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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 "lrat-checker")
(defthm member-equal-monotone
(implies (and (subsetp-equal s1 s2)
(member-equal a s1))
(member-equal a s2)))
(defthm clause-or-assignment-p-forward
(implies (and clause
(clause-or-assignment-p clause))
(literalp (car clause)))
:hints (("Goal" :in-theory (enable clause-or-assignment-p))))
(defthm contradiction-implies-conflicting-literalsp
(implies (and (member-equal lit x)
(member-equal (- lit) x)
(literal-listp x))
(conflicting-literalsp x))
:hints (("Goal" :in-theory (enable literalp) :do-not '(generalize))))
(defthm clause-or-assignment-p-is-not-contradictory
(implies (and (member-equal lit x)
(member-equal (- lit) x))
(not (clause-or-assignment-p x)))
:hints (("Goal"
:in-theory (enable clause-or-assignment-p))))
(defthm is-unit-clause-t-monotone
(implies (and (equal (is-unit-clause clause a1)
t)
(clause-or-assignment-p a2)
(not (equal (is-unit-clause clause a2)
t)))
(not (subsetp-equal a1 a2))))
(defthm subsetp-transitive
(implies (and (subsetp x y)
(subsetp y z))
(subsetp x z)))
(defthm not-member-is-unit-clause
(implies (and (is-unit-clause clause assignment)
(not (equal (is-unit-clause clause assignment)
t)))
(not (member-equal (is-unit-clause clause assignment)
assignment))))
(defthm clause-or-assignment-p-cons-is-unit-clause
(implies (and (clause-or-assignment-p a1)
(clause-or-assignment-p clause)
(is-unit-clause clause a1)
(not (equal (is-unit-clause clause a1)
t)))
(clause-or-assignment-p
(cons (is-unit-clause clause a1)
a1)))
:hints (("Goal" :in-theory (enable clause-or-assignment-p))))
(defthm is-unit-clause-t-monotone-alt
(implies (and (subsetp-equal a1 a2)
(clause-or-assignment-p a2)
(not (equal (is-unit-clause clause a2)
t)))
(not (equal (is-unit-clause clause a1)
t))))
(defthm cons-preserves-subsetp
(implies (subsetp x y)
(subsetp x (cons a y))))
; !! Needed?
(defthm is-unit-clause-implies-not-member-car
(implies (and (is-unit-clause clause a1)
(clause-or-assignment-p a1))
(not (member (car clause) a1)))
:rule-classes :forward-chaining)
(defthm evaluate-clause-nil-implies-is-unit-clause-t
(implies (null (evaluate-clause clause assignment))
(equal (is-unit-clause clause assignment)
t)))
(defthm subsetp-preserves-evaluate-clause-nil
(implies (and (clause-or-assignment-p a2)
(null (evaluate-clause clause a1))
(subsetp a1 a2))
(equal (evaluate-clause clause a2)
nil)))
(defthm is-unit-clause-unchanged
(implies (and (subsetp-equal a1 a2)
(clause-or-assignment-p a2)
(not (equal (is-unit-clause clause a2)
t))
(is-unit-clause clause a1)
(is-unit-clause clause a2))
(equal (equal (is-unit-clause clause a1)
(is-unit-clause clause a2))
t)))
; After modifying unit-propagation so that in the error cases, we use
; unit-propagation-error instead of simply returning assignment, ACL2 failed to
; merge the induction schemes for (unit-propagation formula indices a1) and
; (unit-propagation formula indices a2). So we need an induction hint.
(defun unit-propagation-2 (formula indices a1 a2)
(cond
((endp indices) (list a1 a2))
(t (let* ((pair (hons-get (car indices) formula))
(clause (and pair
(not (deleted-clause-p (cdr pair)))
(cdr pair)))
(unit-literal (and clause
(is-unit-clause clause a1))))
(cond ((not unit-literal)
(unit-propagation-2 formula (cdr indices) a1
(let ((u2 (and clause
(is-unit-clause clause a2))))
(if u2
(cons u2 a2)
a2))))
((eq unit-literal t) ; found contradiction
t)
(t (unit-propagation-2 formula
(cdr indices)
(add-to-set unit-literal
a1)
(add-to-set unit-literal
a2))))))))
(defthm is-unit-clause-superset
(implies (and (subsetp a1 a2)
(syntaxp (term-order a1 a2))
(clause-or-assignment-p a2)
(is-unit-clause clause a1))
(equal (is-unit-clause clause a2)
(cond ((member-equal (is-unit-clause clause a1) a2)
nil)
((member-equal (negate (is-unit-clause clause a1))
a2)
t)
(t (is-unit-clause clause a1))))))
(defthm clause-or-assignment-p-cons
(implies (and (clause-or-assignment-p assignment)
(literalp lit)
(not (member lit assignment))
(not (member (negate lit) assignment)))
(clause-or-assignment-p (cons lit assignment)))
:hints (("Goal" :in-theory (enable clause-or-assignment-p))))
(defthm clause-or-assignment-p-has-literals
(implies (and (clause-or-assignment-p clause)
(not (literalp lit)))
(not (member lit clause))))
(defthm unit-propagation-t-monotone
(implies (and (equal (unit-propagation formula indices a1)
t)
(clause-or-assignment-p a1)
(clause-or-assignment-p a2)
(subsetp-equal a1 a2)
(formula-p formula))
(equal (unit-propagation formula indices a2)
t))
:hints (("Goal" :induct (unit-propagation-2 formula indices a1 a2))))
(defthm unit-propagation-monotonicity
(implies (and (subsetp-equal a1 a2)
(not (equal (unit-propagation formula indices a2)
t))
(clause-or-assignment-p a1)
(clause-or-assignment-p a2)
(formula-p formula))
(and (clause-or-assignment-p a1)
(subsetp (unit-propagation formula indices a1)
(unit-propagation formula indices a2))))
:hints (("Goal" :induct (unit-propagation-2 formula indices a1 a2))))
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