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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")
; We include the following book at the top level, rather than locally to an
; encapsulate, because some of its lemmas other than the last one contribute to
; proofs in this book.
(include-book "sat-drat-claim-1")
(encapsulate
()
(set-enforce-redundancy t)
(defthm sat-drat-claim-1
(implies (and (not (satisfiable (add-proof-clause index clause formula)))
(solution-p assignment formula)
(clause-or-assignment-p clause))
(subsetp (negate-clause-or-assignment clause)
assignment))))
; The following might not be necessary, but it has perhaps been helpful during
; development of the events below.
(local (in-theory (disable verify-clause)))
(encapsulate
()
(local (include-book "sat-drat-claim-2"))
(set-enforce-redundancy t)
(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)))
:rule-classes nil))
(defthm sat-drat-claim-2-for-formula
(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))))
(formula-truep
formula
(flip-literal (car (access add-step entry :clause))
assignment))))
:hints (("Goal"
:in-theory
(enable formula-truep)
:use
((:instance sat-drat-claim-2
(index (formula-truep-witness
formula
(flip-literal (car (access add-step entry
:clause))
assignment)))))))
:rule-classes nil)
(include-book "satisfiable-maybe-shrink-formula")
(defthm sat-drat-key
(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))
(consp (access add-step entry :clause)))
(satisfiable (add-proof-clause
(access add-step entry :index)
(access add-step entry :clause)
new-formula))))
:hints (("Goal"
:in-theory (enable verify-clause)
:restrict
((formula-truep-add-proof-clause
((lit (cadr (car entry)))))
(satisfiable-suff
((assignment (flip-literal (cadr (car entry)) assignment)))))
:use sat-drat-claim-2-for-formula)))
(defthm satisfiable-add-proof-clause-drat
(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)
(satisfiable formula)
(not (equal (unit-propagation formula
(access add-step entry
:rup-indices)
(negate-clause-or-assignment
(access add-step entry
:clause)))
t))
(consp (access add-step entry :clause)))
(satisfiable (add-proof-clause
(access add-step entry :index)
(access add-step entry :clause)
new-formula))))
:hints
(("Goal"
:in-theory (union-theories '(sat-drat-key)
(theory 'minimal-theory))
:expand ((satisfiable formula))
:use ((:instance sat-drat-key
(assignment (satisfiable-witness formula))))))
:rule-classes nil)
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