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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 remove-deleted-clauses-leaves-no-deleted-clauses
(implies (not (equal (cdr (hons-assoc-equal index acc))
*deleted-clause*))
(not (equal (cdr (hons-assoc-equal
index
(remove-deleted-clauses fal acc)))
*deleted-clause*))))
(defthm shrink-formula-has-no-deleted-clauses
(not (equal (cdr (hons-assoc-equal index
(shrink-formula fal)))
*deleted-clause*))
:hints (("Goal" :in-theory (enable shrink-formula))))
(in-theory (disable fast-alist-clean))
(defthm hons-assoc-equal-remove-deleted-clauses-for-non-member
(implies (not (member-equal index (strip-cars fal)))
(equal (hons-assoc-equal index (remove-deleted-clauses fal acc))
(hons-assoc-equal index acc))))
(defthm hons-assoc-equal-remove-deleted-clauses
(implies (no-duplicatesp-equal (strip-cars fal))
(equal (hons-assoc-equal index (remove-deleted-clauses fal acc))
(if (equal (cdr (hons-assoc-equal index fal))
*deleted-clause*)
(hons-assoc-equal index acc)
(or (hons-assoc-equal index fal)
(hons-assoc-equal index acc))))))
(defthm member-strip-cars-is-hons-assoc-equal
(implies (alistp x)
(iff (member a (strip-cars x))
(hons-assoc-equal a x))))
(defthm no-duplicatesp-strip-cars-fast-alist-fork
(implies (alistp ans)
(iff (no-duplicatesp (strip-cars (fast-alist-fork alist ans)))
(no-duplicatesp (strip-cars ans)))))
(defthm alistp-forward-to-null-cdr-last
(implies (alistp x)
(equal (cdr (last x))
nil))
:rule-classes :forward-chaining)
(defthm no-duplicatesp-strip-cars-fast-alist-clean
(implies (alistp fal)
(no-duplicatesp (strip-cars (fast-alist-clean fal))))
:hints (("Goal" :in-theory (enable fast-alist-clean))))
(defthm hons-assoc-equal-fast-alist-fork
(equal (hons-assoc-equal key (fast-alist-fork fal ans))
(or (hons-assoc-equal key ans)
(hons-assoc-equal key fal))))
(defthm hons-assoc-equal-fast-alist-clean
(implies (alistp fal)
(equal (hons-assoc-equal key (fast-alist-clean fal))
(hons-assoc-equal key fal)))
:hints (("Goal" :in-theory (enable fast-alist-clean))))
(defthm hons-assoc-equal-shrink-formula
(implies (alistp fal)
(equal (hons-assoc-equal index (shrink-formula fal))
(if (equal (cdr (hons-assoc-equal index fal))
*deleted-clause*)
nil
(hons-assoc-equal index fal))))
:hints (("Goal" :in-theory (enable shrink-formula))))
(local (in-theory (enable formula-p)))
(defthm formula-true-p-maybe-shrink-formula-forward
(implies
(formula-p formula)
(implies (formula-truep (mv-nth 2 (maybe-shrink-formula ncls ndel formula
factor))
assignment)
(formula-truep formula assignment)))
:hints (("Goal"
:in-theory (enable maybe-shrink-formula)
:expand ((formula-truep formula assignment))
:restrict ((formula-truep-necc
((index (formula-truep-witness formula assignment)))))))
:rule-classes nil)
(defthm formula-true-p-maybe-shrink-formula-backward
(implies
(formula-p formula)
(implies (formula-truep formula assignment)
(formula-truep (mv-nth 2 (maybe-shrink-formula ncls ndel formula
factor))
assignment)))
:hints (("Goal"
:in-theory (enable maybe-shrink-formula)
:expand ((formula-truep (shrink-formula formula)
assignment))))
:rule-classes nil)
(defthm formula-true-p-maybe-shrink-formula
(implies
(formula-p formula)
(equal (formula-truep (mv-nth 2 (maybe-shrink-formula ncls ndel formula
factor))
assignment)
(formula-truep formula assignment)))
:hints (("Goal" :use (formula-true-p-maybe-shrink-formula-forward
formula-true-p-maybe-shrink-formula-backward))))
(defthm satisfiable-maybe-shrink-formula-forward
(implies
(formula-p formula)
(implies
(satisfiable (mv-nth 2 (maybe-shrink-formula ncls ndel formula factor)))
(satisfiable formula)))
:hints
(("Goal"
:expand ((satisfiable (mv-nth 2 (maybe-shrink-formula ncls ndel formula factor))))
:restrict
((satisfiable-suff
((assignment (satisfiable-witness
(mv-nth 2 (maybe-shrink-formula ncls ndel formula factor)))))))))
:rule-classes nil)
(defthm satisfiable-maybe-shrink-formula-backward
(implies
(formula-p formula)
(implies
(satisfiable formula)
(satisfiable (mv-nth 2 (maybe-shrink-formula ncls ndel formula factor)))))
:hints
(("Goal"
:expand ((satisfiable formula))
:restrict
((satisfiable-suff
((assignment (satisfiable-witness formula))))))))
(defthm satisfiable-maybe-shrink-formula
(implies
(formula-p formula)
(equal (satisfiable (mv-nth 2 (maybe-shrink-formula ncls ndel formula factor)))
(satisfiable formula)))
:hints (("Goal" :use (satisfiable-maybe-shrink-formula-forward
satisfiable-maybe-shrink-formula-backward))))
(defthm satisfiable-cons-mv-nth-2-maybe-shrink-formula-lemma
(implies
(and
(satisfiable (cons (cons index clause) formula))
(formula-p formula)
(satisfiable formula))
(satisfiable (cons (cons index clause)
(mv-nth 2
(maybe-shrink-formula ncls ndel formula
factor)))))
; I really don't like to give :instructions. However, it was taking a lot of
; time on this one to eliminate proof-building commands in favor of :hints, so
; I'm going to live with this. It's really not so horrible to use
; :instructions when nested quantifiers are involved (satisfiable is a defun-sk
; that calls formula-truep, which is a defun-sk).
:instructions
((:in-theory (enable maybe-shrink-formula))
(:bash ("Goal" :expand ((satisfiable (cons (cons index clause) formula)))))
(:rewrite
satisfiable-suff
((assignment (satisfiable-witness (cons (cons index clause) formula)))))
(:bash ("Goal" :expand
((formula-truep
(cons (cons index clause)
(shrink-formula formula))
(satisfiable-witness (cons (cons index clause) formula))))))
(:contrapose 2)
(:dv 1)
(:rewrite formula-truep-necc ((index index)))
:prove
(:generalize
((formula-truep-witness
(cons (cons index clause)
(shrink-formula formula))
(satisfiable-witness (cons (cons index clause) formula)))
index2))
(:contrapose 2)
(:dv 1)
(:rewrite formula-truep-necc ((index index2)))
:prove)
:rule-classes nil)
(defthm satisfiable-cons-mv-nth-2-maybe-shrink-formula
(implies
(and
(consp index/clause)
(satisfiable (cons index/clause formula))
(formula-p formula)
(satisfiable formula))
(satisfiable (cons index/clause
(mv-nth 2
(maybe-shrink-formula ncls ndel formula
factor)))))
:hints
(("Goal"
:use ((:instance satisfiable-cons-mv-nth-2-maybe-shrink-formula-lemma
(index (car index/clause))
(clause (cdr index/clause)))))))
(defthm unit-propagation-maybe-shrink-formula
(implies (formula-p formula)
(equal (unit-propagation
(mv-nth 2 (maybe-shrink-formula ncls ndel formula factor))
indices
clause)
(unit-propagation formula indices clause)))
:hints (("Goal" :in-theory (enable maybe-shrink-formula formula-p))))
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