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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")
(defthm truth-remove-irrelevant-lit-1
(implies (and (equal (evaluate-clause clause assignment)
t)
(not (member lit clause)))
(equal (evaluate-clause clause
(remove-literal lit assignment))
t)))
(defthm truth-remove-irrelevant-lit-cons
(implies (and (equal (evaluate-clause clause assignment)
t)
(not (member lit clause)))
(equal (evaluate-clause clause
(cons lit assignment))
t)))
(defthm truth-remove-irrelevant-lit-2
(implies (and (equal (evaluate-clause clause assignment)
t)
(not (member lit clause))
(not (member (negate lit) clause)))
(equal (evaluate-clause clause
(flip-literal lit assignment))
t))
:hints (("Goal" :in-theory (enable flip-literal))))
(defthm truth-remove-irrelevant-lit-3
(implies (equal (evaluate-clause (remove-literal lit clause)
assignment)
t)
(equal (evaluate-clause clause assignment)
t)))
(defthm subsetp-preserves-assignment
(implies (and (subsetp a2 a1)
(clause-or-assignment-p a1)
(literal-listp a2)
(unique-literalsp a2))
(clause-or-assignment-p a2))
:hints (("Goal" :in-theory (enable clause-or-assignment-p))))
(defthm formula-truep-implies-evaluate-clause-t
; It remains to prove Claim 2. Since A |= F, A |= D.
; This is redundant here (proved in unit-propagation-implies-unsat.lisp).
(implies
(and (formula-truep formula assignment)
(hons-assoc-equal index formula)
(not (deleted-clause-p (cdr (hons-assoc-equal index formula)))))
(equal (evaluate-clause (cdr (hons-assoc-equal index formula))
assignment)
t)))
(defthm subsetp-cons-remove-literal
(subsetp a (cons lit (remove-literal lit a))))
(defthm sat-drat-claim-2-1
(implies (and (not (member (negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula))))
(formula-p formula)
(solution-p assignment formula)
(hons-assoc-equal index formula)
(not (deleted-clause-p
(cdr (hons-assoc-equal index formula)))))
(equal (evaluate-clause
(cdr (hons-assoc-equal index formula))
(flip-literal (car (access add-step entry :clause))
assignment))
t))
:hints (("Goal"
:in-theory (enable flip-literal)
:restrict ((truth-monotone-clause
((a1 (remove-literal (- (cadr (car entry)))
assignment)))))))
:rule-classes nil)
(defthm clause-or-assignmentp-cdr-hons-assoc-equal-for-formula
(let ((clause (cdr (hons-assoc-equal index formula))))
(implies (and (formula-p formula)
(not (deleted-clause-p clause)))
(clause-or-assignment-p clause))))
(defthm sat-drat-claim-2-2-1
(implies (and (member (negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
(equal (evaluate-clause
(remove-literal
(negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
assignment)
t)
(formula-p formula)
(not (deleted-clause-p
(cdr (hons-assoc-equal index formula)))))
(equal (evaluate-clause
(remove-literal
(negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
(flip-literal (car (access add-step entry :clause))
assignment))
t))
:instructions (:bash (:dv 1)
(:then (:rewrite truth-remove-irrelevant-lit-2)
:prove))
:rule-classes nil)
(defthm sat-drat-claim-2-2
(implies (and (member (negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
(equal (evaluate-clause
(remove-literal
(negate (car (access add-step entry :clause)))
(cdr (hons-assoc-equal index formula)))
assignment)
t)
(formula-p formula))
(equal (evaluate-clause
(cdr (hons-assoc-equal index formula))
(flip-literal (car (access add-step entry :clause))
assignment))
t))
:hints (("Goal" :use sat-drat-claim-2-2-1))
:rule-classes nil)
(encapsulate
()
(local (include-book "sat-drat-claim-2-3"))
(set-enforce-redundancy t)
(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)))
:rule-classes nil))
(defthm sat-drat-claim-2
(mv-let (ncls ndel new-formula)
(verify-clause formula entry ncls ndel)
(declare (ignore ndel))
(implies (and 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)))
(hons-assoc-equal index formula)
(not (equal (cdr (hons-assoc-equal index formula))
*deleted-clause*)))
(equal (evaluate-clause
(cdr (hons-assoc-equal index formula))
(flip-literal (car (access add-step entry :clause))
assignment))
t)))
:hints (("Goal"
:in-theory (theory 'minimal-theory)
:use (sat-drat-claim-2-1 sat-drat-claim-2-2 sat-drat-claim-2-3)))
:rule-classes nil)
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