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|
; Copyright (C) 2016, Regents of the University of Texas
; Marijn Heule, Warren A. Hunt, Jr., and Matt Kaufmann
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; See soundness.lisp. Here we prove a key lemma in support of that book.
(in-package "LRAT")
(include-book "satisfiable-add-proof-clause-base")
; possible helpful lemmas:
(include-book "sat-drat-claim-1")
; Temporary, during development:
(local (include-book "tools/remove-hyps" :dir :system))
(defun drat-indices (j alist drat-hints)
; based on RATp1
(if (endp alist)
nil
(let* ((index (caar alist))
(clause (cdar alist)))
(cond
((deleted-clause-p clause)
(drat-indices j (cdr alist) drat-hints))
((eql index (caar drat-hints)) ; perform RAT
(if (equal j index)
(cdar drat-hints)
(drat-indices j (cdr alist) (cdr drat-hints))))
(t (drat-indices j (cdr alist) drat-hints))))))
(set-ignore-ok t)
(defun b-fn (x)
`(let* ((a assignment)
(fal formula)
(d (cdr (hons-assoc-equal index fal)))
(c (access add-step entry :clause))
(lit (car c))
(nlit (negate lit))
(dp (remove-literal nlit d))
(nc (negate-clause-or-assignment c))
(rup-indices (access add-step entry :rup-indices))
(nc-up (unit-propagation formula rup-indices nc))
(rat-a (rat-assignment a nlit d))
(rat-nc (rat-assignment nc-up nlit d))
(entry-index (access add-step entry :index))
(drat-hints (access add-step entry :drat-hints))
(vc (verify-clause formula entry ncls ndel))
(vc-success (car vc))
(new-formula (mv-nth 2 vc))
(drat-indices (drat-indices index new-formula drat-hints)))
,x))
(defmacro b (x)
(b-fn x))
(defun b-lst-fn (lst)
(cond ((endp lst) nil)
(t (cons (b-fn (car lst))
(b-lst-fn (cdr lst))))))
(defmacro b-lst (lst)
(b-lst-fn lst))
(defconst *all-hyps*
; Warning: Keep this in sync with sat-drat-claim-2-3. Actually, never mind:
; this constant is no longer used, and some hypotheses have disappeared from
; sat-drat-claim-2-3.
'((member nlit d)
(not (equal (evaluate-clause dp a)
t))
vc-success
(proof-entry-p entry)
(not (proof-entry-deletion-p entry))
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(consp c)
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(hons-assoc-equal index fal)
(not (deleted-clause-p d))
(not (equal (evaluate-clause d (flip-literal lit a))
t))))
(defun make-claim-fn (hyps body)
(let ((hyps (if (eq hyps :all)
*all-hyps*
hyps)))
`(implies
(and ,@(b-lst-fn hyps))
,(b-fn body))))
; For interactive use:
(defmacro make-claim (hyps x)
(make-claim-fn hyps x))
(defmacro verify-claim (hyps x)
`(verify (make-claim ,hyps ,x)))
(defun defclaim-fn (n hyps body args)
`(defthm ,(if n
(intern$ (coerce (acl2::packn1 (list 'main '- n))
'string)
"LRAT")
'main)
,(make-claim-fn hyps body)
,@args
,@(and (not (assoc-keyword :rule-classes args))
'(:rule-classes nil))))
(defmacro defclaim! (n hyps body &rest args)
(defclaim-fn n hyps body args))
(defmacro defclaim (n hyps body &rest args)
`(local ,(defclaim-fn n hyps body args)))
; Start proof of rat-assignment-not-t.
(defthm member-both-implies-evaluate-clause
(implies (and (member lit c)
(member lit a))
(equal (evaluate-clause c a) t)))
(defthm evaluate-clause-subsetp
(implies (and (equal (evaluate-clause c b) t)
(subsetp b a))
(equal (evaluate-clause c a) t)))
(defthm evaluate-clause-cons
(implies (and (equal (evaluate-clause c (cons lit a))
t)
(not (member lit c)))
(equal (evaluate-clause c a)
t)))
(defthm rat-assignment-not-t
(implies (and (clause-or-assignment-p a)
(clause-or-assignment-p c)
(not (equal (evaluate-clause (remove-literal nlit c)
a)
t)))
(not (equal (rat-assignment a nlit c) t)))
:hints (("Goal"
:in-theory (enable clause-or-assignment-p))))
(defclaim 1
((not (equal (evaluate-clause dp a) t))
(formula-p formula)
(solution-p a formula)
(not (deleted-clause-p d)))
(clause-or-assignment-p rat-a))
(defclaim 2
((formula-p formula)
(solution-p a formula))
(equal (unit-propagation formula rup-indices a)
a)
:rule-classes :rewrite)
(encapsulate
()
(local (include-book "satisfiable-maybe-shrink-formula"))
(defclaim 3-1-1
(vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (satisfiable (add-proof-clause entry-index c new-formula))))
(subsetp nc-up
(unit-propagation formula rup-indices a))
:hints (("Goal"
:use main-1
:in-theory (disable main-2 unit-propagation-identity))))
(defclaim! 3-1
(vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (satisfiable (add-proof-clause entry-index c new-formula))))
(subsetp nc-up
(unit-propagation formula rup-indices a))
:hints (("Goal"
:use main-1
:in-theory (disable main-2 unit-propagation-identity)))))
(defclaim 3
(vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (satisfiable (add-proof-clause entry-index c new-formula))))
(subsetp nc-up a)
:hints (("Goal" :use main-3-1)))
(defun rat-assignment-monotone-induction (a1 a2 nlit clause)
; This is based on rat-assignment, and is relevant when (subsetp a1 a2) and
; (rat-assignment a2 nlit c) is not t.
(cond ((endp clause) (list a1 a2))
((or (eql (car clause) nlit)
(member (negate (car clause)) a1))
(rat-assignment-monotone-induction a1 a2 nlit (cdr clause)))
((member (negate (car clause)) a2)
(rat-assignment-monotone-induction (cons (negate (car clause)) a1)
a2
nlit
(cdr clause)))
(t
(rat-assignment-monotone-induction (cons (negate (car clause)) a1)
(cons (negate (car clause)) a2)
nlit
(cdr clause)))))
(defthm rat-assignment-monotone
(implies (and (subsetp a1 a2)
(not (equal (rat-assignment a2 lit c)
t)))
(subsetp (rat-assignment a1 lit c)
(rat-assignment a2 lit c)))
:hints (("Goal" :induct (rat-assignment-monotone-induction a1 a2 lit c))))
(defthm clause-or-assignment-p-implies-literalp-car
(implies (and (clause-or-assignment-p x)
(consp x))
(literalp (car x)))
:hints (("Goal" :in-theory (enable clause-or-assignment-p)))
:rule-classes :type-prescription)
(defthm rat-assignment-monotone-2
(implies (and (subsetp a1 a2)
(clause-or-assignment-p a1)
(clause-or-assignment-p a2)
(clause-or-assignment-p c)
(not (equal (rat-assignment a2 lit c)
t)))
(clause-or-assignment-p (rat-assignment a1 lit c)))
:hints (("Goal" :induct (rat-assignment-monotone-induction a1 a2 lit c))))
(defclaim 4
(vc-success
(not (equal (evaluate-clause dp a) t))
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(not (deleted-clause-p d)))
(and (clause-or-assignment-p rat-nc)
(subsetp rat-nc rat-a))
:hints (("Goal" :use (main-1 main-3))))
(defun tail-of (x1 x2)
(cond ((equal x1 x2) t)
(t (and (consp x2)
(tail-of x1 (cdr x2))))))
(defthm tail-of-cdr
(implies (and (true-listp x2)
(tail-of x1 x2))
(tail-of (cdr x1) x2)))
(defthm formula-p-tail
; This is a :forward-chaining rule so that we can continue chaining forward
; from formula-p to alistp, etc.
(implies (and (tail-of x1 x2)
(formula-p x2))
(formula-p x1))
:rule-classes :forward-chaining)
(defthm ratp1-implies-unit-propagation-t-lemma
(implies (and (equal (RATp1 alist formula nlit drat-hints assignment)
t)
(formula-p formula)
(drat-hint-listp drat-hints)
(clause-or-assignment-p assignment)
(equal clause
(cdr (hons-assoc-equal index alist)))
(equal new-assignment
(rat-assignment assignment nlit clause))
(not (or (not (member nlit clause))
(deleted-clause-p
(cdr (hons-assoc-equal index formula)))))
(not (equal new-assignment t))
(tail-of alist formula))
(equal (unit-propagation formula
(drat-indices index alist drat-hints)
new-assignment)
t))
:hints (("Goal"
:induct (RATp1 alist formula nlit drat-hints assignment)
:in-theory (enable formula-p))))
(encapsulate
()
(local (include-book "satisfiable-maybe-shrink-formula"))
(local (in-theory (enable maybe-shrink-formula formula-p)))
(defclaim 5-1
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(not (deleted-clause-p d)))
(equal (unit-propagation new-formula drat-indices rat-nc)
t)
:hints (("Goal"
:in-theory (e/d (verify-clause)
(ratp1
; At one point, when the definition of RATp1 produced a Boolean
; type-prescription rule, it was necessary to disable that rule. Although it's
; no longer necessary, we continue to do so in case RATp1 once again becomes
; Boolean-valued.
(:t ratp1)))
:use main-4)))
(defclaim! 5
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(not (deleted-clause-p d)))
(equal (unit-propagation formula drat-indices rat-nc)
t)
:hints (("Goal"
:in-theory
(union-theories '(verify-clause
unit-propagation-maybe-shrink-formula)
(theory 'minimal-theory))
:use main-5-1))))
(defclaim 6
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(not (proof-entry-deletion-p entry))
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula))))
(equal (unit-propagation formula drat-indices rat-a)
t)
:hints (("Goal" :use (main-1 main-4 main-5))))
(defthm negate-clause-or-assignment-self-inverts-lemma
(implies (literal-listp a)
(equal (negate-clause-or-assignment-rec
(negate-clause-or-assignment-rec a b)
c)
(negate-clause-or-assignment-rec b
(append a c)))))
(defthm negate-clause-or-assignment-self-inverts
(implies (clause-or-assignment-p a) ; probably (literal-listp a) suffices
(equal (negate-clause-or-assignment
(negate-clause-or-assignment a))
a))
:hints (("Goal" :in-theory (enable negate-clause-or-assignment))))
(defclaim 7
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(not (deleted-clause-p d)))
(equal (evaluate-clause (negate-clause-or-assignment rat-a) a)
t)
:hints (("Goal"
:use main-6
:restrict
((unit-propagation-correct
((indices
(drat-indices
index
(mv-nth 2 (maybe-shrink-formula ncls ndel formula 1/3))
(cddr entry)))
(formula formula)))))))
(defthm negate-rat-assignment-key-base-lemma
(implies (and (clause-or-assignment-p a)
(subsetp b a)
(not (intersectp-equal acc a)))
(not (equal (evaluate-clause (negate-clause-or-assignment-rec b acc)
a)
t))))
(defthm negate-rat-assignment-key-base
(implies (clause-or-assignment-p a)
(not (equal (evaluate-clause (negate-clause-or-assignment a)
a)
t)))
:hints (("Goal" :in-theory (enable negate-clause-or-assignment))))
(defthm clause-or-assignment-p-implies-negate-car-not-member-cdr
(implies (and (clause-or-assignment-p c)
(consp c))
(not (member (negate (car c)) (cdr c))))
:hints (("Goal" :in-theory (enable clause-or-assignment-p)))
:rule-classes :forward-chaining)
(defthm negate-rat-assignment-key
(implies (and (clause-or-assignment-p a)
(clause-or-assignment-p d)
(equal (evaluate-clause (negate-clause-or-assignment
(rat-assignment a x d))
a)
t))
(equal (evaluate-clause (remove-literal x d) a)
t)))
(defclaim 8
((member nlit d)
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula)))
(not (deleted-clause-p d)))
(equal (evaluate-clause dp a) t)
:hints (("Goal" :use main-7)))
(defclaim 9
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula))))
(equal (evaluate-clause dp b) t)
:hints (("Goal" :use main-8)))
(defclaim nil
((member nlit d)
(not (equal (evaluate-clause dp a) t))
vc-success
(proof-entry-p entry)
(formula-p formula)
(solution-p a formula)
(not (equal nc-up t))
(not (satisfiable (add-proof-clause entry-index c new-formula))))
nil
:hints (("Goal" :use (main-8 main-9))))
(defthm sat-drat-claim-2-3
(mv-let (ncls ndel new-formula)
(verify-clause formula entry ncls ndel)
(declare (ignore ndel))
(implies (and (member (negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
(not (equal (evaluate-clause
(remove-literal
(negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
assignment)
t))
ncls
(proof-entry-p entry)
(not (proof-entry-deletion-p entry))
(formula-p formula)
(solution-p assignment formula)
(not (equal (unit-propagation formula
(access add-step entry
:rup-indices)
(negate-clause-or-assignment
(access add-step entry
:clause)))
t))
(not (satisfiable (add-proof-clause
(access add-step entry :index)
(access add-step entry :clause)
new-formula))))
(equal (evaluate-clause
(cdr (hons-assoc-equal index formula))
(flip-literal (car (access add-step entry :clause))
assignment))
t)))
:hints (("Goal" :use main))
:rule-classes nil)
|
