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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 ./README.
(in-package "LRAT")
(include-book "../stobj-based/lrat-checker")
(defun ordered-literalsp (x)
(declare (xargs :guard (literal-listp x)))
(cond ((endp x) t)
(t (or (null (cdr x))
(and (< (abs (car x)) (abs (cadr x)))
(ordered-literalsp (cdr x)))))))
(defthm ordered-literalsp-implies-not-member
(implies (and (consp x)
(< (abs a) (abs (car x)))
(ordered-literalsp x))
(not (member a x))))
(defthm ordered-literalsp-implies-unique-literalsp
(implies (ordered-literalsp x)
(unique-literalsp x)))
(defthm ordered-literalsp-implies-not-member-negate
(implies (and (consp x)
(< (abs a) (abs (car x)))
(ordered-literalsp x))
(not (member (negate a) x))))
(defthm ordered-literalsp-implies-not-conflicting-literalsp
(implies (ordered-literalsp x)
(not (conflicting-literalsp x))))
(defun lrat-clausep (x)
(declare (xargs :guard t))
(or (null x)
(and (literal-listp x)
(not (member (car x) (cdr x)))
(not (member (negate (car x)) (cdr x)))
(ordered-literalsp (cdr x)))))
; !! Consider introducing/using ordered versions of clause-max-var,
; formula-max-var, and proof-max-var.
(defun lrat-add-step-p (x)
(declare (xargs :guard t
:guard-hints (("Goal" :use (:guard-theorem add-step-p)))))
(and (weak-add-step-p x)
(posp (access add-step x :index))
(lrat-clausep (access add-step x :clause))
(index-listp (access add-step x :rup-indices))
(drat-hint-listp (access add-step x :drat-hints))))
(defthm lrat-add-step-p-implies-add-step-p
(implies (lrat-add-step-p proof)
(add-step-p proof))
:rule-classes :forward-chaining)
(defun lrat-proof-entry-p (entry)
(declare (xargs :guard t))
(cond ((and (consp entry)
(eq (car entry) t)) ; deletion
(index-listp (cdr entry)))
(t (lrat-add-step-p entry))))
(defthm lrat-proof-entry-p-implies-proof-entry-p
(implies (lrat-proof-entry-p proof)
(proof-entry-p proof))
:rule-classes :forward-chaining)
(defun lrat-proofp (proof)
(declare (xargs :guard t))
(if (atom proof)
(null proof)
(and (lrat-proof-entry-p (car proof))
(lrat-proofp (cdr proof)))))
(defthm lrat-proofp-implies-proofp
(implies (lrat-proofp proof)
(proofp proof))
:rule-classes :forward-chaining)
(defun lrat-valid-proofp$ (formula proof a$)
(declare (xargs :stobjs a$ ; not necessarily satisfying a$p
:guard (formula-p formula)
:guard-hints (("Goal" :use (:guard-theorem valid-proofp$)))))
(let* ((formula (make-fast-alist formula))
(max-var (and (lrat-proofp proof)
(proof-max-var proof
(formula-max-var formula 0)))))
(cond ((varp max-var)
(let ((a$ (initialize-a$ max-var a$)))
(mv-let (v a$)
(verify-proof$ formula proof a$)
(mv v
(proof-contradiction-p proof)
a$))))
(t (mv nil nil a$)))))
(defun lrat-valid-proofp$-top (formula proof incomplete-okp)
(declare (xargs :guard t))
(and (formula-p formula)
(with-local-stobj a$
(mv-let (v c a$)
(lrat-valid-proofp$ formula proof a$)
(and v
(or incomplete-okp
c))))))
(in-theory (disable verify-proof$))
(encapsulate
()
(local (defthm lrat-valid-proofp$-is-valid-proofp$
(implies (car (lrat-valid-proofp$ formula proof a$))
(equal (valid-proofp$ formula proof a$)
(lrat-valid-proofp$ formula proof a$)))))
(in-theory (disable lrat-valid-proofp$ valid-proofp$))
(defthm lrat-valid-proofp$-top-implies-valid-proofp$-top
(implies (lrat-valid-proofp$-top formula proof incomplete-okp)
(valid-proofp$-top formula proof incomplete-okp))
:rule-classes :forward-chaining))
(defun lrat-refutation-p$ (proof formula)
(declare (xargs :guard t))
(lrat-valid-proofp$-top formula proof nil))
(in-theory (disable valid-proofp$-top lrat-valid-proofp$-top))
(defthm lrat-refutation-p$-implies-refutation-p$
(implies (lrat-refutation-p$ proof formula)
(refutation-p$ proof formula))
:rule-classes :forward-chaining)
(in-theory (disable satisfiable formula-p lrat-refutation-p$))
(local (include-book "../stobj-based/soundness"))
(defthm main-theorem
(implies (and (formula-p formula)
(lrat-refutation-p$ proof formula))
(not (satisfiable formula))))
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