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; Reasoning about evaluators and falsifiers
; Copyright (C) 2010 Centaur Technology
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.com/
;
; License: (An MIT/X11-style license)
;
; Permission is hereby granted, free of charge, to any person obtaining a
; copy of this software and associated documentation files (the "Software"),
; to deal in the Software without restriction, including without limitation
; the rights to use, copy, modify, merge, publish, distribute, sublicense,
; and/or sell copies of the Software, and to permit persons to whom the
; Software is furnished to do so, subject to the following conditions:
;
; The above copyright notice and this permission notice shall be included in
; all copies or substantial portions of the Software.
;
; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
; IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
; AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
; DEALINGS IN THE SOFTWARE.
;
; Original author: Sol Swords <sswords@centtech.com>
(in-package "ACL2")
;; (include-book "misc/untranslate-patterns" :dir :system)
(include-book "join-thms")
;; This is a sadly incomplete book for reasoning about evaluators and
;; "falsifier" functions. An evaluator ev gives a value to a term X
;; under an alist A, and a falsifier evf provides for a given X an
;; alist A under which it evaluates to NIL, if one exists. Therefore,
;; if (ev X (evf X)) is non-nil, then (ev X A) is nonnil for all A,
;; and we say X is a theorem under evaluator ev.
;; Reasoning about this concept gets much more complicated when one
;; considers evaluations of terms under falsifiers of other terms.
;; For example, (ev (and A B) (evf (and A B))) says that (and A B) is
;; a theorem, meaning that A and B are both theorems. But commonly,
;; rewrite rules will decompose that term into
;; (and (ev A (evf (and A B)))
;; (ev B (evf (and A B))))
;; But separately, each of (ev A (evf (and A B))) and
;; (ev B (evf (and A B))) don't tell us much. We'd rather rewrite the
;; original form to
;; (and (ev A (evf A))
;; (ev B (evf B))).
(defmacro def-ev-theoremp (ev)
(let* ((falsify (intern-in-package-of-symbol
(concatenate 'string (symbol-name ev)
"-FALSIFY")
ev))
(theoremp (intern-in-package-of-symbol
(concatenate 'string (symbol-name ev)
"-THEOREMP")
ev))
(subst
`((ev . ,ev)
(falsify . ,falsify)
(theoremp . ,theoremp)
(disjoin-cons-unless-theoremp
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-DISJOIN-CONS-UNLESS-THEOREMP")
ev))
(disjoin-when-consp-unless-theoremp
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-DISJOIN-WHEN-CONSP-UNLESS-THEOREMP")
ev))
(conjoin-cons
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-CONJOIN-CONS")
ev))
(conjoin-append
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-CONJOIN-APPEND")
ev))
(conjoin-clauses-cons
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-CONJOIN-CLAUSES-CONS")
ev))
(conjoin-clauses-append
. ,(intern-in-package-of-symbol
(concatenate 'string (symbol-name theoremp)
"-CONJOIN-CLAUSES-APPEND")
ev))
(disjoin-cons . ,(ev-mk-rulename ev 'disjoin-cons))
(disjoin-when-consp . ,(ev-mk-rulename ev 'disjoin-when-consp))))
(event
(sublis
subst
'(encapsulate nil
(local (in-theory nil))
(def-join-thms ev)
(defchoose falsify (a) (x)
(not (ev x a)))
(defmacro theoremp (x)
`(ev ,x (falsify ,x)))
(defthm conjoin-cons
(iff (theoremp (conjoin (cons a b)))
(and (theoremp a)
(theoremp (conjoin b))))
:hints (("goal" :use
((:instance
falsify
(x (conjoin (cons a b)))
(a (falsify a)))
(:instance
falsify
(x a)
(a (falsify (conjoin (cons a b)))))
(:instance
falsify
(x (conjoin (cons a b)))
(a (falsify (conjoin b))))
(:instance
falsify
(x (conjoin b))
(a (falsify (conjoin (cons a b)))))))))
(defthm conjoin-append
(iff (theoremp (conjoin (append a b)))
(and (theoremp (conjoin a))
(theoremp (conjoin b))))
:hints(("Goal" :in-theory (enable append endp car-cdr-elim))))
(defthm conjoin-clauses-cons
(iff (theoremp
(conjoin-clauses (cons cl1 clrest)))
(and (theoremp (disjoin cl1))
(theoremp (conjoin-clauses clrest))))
:hints(("Goal" :in-theory (enable conjoin-clauses disjoin-lst
car-cons cdr-cons))))
(defthm conjoin-clauses-append
(iff (theoremp
(conjoin-clauses (append cls1 cls2)))
(and (theoremp (conjoin-clauses cls1))
(theoremp (conjoin-clauses cls2))))
:hints (("goal" :in-theory (enable append endp car-cdr-elim)
:induct (append cls1 cls2))))
(defthm disjoin-cons-unless-theoremp
(implies (and (equal aa (falsify (disjoin (cons x y))))
(syntaxp (not (equal a aa))))
(iff (ev (disjoin (cons x y)) a)
(or (ev x a)
(ev (disjoin y) a)))))
(defthmd disjoin-when-consp-unless-theoremp
(implies (syntaxp (not (equal a `(falsify ,x))))
(implies (consp x)
(iff (ev (disjoin x) a)
(or (ev (car x) a)
(ev (disjoin (cdr x)) a)))))
:hints(("Goal" :in-theory (enable disjoin-when-consp))))
(in-theory (disable disjoin-cons))))))
event))
(local
(progn
(defevaluator test-ev test-ev-lst ((if a b c)))
(def-ev-theoremp test-ev)))
#||
(defevaluator evthmp-ev evthmp-ev-lst
((if a b c) (not a)))
(def-join-thms evthmp-ev)
(defchoose evthmp-ev-falsify (a) (x)
(not (evthmp-ev x a)))
(defmacro evthmp-ev-theoremp (x)
`(evthmp-ev ,x (evthmp-ev-falsify ,x)))
(defthm evthmp-ev-theoremp-conjoin-cons
(iff (evthmp-ev-theoremp (conjoin (cons a b)))
(and (evthmp-ev-theoremp a)
(evthmp-ev-theoremp (conjoin b))))
:hints (("goal" :use
((:instance
evthmp-ev-falsify
(x (conjoin (cons a b)))
(a (evthmp-ev-falsify a)))
(:instance
evthmp-ev-falsify
(x a)
(a (evthmp-ev-falsify (conjoin (cons a b)))))
(:instance
evthmp-ev-falsify
(x (conjoin (cons a b)))
(a (evthmp-ev-falsify (conjoin b))))
(:instance
evthmp-ev-falsify
(x (conjoin b))
(a (evthmp-ev-falsify (conjoin (cons a b)))))))))
||#
;; (defthmd evthmp-ev-theoremp-remove-first-lit-when-false
;; (implies (evthmp-ev-theoremp (list 'not lit))
;; (iff (evthmp-ev-theoremp (disjoin (cons lit clause)))
;; (evthmp-ev-theoremp (disjoin clause))))
;; :hints (("Goal" :use
;; ((:instance evthmp-ev-falsify
;; (x (disjoin clause))
;; (a (evthmp-ev-falsify (disjoin (cons lit clause)))))
;; (:instance evthmp-ev-falsify
;; (x (list 'not lit))
;; (a (evthmp-ev-falsify (disjoin clause))))
;; (:instance evthmp-ev-falsify
;; (x (disjoin (cons lit clause)))
;; (a (evthmp-ev-falsify (disjoin clause)))))))
;; :otf-flg t)
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