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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.
(in-package "LRAT")
(include-book "lrat-checker")
(include-book "satisfiable-maybe-shrink-formula")
(defun extend-with-proof (ncls ndel formula proof)
; This function is essentially verify-proof, except that instead of returning a
; Boolean, it returns the extended formula. And there is one other difference:
; We stop if/when we hit the empty clause in proof, since we want to prove that
; the empty clause gives us an unsat formula, but later steps might delete the
; empty clause!
(cond
((atom proof) formula)
(t
(let* ((entry (car proof))
(delete-flg (proof-entry-deletion-p entry)))
(cond
(delete-flg
(let* ((indices (proof-entry-deletion-indices entry))
(new-formula (delete-clauses indices formula))
(len (length indices))
(ncls
; We expect that (<= len ncls). It is tempting to assert that here (with
; assert$), but it's not necessary so we avoid the overhead (mostly in proof,
; but perhaps also a bit in execution).
(- ncls len))
(ndel (+ ndel len)))
(extend-with-proof ncls ndel new-formula (cdr proof))))
(t ; addition
(let ((clause (access add-step entry :clause)))
(mv-let (ncls ndel formula)
(verify-clause formula entry ncls ndel)
(let* ((index (access add-step entry :index))
(new-formula (add-proof-clause index clause formula)))
(if (null clause)
new-formula ; see comment above
(extend-with-proof
(1+ ncls)
ndel
new-formula
(cdr proof))))))))))))
(defthm formula-p-delete-clauses
(implies (and (formula-p fal)
(index-listp index-list))
(formula-p (delete-clauses index-list fal))))
(defthm formula-p-add-proof-clause
(implies (and (posp index)
(clause-or-assignment-p clause)
(formula-p formula))
(formula-p (add-proof-clause index clause formula))))
(encapsulate
()
(local
(defthm hons-assoc-equal-delete-clauses-member-equal-case
(implies (equal (hons-assoc-equal index fal)
(cons index *deleted-clause*))
(equal (hons-assoc-equal index (delete-clauses index-list fal))
(cons index *deleted-clause*)))))
(local
(defthm hons-assoc-equal-delete-clauses
(equal (hons-assoc-equal index (delete-clauses index-list fal))
(if (member-equal index index-list)
(cons index *deleted-clause*)
(hons-assoc-equal index fal)))))
(local
(defthm delete-clauses-preserves-formula-truep
(implies (and (formula-p formula)
(clause-or-assignment-p assignment)
(formula-truep formula assignment))
(formula-truep (delete-clauses index-list formula)
assignment))
:hints (("Goal"
:in-theory (disable formula-truep)
:expand ((formula-truep (delete-clauses index-list formula)
assignment))
:restrict ((formula-truep-necc
((index (formula-truep-witness
(delete-clauses index-list formula)
assignment)))))))))
(defthm delete-clauses-preserves-satisfiable
(implies (and (formula-p formula)
(satisfiable formula))
(satisfiable (delete-clauses index-list formula)))
:hints (("Goal"
:in-theory (disable satisfiable formula-truep)
:expand ((satisfiable formula))
:restrict ((satisfiable-suff
((assignment (satisfiable-witness formula)))))))))
(encapsulate
()
(local (include-book "satisfiable-add-proof-clause"))
(defthm satisfiable-add-proof-clause
(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))
(satisfiable (add-proof-clause
(access add-step entry :index)
(access add-step entry :clause)
new-formula))))))
(in-theory (disable add-proof-clause verify-clause))
(defthm formula-truep-cons-shrink-formula
(implies (and (formula-p formula)
(clause-or-assignment-p assignment)
(formula-truep (cons pair (shrink-formula formula))
assignment))
(formula-truep (cons pair formula)
assignment))
:hints (("Goal"
:expand ((formula-truep (cons pair formula)
assignment))
:restrict ((formula-truep-necc
((index (formula-truep-witness (cons pair formula)
assignment))))))))
(defthm satisfiable-cons-shrink-formula
(implies (formula-p formula)
(implies (satisfiable (cons pair (shrink-formula formula)))
(satisfiable (cons pair formula))))
:hints (("Goal"
:expand ((satisfiable (cons pair (shrink-formula formula))))
:restrict
((satisfiable-suff
((assignment
(satisfiable-witness (cons pair
(shrink-formula formula))))))))))
(encapsulate
()
(local
(defthm not-satisfiable-add-proof-clause-nil
(implies (and (formula-p formula)
(posp index))
(not (satisfiable (add-proof-clause index nil formula))))
:hints (("Goal"
:in-theory (enable add-proof-clause satisfiable)
:restrict ((formula-truep-necc ((index index))))))
:rule-classes nil))
(defthm verify-clause-implies-unsatisfiable
(implies (and (formula-p formula)
(proof-entry-p entry)
(equal (access add-step entry :clause) nil)
(satisfiable formula))
(not (car (verify-clause formula entry ncls ndel))))
:hints (("Goal"
:in-theory (enable verify-clause add-proof-clause
maybe-shrink-formula)
:use ((:instance not-satisfiable-add-proof-clause-nil
(index (access add-step entry :index)))
(:instance satisfiable-add-proof-clause
(entry (change add-step entry
:clause nil))))))))
(defthm formula-p-mv-nth-2-verify-clause
(implies (formula-p formula)
(formula-p (mv-nth 2
(verify-clause formula entry ncls ndel))))
:hints (("Goal" :in-theory (enable verify-clause maybe-shrink-formula))))
(defthm extend-with-proof-preserves-satisfiable
(implies (and (formula-p formula)
(proofp proof)
(verify-proof-rec ncls ndel formula proof)
(satisfiable formula))
(satisfiable (extend-with-proof ncls ndel formula proof)))
:rule-classes nil)
(defthm not-formula-truep-add-nil-clause
(not (formula-truep (cons (list (car (car (car proof))))
fal)
assignment))
:hints (("Goal" :restrict ((formula-truep-necc
((index (car (car (car proof))))))))))
(defthm proof-contradiction-p-implies-false
(implies (and (formula-p formula)
(proofp proof)
(proof-contradiction-p proof))
(equal (formula-truep (extend-with-proof ncls ndel formula proof)
assignment)
nil))
:hints (("Goal" :in-theory (enable verify-clause add-proof-clause
maybe-shrink-formula))))
(defthm proof-contradiction-p-implies-not-satisfiable
(implies (and (formula-p formula)
(proofp proof)
(proof-contradiction-p proof))
(not (satisfiable (extend-with-proof ncls ndel formula proof))))
:hints (("Goal" :in-theory (e/d (satisfiable) (extend-with-proof)))))
(defthm main-theorem-list-based
(implies (and (formula-p formula)
(refutation-p proof formula))
(not (satisfiable formula)))
:hints (("Goal" :use ((:instance extend-with-proof-preserves-satisfiable
(ncls (len (fast-alist-fork formula nil)))
(ndel 0))))))
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