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|
; Milawa - A Reflective Theorem Prover
; Copyright (C) 2005-2009 Kookamara LLC
;
; Contact:
;
; Kookamara LLC
; 11410 Windermere Meadows
; Austin, TX 78759, USA
; http://www.kookamara.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: Jared Davis <jared@kookamara.com>
(in-package "MILAWA")
(include-book "trace-okp")
(set-verify-guards-eagerness 2)
(set-case-split-limitations nil)
(set-well-founded-relation ord<)
(set-measure-function rank)
(defthm revappend-under-iff
(iff (revappend x acc)
(or (consp x)
acc))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm consp-of-revappend
(equal (consp (revappend x acc))
(or (consp x)
(consp acc)))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm memberp-of-revappend
(equal (memberp a (revappend x acc))
(or (memberp a x)
(memberp a acc)))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm subsetp-of-revappend-one
(equal (subsetp x (revappend x acc))
t)
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm subsetp-of-revappend-two
(equal (subsetp acc (revappend x acc))
t)
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm true-listp-of-revappend
(equal (true-listp (revappend x acc))
(true-listp acc))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm logic.formula-listp-of-revappend
(implies (force (and (logic.formula-listp x)
(logic.formula-listp acc)))
(equal (logic.formula-listp (revappend x acc))
t))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(defthm logic.formula-list-atblp-of-revappend
(implies (force (and (logic.formula-list-atblp x atbl)
(logic.formula-list-atblp acc atbl)))
(equal (logic.formula-list-atblp (revappend x acc) atbl)
t))
:hints(("Goal" :in-theory (e/d (revappend)
(forcing-revappend-removal)))))
(definlined fast-merge (x y)
;; This is never worse and is sometimes faster than revappend. But unlike
;; revappend is does not produce a very predictable result. You may be able
;; to use it when the only thing you care about is that the joined list has
;; all the members of x and y.
(declare (xargs :guard (and (true-listp x)
(true-listp y))))
(if (consp x)
(if (consp y)
;; This is just an inlined call of revappend.
(revappend (cdr x) (cons (car x) y))
x)
y))
(defthm consp-of-fast-merge
(equal (consp (fast-merge x y))
(or (consp x)
(consp y)))
:hints(("Goal" :in-theory (e/d (fast-merge)
(forcing-revappend-removal)))))
(defthm true-listp-of-fast-merge
(implies (force (and (true-listp x)
(true-listp y)))
(equal (true-listp (fast-merge x y))
t))
:hints(("Goal" :in-theory (enable fast-merge))))
(defthm memberp-of-fast-merge
(equal (memberp a (fast-merge x y))
(or (memberp a x)
(memberp a y)))
:hints(("Goal" :in-theory (e/d (fast-merge)
(forcing-revappend-removal)))))
(defthm subsetp-of-fast-merge-one
(equal (subsetp x (fast-merge x y))
t)
:hints(("Goal" :in-theory (e/d (fast-merge)
(forcing-revappend-removal)))))
(defthm subsetp-of-fast-merge-two
(equal (subsetp y (fast-merge x y))
t)
:hints(("Goal"
:in-theory (e/d (fast-merge)
(forcing-revappend-removal
subsetp-of-revappend-two))
:use ((:instance subsetp-of-revappend-two
(x x)
(acc y)))
:expand (revappend x y))))
(defthm logic.formula-listp-of-fast-merge
(implies (force (and (logic.formula-listp x)
(logic.formula-listp y)))
(equal (logic.formula-listp (fast-merge x y))
t))
:hints(("Goal" :in-theory (e/d (fast-merge)
(forcing-revappend-removal)))))
(defthm logic.formula-list-atblp-of-fast-merge
(implies (force (and (logic.formula-list-atblp x atbl)
(logic.formula-list-atblp y atbl)))
(equal (logic.formula-list-atblp (fast-merge x y) atbl)
t))
:hints(("Goal" :in-theory (e/d (fast-merge)
(forcing-revappend-removal)))))
(defthm fast-merge-when-not-consp-left
(implies (not (consp x))
(equal (fast-merge x y)
y))
:hints(("Goal" :in-theory (enable fast-merge))))
(defthm fast-merge-with-nil-left
(equal (fast-merge nil x)
x)
:hints(("Goal" :in-theory (enable fast-merge))))
(defthm fast-merge-when-not-consp-right
(implies (not (consp y))
(equal (fast-merge x y)
(if (consp x)
x
y)))
:hints(("Goal" :in-theory (enable fast-merge))))
(defthm fast-merge-with-nil-right
(equal (fast-merge x nil)
(if (consp x)
x
nil))
:hints(("Goal" :in-theory (enable fast-merge))))
;; Collecting forced goals from traces.
;;
;; I originally implemented the forced-goals collection with the following very
;; fast, tail-recursive, accumulator-style routine:
;;
;; (defund rw.flag-collect-forced-goals (flag x acc)
;; (declare (xargs :guard (if (equal flag 'term)
;; (rw.tracep x)
;; (rw.trace-listp x))
;; :measure (two-nats-measure (rank x) (if (equal flag 'term) 1 0))))
;; (if (equal flag 'term)
;; (cond ((equal (rw.trace->method x) 'force)
;; (cons (rw.trace-formula x) acc))
;; (t
;; (rw.flag-collect-forced-goals 'list (rw.trace->subtraces x) acc)))
;; (if (consp x)
;; (rw.flag-collect-forced-goals 'term
;; (car x)
;; (rw.flag-collect-forced-goals 'list (cdr x) acc))
;; acc)))
;;
;; This routine is provably equal to:
;;
;; (defund rw.slow-collect-forced-goals (flag x)
;; (declare (xargs :guard (if (equal flag 'term)
;; (rw.tracep x)
;; (rw.trace-listp x))
;; :measure (two-nats-measure (rank x) (if (equal flag 'term) 1 0))))
;; (if (equal flag 'term)
;; (cond ((equal (rw.trace->method x) 'force)
;; (list (rw.trace-formula x)))
;; (t
;; (rw.slow-collect-forced-goals 'list (rw.trace->subtraces x))))
;; (if (consp x)
;; (app (rw.slow-collect-forced-goals 'term (car x))
;; (rw.slow-collect-forced-goals 'list (cdr x)))
;; nil)))
;;
;; But now I use a less-optimized routine based on revappend. Why? The
;; problem is that we want to compute exactly the same forced goals on fast
;; traces as for regular traces. But in the fast traces, we have to collect
;; the forced goals incrementally as we construct the trace, and we do not have
;; the benefit of an accumulator. This means that the fast rewriter would have
;; to aggregate its forced goals with app (or fast-app), as above in
;; slow-collect-forced-goals. But we would prefer to make the fast rewriter
;; faster by using revappend, even though it means making the regular rewriter
;; slightly slower.
(defund rw.flag-collect-forced-goals (flag x)
(declare (xargs :guard (if (equal flag 'term)
(rw.tracep x)
(rw.trace-listp x))
:measure (two-nats-measure (rank x) (if (equal flag 'term) 1 0))
:verify-guards nil))
(if (equal flag 'term)
(cond ((equal (rw.trace->method x) 'force)
(list (rw.trace-formula x)))
(t
(rw.flag-collect-forced-goals 'list (rw.trace->subtraces x))))
(if (consp x)
(fast-merge (rw.flag-collect-forced-goals 'term (car x))
(rw.flag-collect-forced-goals 'list (cdr x)))
nil)))
(defthm true-listp-of-rw.flag-collect-forced-goals
(equal (true-listp (rw.flag-collect-forced-goals flag x))
t)
:hints(("Goal" :in-theory (enable rw.flag-collect-forced-goals))))
(verify-guards rw.flag-collect-forced-goals)
(definlined rw.collect-forced-goals (x)
(declare (xargs :guard (rw.tracep x)))
(rw.flag-collect-forced-goals 'term x))
(definlined rw.collect-forced-goals-list (x)
(declare (xargs :guard (rw.trace-listp x)))
(rw.flag-collect-forced-goals 'list x))
(defthmd definition-of-rw.collect-forced-goals
(equal (rw.collect-forced-goals x)
(cond ((equal (rw.trace->method x) 'force)
(list (rw.trace-formula x)))
(t
(rw.collect-forced-goals-list (rw.trace->subtraces x)))))
:rule-classes :definition
:hints(("Goal" :in-theory (enable rw.collect-forced-goals
rw.collect-forced-goals-list
rw.flag-collect-forced-goals))))
(defthmd definition-of-rw.collect-forced-goals-list
(equal (rw.collect-forced-goals-list x)
(if (consp x)
(fast-merge (rw.collect-forced-goals (car x))
(rw.collect-forced-goals-list (cdr x)))
nil))
:rule-classes :definition
:hints(("Goal" :in-theory (enable rw.collect-forced-goals
rw.collect-forced-goals-list
rw.flag-collect-forced-goals))))
(defthm rw.flag-collect-forced-goals-of-term
(equal (rw.flag-collect-forced-goals 'term x)
(rw.collect-forced-goals x))
:hints(("Goal" :in-theory (enable rw.collect-forced-goals))))
(defthm rw.flag-collect-forced-goals-of-list
(equal (rw.flag-collect-forced-goals 'list x)
(rw.collect-forced-goals-list x))
:hints(("Goal" :in-theory (enable rw.collect-forced-goals-list))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition rw.collect-forced-goals))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition rw.collect-forced-goals-list))))
(ACL2::theory-invariant (not (ACL2::active-runep '(:definition rw.flag-collect-forced-goals))))
(defthm rw.collect-forced-goals-list-when-not-consp
(implies (not (consp x))
(equal (rw.collect-forced-goals-list x)
nil))
:hints(("Goal" :in-theory (enable definition-of-rw.collect-forced-goals-list))))
(defthm rw.collect-forced-goals-list-of-cons
(equal (rw.collect-forced-goals-list (cons a x))
(fast-merge (rw.collect-forced-goals a)
(rw.collect-forced-goals-list x)))
:hints(("Goal" :in-theory (enable definition-of-rw.collect-forced-goals-list))))
(defthms-flag
:thms ((term true-listp-of-rw.collect-forced-goals
(equal (true-listp (rw.collect-forced-goals x))
t))
(t true-listp-of-rw.collect-forced-goals-list
(equal (true-listp (rw.collect-forced-goals-list x))
t)))
:hints (("Goal"
:induct (rw.trace-induction flag x)
:in-theory (enable definition-of-rw.collect-forced-goals))))
(defthms-flag
:thms ((term forcing-logic.formula-listp-of-rw.collect-forced-goals
(implies (force (rw.tracep x))
(equal (logic.formula-listp (rw.collect-forced-goals x))
t)))
(t forcing-logic.formula-listp-of-rw.collect-forced-goals-list
(implies (force (rw.trace-listp x))
(equal (logic.formula-listp (rw.collect-forced-goals-list x))
t))))
:hints (("Goal"
:induct (rw.trace-induction flag x)
:in-theory (enable definition-of-rw.collect-forced-goals))))
(defthms-flag
:shared-hyp (force (and (equal (cdr (lookup 'equal atbl)) 2)
(equal (cdr (lookup 'iff atbl)) 2)))
:thms ((term forcing-logic.formula-list-atblp-of-rw.collect-forced-goals
(implies (force (and (rw.tracep x)
(rw.trace-atblp x atbl)))
(equal (logic.formula-list-atblp (rw.collect-forced-goals x) atbl)
t)))
(t forcing-logic.formula-list-atblp-of-rw.collect-forced-goals-list
(implies (force (and (rw.trace-listp x)
(rw.trace-list-atblp x atbl)))
(equal (logic.formula-list-atblp (rw.collect-forced-goals-list x) atbl)
t))))
:hints (("Goal"
:induct (rw.trace-induction flag x)
:in-theory (enable definition-of-rw.collect-forced-goals))))
(defthm memberp-of-rw.trace-conclusion-formula-in-rw.collect-forced-goals
(implies (force (equal (rw.trace->method x) 'force))
(equal (memberp (rw.trace-formula x) (rw.collect-forced-goals x))
t))
:hints(("Goal" :in-theory (enable definition-of-rw.collect-forced-goals))))
(defthm forcing-subsetp-of-rw.collect-forced-goals-list-of-subtraces
(implies (force (and (rw.tracep x)
(rw.trace-okp x defs)))
(subsetp (rw.collect-forced-goals-list (rw.trace->subtraces x))
(rw.collect-forced-goals x)))
:hints(("Goal" :in-theory (enable definition-of-rw.collect-forced-goals
definition-of-rw.trace-okp
rw.trace-step-okp
rw.force-tracep))))
(defund rw.collect-forced-goals-list-list (x)
(declare (xargs :guard (rw.trace-list-listp x)))
(if (consp x)
(fast-merge (rw.collect-forced-goals-list (car x))
(rw.collect-forced-goals-list-list (cdr x)))
nil))
(defthm true-listp-of-rw.collect-forced-goals-list-list
(equal (true-listp (rw.collect-forced-goals-list-list x))
t)
:hints(("Goal" :in-theory (enable rw.collect-forced-goals-list-list))))
(defthm rw.collect-forced-goals-list-list-when-not-consp
(implies (not (consp x))
(equal (rw.collect-forced-goals-list-list x)
nil))
:hints(("Goal" :in-theory (enable rw.collect-forced-goals-list-list))))
(defthm rw.collect-forced-goals-list-list-of-cons
(equal (rw.collect-forced-goals-list-list (cons a x))
(fast-merge (rw.collect-forced-goals-list a)
(rw.collect-forced-goals-list-list x)))
:hints(("Goal" :in-theory (enable rw.collect-forced-goals-list-list))))
(defthm forcing-rw.formula-listp-of-rw.collect-forced-goals-list-list
(implies (force (rw.trace-list-listp x))
(equal (logic.formula-listp (rw.collect-forced-goals-list-list x))
t))
:hints(("Goal" :induct (cdr-induction x))))
;; BOZO errr, don't have a trace-list-list-atblp, maybe we won't need it.
;; (defthm forcing-rw.formula-list-atblp-of-rw.collect-forced-goals-list-list
;; (implies (force (rw.trace-list-list-atblp x atbl))
;; (equal (logic.formula-list-atblp (rw.collect-forced-goals-list-list x) atbl)
;; t))
;; :hints(("Goal" :induct (cdr-induction x))))
|
