
defthm). If @(':verify-guards') is supplied then it must be @('t') or
@('nil'), indicating that a call of @(tsee verify-guards) on @('name') will or
won't be attempted, respectively. If @(':verify-guards') is omitted, then its
value is normally treated as @('t'); but it is treated as @('nil') if the
@(see default-verify-guards-eagerness) is 0 (rather than either 2 or its usual
value of 1, as we will assume for the rest of this documentation topic).
Finally, if @(':guard-hints') is supplied and @('verify-guards') is attempted
on @('name'), then the specified @('guard-hints') will become the value of
@(':hints') for that @('verify-guards') event.</p>
<p>We now consider in some detail the example displayed above. Consider it
again, but this time with @('defthm') instead of @('defthmg').</p>
@({
(defthm true-listp-member-equal
(implies (true-listp x)
(true-listp (member-equal a x))))
})
<p>The proof succeeds, after which we might try to call @(tsee verify-guards).
But @('verify-guards') would guarantee that the indicated form will evaluate
without a Lisp guard violation for all values of @('a') and @('x'), and that's
not always the case! Suppose for example that @('x') is @('17'). Since
@('implies') is an ordinary function, evaluation will take place for both its
arguments, even though @('(true-listp x)') is false. The call
@('(member-equal a 17)') will cause a guard violation, regardless of the value
of @('a'), since the guard for @(tsee member-equal) requires that its second
argument satisfy @(tsee true-listp).</p>
<p>A way to allow guard verification for such a theorem is to replace
@('implies') by the macro, @(tsee impliez), whose calls expand to calls of
@('IF'), for example as follows (see @(tsee trans1)).</p>
@({
ACL2 !>:trans1 (impliez (true-listp x)
(true-listp (member-equal a x)))
(IF (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X))
T)
ACL2 !>
})
<p>When @('x') is 17, evaluation of the form above (either the @('impliez')
version or its expansion to an @('IF') call) will <i>not</i> lead to
evaluation of the @('member-equal') call. Guard verification will then be
possible.</p>
<p>But simply changing @('implies') to @('impliez') doesn't work.</p>
@({
ACL2 !>(defthm true-listp-member-equal
(impliez (true-listp x)
(true-listp (member-equal a x))))
ACL2 Error in ( DEFTHM TRUE-LISTP-MEMBER-EQUAL ...): A :REWRITE rule
generated from TRUE-LISTP-MEMBER-EQUAL is illegal because it rewrites
the IF-expression (IF (TRUE-LISTP X) (TRUE-LISTP (MEMBER-EQUAL A X)) 'T).
For general information about rewrite rules in ACL2, see :DOC rewrite.
})
<p>The error message is basically telling us that we need @('implies'), not
@('impliez') (or @('IF')), in order to store the indicated theorem as a @(see
rewrite) rule, which is the default. We can overcome this problem by
supplying an explicit @(':')@(tsee corollary) equal to the original theorem,
as follows.</p>
@({
(defthm true-listp-member-equal
(impliez (true-listp x)
(true-listp (member-equal a x)))
:rule-classes
((:rewrite :corollary
(implies (true-listp x)
(true-listp (member-equal a x))))))
})
<p>Now the intended rewrite rule is stored, and also we can verify guards,
since the guard proof obligation is based on the body of the @('defthm')
event (with @('impliez')), not the corollary.</p>
@({
(verify-guards true-listp-member-equal)
})
<p>The purpose of @('defthmg') is to automate the process described above.
Our example @('defthmg') call generates a @(tsee progn) containing the
@('defthm') and @('verify-guards') forms displayed just above (except for the
addition of suitable @(':')@(tsee hints) to streamline the process).</p>
@({
(DEFTHM TRUE-LISTP-MEMBER-EQUAL
(IMPLIEZ (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X)))
:RULE-CLASSES
((:REWRITE
:COROLLARY (IMPLIES (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X)))
:HINTS ((\"Goal\" :IN-THEORY (THEORY 'MINIMAL-THEORY))))))
(VERIFY-GUARDS TRUE-LISTP-MEMBER-EQUAL)
})
<p>If @(':')@(tsee rule-classes) are supplied explicitly, these will be
handled appropriately: for each rule class, if a @(':corollary') is supplied
explicitly then that rule class is not changed, and otherwise a
@(':corollary') is specified to be the original theorem (hence with
@('implies'), not changed to @('impliez')), and the @(':in-theory') hint
displayed just above will be generated in order to make the proof of the
corollary very fast.</p>
<p>The following more complex (but rather nonsensical) example illustrates the
various arguments of @('defthmg').</p>
@({
(defthmg true-listp-member-equal
(implies (true-listp x)
(true-listp (member-equal a x)))
:verify-guards t
:guard-hints ((\"Goal\" :use car-cons))
:hints ((\"Goal\" :induct (member-equal a x)))
:rule-classes
(:rewrite
(:rewrite ; awful rule with free variable
:corollary (implies (not (true-listp (member-equal a x)))
(not (true-listp x))))
:type-prescription)
:otf-flg t)
})
<p>Here are the two events generated after successfully evaluating the form
above. Notice that the rule class with an explicit @(':corollary') is not
modified.</p>
@({
(DEFTHM TRUE-LISTP-MEMBER-EQUAL
(IMPLIEZ (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X)))
:RULE-CLASSES
((:REWRITE
:COROLLARY (IMPLIES (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X)))
:HINTS ((\"Goal\" :IN-THEORY (THEORY 'MINIMAL-THEORY))))
(:REWRITE
:COROLLARY (IMPLIES (NOT (TRUE-LISTP (MEMBER-EQUAL A X)))
(NOT (TRUE-LISTP X))))
(:TYPE-PRESCRIPTION
:COROLLARY (IMPLIES (TRUE-LISTP X)
(TRUE-LISTP (MEMBER-EQUAL A X)))
:HINTS ((\"Goal\" :IN-THEORY (THEORY 'MINIMAL-THEORY)))))
:HINTS ((\"Goal\" :INDUCT (MEMBER-EQUAL A X)))
:OTF-FLG T)
(VERIFY-GUARDS TRUE-LISTP-MEMBER-EQUAL
:HINTS ((\"Goal\" :USE CAR-CONS)))
})")
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