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; Copyright (C) 2017, 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 cube.lisp.
; Originally, our development of verify-for-cube-soundness.lisp included some
; rather general lemmas. We initially created the present book by moving those
; lemmas, along with some supporting definitions, into this book, so that they
; could be reused in development of the book
; verify-for-cube-soundness-main.lisp.
(in-package "LRAT")
(include-book "cube")
(defun falsify-clause (clause assignment)
(cond ((atom clause) assignment)
((equal (evaluate-literal (car clause) assignment) 0)
(falsify-clause (cdr clause)
(cons (negate (car clause)) assignment)))
(t (falsify-clause (cdr clause) assignment))))
; Start proof of evaluate-clause-falsify-clause
(defthm falsify-clause-preserves-member
(implies (member-equal lit assignment)
(member-equal lit (falsify-clause clause assignment))))
(defthm member-falsify-clause
(implies (and (literalp lit)
(clause-or-assignment-p clause)
(member-equal lit clause)
(not (member-equal lit assignment)))
(member-equal (negate lit)
(falsify-clause clause assignment))))
(defthm not-member-falsify-clause-1-1
(implies (and (literalp lit)
(clause-or-assignment-p clause))
(implies (member-equal lit
(falsify-clause clause assignment))
(or (member-equal (- lit) clause)
(member-equal lit assignment))))
:rule-classes nil)
(defthm conflicting-literals-p-suff
(implies (and (not (conflicting-literalsp clause))
(literalp lit)
(member lit clause))
(not (member (negate lit) clause))))
(defthm not-member-falsify-clause-1
(implies (and (literalp lit)
(clause-or-assignment-p clause)
(member-equal lit clause))
(implies (member-equal lit
(falsify-clause clause assignment))
(member-equal lit assignment)))
:hints (("Goal"
:use not-member-falsify-clause-1-1
:do-not-induct t))
:rule-classes nil)
(defthm not-member-falsify-clause
(implies (and (literalp lit)
(clause-or-assignment-p clause)
(member-equal lit clause))
(iff (member-equal lit
(falsify-clause clause assignment))
(member-equal lit assignment)))
:hints (("Goal" :use (not-member-falsify-clause-1
falsify-clause-preserves-member))))
(defthm evaluate-clause-falsify-clause
(implies (and (subsetp-equal clause1 clause2)
(clause-or-assignment-p clause1)
(clause-or-assignment-p clause2)
(not (equal (evaluate-clause clause1 assignment)
t)))
(equal (evaluate-clause clause1 (falsify-clause clause2 assignment))
nil)))
(encapsulate
()
(local (include-book "../list-based/truth-monotone"))
(defthm truth-monotone
(implies (and (subsetp-equal a1 a2)
(equal (formula-truep formula a1) t))
(equal (formula-truep formula a2) t)))
(defthm truth-monotone-alt
(implies (and (formula-truep formula a1)
(subsetp-equal a1 a2))
(equal (formula-truep formula a2) t))))
; Start proof of cube-soundness-2.
(defthm literal-listp-falsify-clause
(implies (and (literal-listp clause)
(literal-listp assignment))
(literal-listp (falsify-clause clause assignment))))
(defthm unique-literalsp-falsify-clause
(implies (unique-literalsp assignment)
(unique-literalsp (falsify-clause clause assignment))))
(defthm not-conflicting-literalsp-falsify-clause
(implies (and (literal-listp clause)
(not (conflicting-literalsp assignment)))
(not (conflicting-literalsp (falsify-clause clause assignment)))))
(defthm cons-preserves-subsetp
(implies (subsetp-equal x y)
(subsetp-equal x (cons a y))))
(defthm subsetp-x-x
(subsetp-equal x x))
(defthm subset-equal-falsify-clause
(subsetp-equal assignment
(falsify-clause clause assignment)))
(defun cube-to-formula (cube next-index)
; This variant of extend-formula-with-cube1 is useful for reasoning, below.
(cond ((endp cube) nil)
(t (acons next-index
(list (car cube))
(cube-to-formula
(cdr cube)
(1+ next-index))))))
(defthm extend-formula-with-cube1-rewrite
(implies (and (clause-or-assignment-p cube)
(posp next-index))
(equal (extend-formula-with-cube1 formula cube next-index)
(revappend (cube-to-formula cube next-index) formula))))
(defthm hons-assoc-equal-revappend-1-1
(implies (and (consp pair)
(not (member-equal (car pair) (strip-cars x1)))
(not (member-equal (car pair) (strip-cars x2))))
(equal (hons-assoc-equal (car pair)
(revappend x1 (append x2 (cons pair y))))
pair))
:hints (("Goal" :induct (revappend x1 x2)))
:rule-classes nil)
(defthm hons-assoc-equal-revappend-1
(implies (and (consp pair)
(not (member-equal (car pair) (strip-cars x))))
(equal (hons-assoc-equal (car pair)
(revappend x (cons pair y)))
pair))
:hints (("Goal" :use (:instance hons-assoc-equal-revappend-1-1
(x1 x) (x2 nil)))))
(defthm hons-assoc-equal-revappend
(implies (no-duplicatesp (strip-cars x))
(equal (hons-assoc-equal i (revappend x y))
(or (hons-assoc-equal i x)
(hons-assoc-equal i y)))))
(defthm hons-assoc-equal-cube-to-formula-implies-large-index
(implies (and (natp next-index)
(hons-assoc-equal i (cube-to-formula cube next-index)))
(and (natp i)
(>= i next-index)))
:rule-classes nil)
(defthm hons-assoc-equal-formula-implies-small-index
(implies (and (ordered-formula-p formula)
(hons-assoc-equal i formula))
(and (natp i)
(<= i (formula-max-index formula))))
:hints (("Goal" :in-theory (enable ordered-formula-p)))
:rule-classes nil)
(defthm hons-assoc-equal-formula-implies-not-hons-assoc-equal-cube-to-formula
(implies (and (ordered-formula-p formula)
(hons-assoc-equal i formula))
(not (hons-assoc-equal
i
(cube-to-formula cube
(+ 1 (formula-max-index formula))))))
:otf-flg t
:hints (("Goal"
:do-not-induct t
:in-theory (disable ordered-formula-p-implies-formula-p)
:use
((:instance hons-assoc-equal-cube-to-formula-implies-large-index
(next-index (+ 1 (formula-max-index formula))))
hons-assoc-equal-formula-implies-small-index
ordered-formula-p-implies-formula-p))))
(defthm no-duplicatesp-strip-cars-cube-to-formula-1
(implies (< i j)
(not (member-equal i (strip-cars (cube-to-formula cube j))))))
(defthm no-duplicatesp-strip-cars-cube-to-formula
(implies (natp j)
(no-duplicatesp-equal
(strip-cars (cube-to-formula cube j)))))
(local (defthm ordered-formula-p-implies-formula-p-forward
(implies (ordered-formula-p formula)
(formula-p formula))
:rule-classes :forward-chaining))
(local (in-theory (disable ordered-formula-p-implies-formula-p)))
(in-theory (disable formula-max-index))
(defthm hons-assoc-equal-extend-formula-with-cube-1
(implies (and (ordered-formula-p formula)
(clause-or-assignment-p cube)
(hons-assoc-equal i formula))
(equal (hons-assoc-equal i
(extend-formula-with-cube formula cube))
(hons-assoc-equal i formula)))
:hints (("Goal" :in-theory (enable extend-formula-with-cube)))
:rule-classes nil)
(defthm hons-assoc-equal-cube-to-formula-implies-member-equal
(implies (and (literal-listp cube)
(hons-assoc-equal i
(cube-to-formula cube j)))
(member-equal
(cadr (hons-assoc-equal i (cube-to-formula cube j)))
cube)))
(defthm not-cddr-hons-assoc-equal-cube-to-formula
(not (cddr (hons-assoc-equal i (cube-to-formula cube j)))))
(defthm consp-cdr-hons-assoc-equal-cube-to-formula
(implies (hons-assoc-equal i (cube-to-formula cube j))
(consp (cdr (hons-assoc-equal i (cube-to-formula cube j))))))
(defthm hons-assoc-equal-extend-formula-with-cube-2
(let ((pair (hons-assoc-equal i
(extend-formula-with-cube formula cube))))
(implies (and (formula-p formula)
(clause-or-assignment-p cube)
pair
(not (hons-assoc-equal i formula)))
(let ((clause (cdr pair)))
(and (consp clause)
(null (cdr clause))
(member-equal (car clause) cube)))))
:hints (("Goal"
:do-not-induct t
:in-theory (enable extend-formula-with-cube)))
:otf-flg t
:rule-classes nil)
(defthm literal-listp-negate-cube
(implies (literal-listp x)
(literal-listp (negate-cube x))))
(defthm member-equal-negate-cube
(implies (literal-listp x)
(iff (member-equal lit (negate-cube x))
(member-equal (negate lit) x))))
(defthm conflicting-literalsp-negate-cube
(implies (literal-listp x)
(iff (conflicting-literalsp (negate-cube x))
(conflicting-literalsp x))))
(defthm unique-literalsp-negate-cube
(implies (literal-listp x)
(equal (unique-literalsp (negate-cube x))
(unique-literalsp x))))
(defthm member-negate-implies-member-assignment
(implies (and (equal (evaluate-clause clause assignment)
nil)
(clause-or-assignment-p clause)
(force (literalp lit))
(member-equal (negate lit) clause))
(member-equal lit assignment)))
(defthm formula-truep-extend-formula-with-cube
; There is lots of induction under forcing rounds; maybe a little work could
; eliminate that.
(implies
(and (ordered-formula-p formula)
(clause-or-assignment-p clause)
(clause-or-assignment-p assignment))
(implies (and (formula-truep formula assignment)
(equal (evaluate-clause clause assignment)
nil))
(formula-truep (extend-formula-with-cube formula
(negate-cube clause))
assignment)))
:hints (("Goal"
:use
((:instance
hons-assoc-equal-extend-formula-with-cube-1
(i (formula-truep-witness
(extend-formula-with-cube formula (negate-cube clause))
assignment))
(cube (negate-cube clause)))
(:instance
hons-assoc-equal-extend-formula-with-cube-2
(i (formula-truep-witness
(extend-formula-with-cube formula (negate-cube clause))
assignment))
(cube (negate-cube clause))))
:expand ((formula-truep
(extend-formula-with-cube formula (negate-cube clause))
assignment))
:restrict ((formula-truep-necc
((index
(formula-truep-witness
(extend-formula-with-cube formula (negate-cube clause))
assignment))))
(member-negate-implies-member-assignment
((clause clause)))))))
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