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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 ../README.
; This modification of the list-based lrat-checker.lisp uses stobjs to speed up
; the handling of assignments, in particular, evaluation of literals and
; clauses.
(in-package "LRAT")
(include-book "lrat-checker")
(local (include-book "../list-based/soundness"))
(in-theory (disable push-literal
negate-clause-or-assignment negate-clause
unit-propagation unit-propagation$
verify-clause verify-clause$
verify-proof-rec verify-proof$-rec
verify-proof verify-proof$
clausep clausep$
formula-p formula-p$))
(local (in-theory (disable nth update-nth)))
(defthm arr-matches-stk-implies-booleanp-alt
(implies (and (a$p a$)
(natp i)
(< i (a$ptr a$)))
(booleanp (nth (nth i (nth *a$stki* a$))
(nth *a$arri* a$))))
:hints (("Goal"
:in-theory '(a$p a$arr-length a$p-weak a$ptr a$ptrp natp)
:use arr-matches-stk-implies-booleanp)))
(defun list-assignment-rec (i a$)
(declare (xargs :stobjs a$
:guard (and (a$p a$)
(natp i)
(<= i (a$ptr a$)))))
(cond ((zp i)
nil)
(t (let* ((i (1- i))
(var (a$stki i a$))
(val (a$arri var a$)))
(cons (if (equal val nil)
(negate var)
var)
(list-assignment-rec i a$))))))
(defund list-assignment (a$)
(declare (xargs :stobjs a$
:guard (a$p a$)))
(list-assignment-rec (a$ptr a$) a$))
; Test:
(defthm test-list-assignment
(equal (with-local-stobj a$
(mv-let (ast a$)
(b* ((a$ (initialize-a$ 10 a$))
((mv - a$) (push-literal 3 a$))
((mv - a$) (push-literal -5 a$))
((mv - a$) (push-literal -1 a$))
(ast (list-assignment a$)))
(mv ast a$))
ast))
'(-1 -5 3))
:rule-classes nil)
(defthm list-assignment-rec-update-nth-a$ptr
(equal (list-assignment-rec i
(update-nth *a$ptr* j a$))
(list-assignment-rec i a$)))
(defthm list-assignment-rec-update-nth-a$stk
(implies (and (natp i)
(integerp j)
(<= i j)
(equal stk (nth *a$stki* a$)))
(equal (list-assignment-rec
i
(update-nth *a$stki*
(update-nth j var stk)
a$))
(list-assignment-rec i a$))))
(defthm list-assignment-rec-update-nth-a$arr
(implies (and (varp$ var a$)
(not (find-var-on-stk var ptr a$)))
(equal
(list-assignment-rec ptr
(update-nth *a$arri*
(update-nth var val (nth *a$arri* a$))
a$))
(list-assignment-rec ptr a$)))
:hints (("Goal" :in-theory (enable list-assignment-rec))))
(defthm list-assignment-update-nth-a$ptr
(implies (and (a$p a$)
(force (varp$ var a$))
(equal ptr (nth *a$ptr* a$))
(booleanp val)
(not (find-var-on-stk var ptr a$)))
(equal (list-assignment
(update-nth *a$ptr*
(+ 1 ptr)
(update-nth
*a$stki*
(update-nth ptr
var
(nth *a$stki* a$))
(update-nth *a$arri*
(update-nth var
val
(nth *a$arri* a$))
a$))))
(cons (if val var (negate var))
(list-assignment a$))))
:hints (("Goal" :in-theory (enable list-assignment list-assignment-rec))))
(defun find-lst-on-stk (lst a$)
(declare (xargs :stobjs a$ :verify-guards nil))
(cond ((atom lst)
nil)
(t (or (find-var-on-stk (abs (car lst)) (a$ptr a$) a$)
(find-lst-on-stk (cdr lst) a$)))))
(defthm find-var-on-stk-update-stk-past-ptr-better
(implies (and (integerp j)
(<= i j)
(equal stk (nth *a$stki* a$)))
(equal (find-var-on-stk
var i
(update-nth *a$stki*
(update-nth j lit stk)
a$))
(find-var-on-stk var i a$))))
(defthm find-lst-on-stk-update
(implies (and (a$p-weak a$)
(equal ptr (nth *a$ptr* a$))
(not (member var lst))
(not (member (negate var) lst))
(equal stk (nth *a$stki* a$)))
(equal (find-lst-on-stk
lst
(update-nth *a$ptr* (+ 1 ptr)
(update-nth *a$stki*
(update-nth ptr var stk)
(update-nth *a$arri*
(update-nth var
val
(nth *a$arri* a$))
a$))))
(find-lst-on-stk lst a$)))
:hints (("Goal" :induct (find-lst-on-stk lst a$))))
(defthm a$-implies-a$-weak
(implies (a$p a$)
(a$p-weak a$))
:hints (("Goal" :in-theory (enable a$p))))
(defthm literal-listp$-update-a$ptr
(equal (literal-listp$ lst (update-nth *a$ptr* ptr a$))
(literal-listp$ lst a$))
:hints (("Goal" :in-theory (enable literal-listp$))))
(defthm literal-listp$-update-a$stk
(equal (literal-listp$ lst (update-nth *a$stki* stk a$))
(literal-listp$ lst a$))
:hints (("Goal" :in-theory (enable literal-listp$))))
(defthm literal-listp$-update-a$arr
(implies (and (natp i)
(< i (len (nth *a$arri* a$))))
(equal (literal-listp$ lst
(update-nth *a$arri*
(update-nth i
val
(nth *a$arri* a$))
a$))
(literal-listp$ lst a$)))
:hints (("Goal" :in-theory (enable literal-listp$))))
(defthm a$-implies-array-matches-stack
(implies (a$p a$)
(arr-matches-stk (len (nth *a$arri* a$)) a$))
:hints (("Goal" :in-theory (enable a$p))))
; Working towards proof of verify-clause-equiv.
(defthm negate-clause-equiv-0-lemma
(implies (and (a$p a$)
(clausep$ clause a$)
(not (find-lst-on-stk clause a$)))
(equal (car (negate-clause clause a$))
nil))
:hints (("Goal"
:induct (negate-clause clause a$)
:in-theory (enable negate-clause
clausep$
literal-listp$
push-literal))))
(defthm a$ptr-0-implies-not-find-lst-on-stk
(implies (equal (a$ptr a$) 0)
(not (find-lst-on-stk clause a$))))
(defthm negate-clause-equiv-0
(implies (and (a$p a$)
(= (a$ptr a$) 0)
(clausep$ clause a$))
(equal (car (negate-clause clause a$))
nil)))
(defthm negate-clause-nil
(equal (negate-clause nil a$)
(mv nil a$))
:hints (("Goal" :in-theory (enable negate-clause))))
; Start proof of unit-propagation-equiv
; Start proof of is-unit-clause-equiv
; Start proof of member-equal-list-assignment
(defthm booleanp-nth-stk
(implies (and (a$p a$)
(force (natp i))
(< i (nth *a$ptr* a$)))
(booleanp (nth (nth i (nth *a$stki* a$))
(nth *a$arri* a$))))
:rule-classes :type-prescription)
(defthm member-equal-list-assignment-forward
(implies (and (a$p a$)
(literalp$ lit a$)
(<= ptr (nth *a$ptr* a$)))
(implies (member-equal lit (list-assignment-rec ptr a$))
(equal (nth (abs lit) (nth *a$arri* a$))
(equal (abs lit) lit))))
:rule-classes nil)
(defthm member-equal-list-assignment-reverse-1
(implies (and (literalp$ lit a$)
(find-var-on-stk (abs lit) ptr a$)
(equal (nth (abs lit) (nth *a$arri* a$))
(< 0 lit)))
(member-equal lit (list-assignment-rec ptr a$))))
(defthm member-equal-list-assignment-reverse
(implies (and (a$p a$)
(literalp$ lit a$))
(implies (equal (nth (abs lit) (nth *a$arri* a$))
(equal (abs lit) lit))
(member-equal lit (list-assignment a$))))
:hints (("Goal" :in-theory (enable list-assignment)))
:rule-classes nil)
(defthm member-equal-list-assignment
(implies (and (a$p a$)
(literalp$ lit a$))
(iff (member-equal lit (list-assignment a$))
(equal (nth (abs lit) (nth *a$arri* a$))
(equal (abs lit) lit))))
:hints (("Goal"
:in-theory (enable list-assignment)
:use ((:instance member-equal-list-assignment-forward
(ptr (nth *a$ptr* a$)))
member-equal-list-assignment-reverse))))
(defthm evaluate-clause-equiv
(implies (and (a$p a$)
(clausep$ clause a$))
(equal (evaluate-clause$ clause a$)
(evaluate-clause clause (list-assignment a$))))
:hints (("Goal" :in-theory (enable evaluate-clause evaluate-clause$))))
(defthm is-unit-clause-equiv
(implies (and (a$p a$)
(clausep$ clause a$))
(equal (is-unit-clause$ clause a$)
(is-unit-clause clause (list-assignment a$))))
:hints (("Goal" :in-theory (enable is-unit-clause$ is-unit-clause))))
(defthm list-assignment-push-unit-literal
(let ((lit (is-unit-clause clause (list-assignment a$))))
(implies (and (a$p a$)
(clausep$ clause a$)
lit
(not (equal lit t)))
(equal (list-assignment
(mv-nth
1
(push-literal lit a$)))
(cons lit (list-assignment a$)))))
:hints (("Goal" :in-theory (enable push-literal))))
(in-theory ; for unit-propagation-equiv
(disable member-equal-list-assignment))
(defthm literalp$-is-unit-clause
(implies (and (a$p a$)
(clausep$ clause a$)
(is-unit-clause$ clause a$)
(not (equal (is-unit-clause$ clause a$) t)))
(literalp$ (is-unit-clause clause (list-assignment a$))
a$))
:hints (("Goal"
:in-theory (disable literalp$-is-unit-clause$)
:use literalp$-is-unit-clause$)))
(defthm unit-propagation-equiv
(implies
(and (a$p a$)
(formula-p$ formula a$))
(and (equal (car (unit-propagation$ formula indices a$))
(equal (unit-propagation formula
indices
(list-assignment a$))
t))
(implies (not (car (unit-propagation$ formula indices a$)))
(equal (list-assignment
(mv-nth 1 (unit-propagation$ formula indices a$)))
(unit-propagation formula indices
(list-assignment a$))))))
:hints (("Goal"
:induct t
:expand ((unit-propagation$ formula indices a$)
(unit-propagation formula indices (list-assignment a$)))
:in-theory (enable unit-propagation unit-propagation$))))
(defthm list-assignment-base
(implies (equal (nth *a$ptr* a$) 0)
(equal (list-assignment a$)
nil))
:hints (("Goal" :in-theory (enable list-assignment))))
; Start proof of negate-clause-equiv-1
(defthm negate-clause-equiv-1-lemma
(implies (and (a$p a$)
(clausep$ clause a$)
(not (intersectp-equal clause (list-assignment a$)))
(not (intersectp-complementary clause (list-assignment a$))))
(equal (list-assignment (mv-nth 1 (negate-clause clause a$)))
(negate-clause-or-assignment-rec clause
(list-assignment a$))))
:hints (("Goal" :in-theory (enable negate-clause push-literal clausep$
member-equal-list-assignment literalp$))))
(defthm negate-clause-equiv-1
(implies (and (a$p a$)
(= (a$ptr a$) 0)
(clausep$ clause a$))
(equal (list-assignment (mv-nth 1 (negate-clause clause a$)))
(negate-clause-or-assignment clause)))
:hints (("Goal" :in-theory (enable negate-clause-or-assignment))))
; Start proof of ratp1-equiv
(defthm rat-assignment-equiv-no-error
(implies (and (a$p a$)
(literalp$ nlit a$)
(clausep$ clause a$)
(null (car (rat-assignment$ a$ nlit clause))))
(and (not (equal (rat-assignment$ (list-assignment a$)
nlit
clause)
t))
(equal (list-assignment
(mv-nth 1 (rat-assignment$ a$ nlit clause)))
(rat-assignment
(list-assignment a$) nlit clause))))
:hints (("Goal" :in-theory (enable rat-assignment$
rat-assignment
push-literal
member-equal-list-assignment
clausep$
))))
(defthm rat-assignment-equiv-error
(implies (and (a$p a$)
(literalp$ nlit a$)
(clausep$ clause a$))
(equal (car (rat-assignment$ a$ nlit clause))
(equal (rat-assignment
(list-assignment a$) nlit clause)
t)))
:hints (("Goal" :in-theory (enable rat-assignment$
rat-assignment
push-literal
member-equal-list-assignment
clausep$
))))
(defthm a$=-pop-literals-mv-nth-1-rat-assignment$
(implies (and (a$p a$)
(clausep$ clause a$))
(a$= (pop-literals (nth *a$ptr* a$)
(mv-nth 1
(rat-assignment$ a$ nlit clause)))
a$))
:hints (("Goal" :in-theory (enable pop-literals rat-assignment$))))
(defthm a$=-pop-literals-mv-nth-1-unit-propagation$
(implies (and (a$p a$)
(formula-p$ formula a$))
(a$= (pop-literals (nth *a$ptr* a$)
(mv-nth 1
(unit-propagation$ formula indices a$)))
a$))
:hints (("Goal" :in-theory (enable pop-literals unit-propagation$))))
(defthm lst=-implies-nth-equal
(implies (and (lst= ptr x y)
(integerp ptr)
(natp i)
(< i ptr))
(equal (nth i x)
(nth i y))))
(defthm a$=-implies-equal-list-assignment-rec
(implies (and (a$= a$1 a$2)
(<= ptr (nth *a$ptr* a$1)))
(equal (list-assignment-rec ptr a$1) (list-assignment-rec ptr a$2)))
:hints (("Goal" :in-theory (enable a$=)))
:rule-classes nil)
(defthm a$=-implies-equal-list-assignment-1
(implies (a$= a$1 a$2)
(equal (list-assignment a$1) (list-assignment a$2)))
:hints (("Goal"
:in-theory (enable a$= list-assignment)
:use ((:instance a$=-implies-equal-list-assignment-rec
(ptr (nth *a$ptr* a$1))))))
:rule-classes (:congruence))
(defthm ratp1-equiv
(implies (and (a$p a$)
(formula-p$ alist a$)
(formula-p$ formula a$)
(literalp$ nlit a$)
(drat-hint-listp drat-hints))
(and (equal (car (RATp1$ alist formula nlit drat-hints a$))
(RATp1 alist formula nlit drat-hints
(list-assignment a$)))
(a$= (mv-nth 1 (RATp1$ alist formula nlit drat-hints a$))
a$)))
:hints (("Goal"
:induct (ratp1$ alist formula nlit drat-hints a$)
:expand ((ratp1$ alist formula nlit drat-hints a$)
(ratp1 alist formula
nlit drat-hints (list-assignment a$)))
:in-theory (enable ratp1$ ratp1 formula-p$))))
(defthm verify-clause-equiv
(implies (force (and (a$p a$)
(formula-p$ formula a$)
(equal (nth *a$ptr* a$) 0)
(integerp ncls)
(natp ndel)
(not (equal (car add-step) t))
(add-step-p add-step)
(clausep$ (access add-step add-step :clause) a$)))
(and (equal (car (verify-clause$ formula add-step ncls ndel a$))
(car (verify-clause formula add-step ncls ndel)))
(equal (mv-nth 1 (verify-clause$ formula add-step ncls ndel a$))
(mv-nth 1 (verify-clause formula add-step ncls ndel)))
(equal (mv-nth 2 (verify-clause$ formula add-step ncls ndel a$))
(mv-nth 2 (verify-clause formula add-step ncls ndel)))
(a$= (mv-nth 3 (verify-clause$ formula add-step ncls ndel a$))
a$)))
:hints (("Goal" :in-theory (enable verify-clause verify-clause$))))
(defthm formula-p$-add-proof-clause
(implies (force (and (a$p a$)
(posp index)
(clausep$ clause a$)
(formula-p$ formula a$)))
(formula-p$ (add-proof-clause index clause formula)
a$))
:hints (("Goal" :in-theory (union-theories '(formula-p$ add-proof-clause)
(current-theory :here)))))
(defthm a$p-mv-nth-3-verify-clause$
(implies (and (a$p a$)
(formula-p$ formula a$)
(clausep$ (access add-step add-step :clause) a$))
(a$p (mv-nth 3 (verify-clause$ formula add-step ncls ndel a$))))
:hints (("Goal" :in-theory (enable verify-clause$))))
(defthm integerp-car-verify-clause
(implies (force (integerp ncls))
(or (integerp (car (verify-clause formula add-step ncls ndel)))
(equal (car (verify-clause formula add-step ncls ndel))
nil)))
:hints (("Goal" :in-theory (enable verify-clause)))
:rule-classes :type-prescription)
(defthm natp-mv-nth-1-verify-clause
(implies (force (natp ndel))
(or (natp (mv-nth 1 (verify-clause formula add-step ncls ndel)))
(equal (mv-nth 1 (verify-clause formula add-step ncls ndel))
nil)))
:hints (("Goal" :in-theory (enable verify-clause)))
:rule-classes :type-prescription)
(defthm natp-mv-nth-1-verify-clause-extra
(implies (and (natp ndel)
(car (verify-clause formula add-step ncls ndel)))
(natp (mv-nth 1 (verify-clause formula add-step ncls ndel))))
:hints (("Goal" :in-theory (enable verify-clause)))
:rule-classes :type-prescription)
(in-theory (enable add-step-p))
(defthm a$ptr-0-for-verify-clause$
(implies (and (a$p a$)
(equal (nth *a$ptr* a$) 0)
(formula-p$ formula a$)
(clausep$ (access add-step add-step :clause) a$)
(car (verify-clause$ formula add-step ncls ndel a$)))
(equal (nth *a$ptr*
(mv-nth 3 (verify-clause$ formula add-step ncls ndel a$)))
0))
:hints (("Goal" :in-theory (enable verify-clause$))))
(defthm formula-p$-for-verify-clause$
(implies (and (a$p a$)
(formula-p$ formula a$)
(clausep$ (access add-step add-step :clause) a$)
(car (verify-clause formula add-step ncls ndel)))
(formula-p$ (mv-nth 2
(verify-clause formula add-step ncls ndel))
(mv-nth 3
(verify-clause$ formula add-step ncls ndel
a$))))
:hints (("Goal" :in-theory (enable verify-clause verify-clause$))))
(defthm verify-proof-equiv
(implies (and (formula-p$ formula a$)
(proofp$ proof a$)
(a$p a$)
(equal (nth *a$ptr* a$) 0)
(integerp ncls)
(natp ndel))
(and (equal (car (verify-proof$-rec ncls ndel formula proof a$))
(verify-proof-rec ncls ndel formula proof))
(a$= (mv-nth 1 (verify-proof$-rec ncls ndel formula proof a$))
a$)))
:hints (("Goal"
:induct (verify-proof$-rec ncls ndel formula proof a$)
:in-theory (enable verify-proof-rec verify-proof$-rec))))
(defthm equiv-main
(implies (and (formula-p formula)
(proofp proof)
(< 0
(proof-max-var proof (formula-max-var formula 0)))
(car (verify-proof$-rec
(len (fast-alist-fork formula nil))
0 formula proof
(initialize-a$
(proof-max-var proof (formula-max-var formula 0))
'(0 (nil) (0)))))
(proof-contradiction-p proof))
(verify-proof-rec (len (fast-alist-fork formula nil))
0 formula proof))
:hints (("Goal"
:restrict
((verify-proof-equiv
((a$ (initialize-a$
(proof-max-var proof (formula-max-var formula 0))
'(0 (nil) (0))))))))))
(defthm refutation-p-equiv
(implies (and (formula-p formula)
(refutation-p$ proof formula))
(refutation-p proof formula))
:hints (("Goal" :in-theory (enable verify-proof verify-proof$))))
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