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; UBDD Library
; Copyright (C) 2008-2011 Warren Hunt and Bob Boyer
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Significantly revised in 2008 by Jared Davis and Sol Swords.
; Now maintained by Centaur Technology.
;
; Contact:
; Centaur Technology Formal Verification Group
; 7600-C N. Capital of Texas Highway, Suite 300, Austin, TX 78731, USA.
; http://www.centtech.com/
; deps.lisp - tracking variable dependencies of BDDs
(in-package "ACL2")
(include-book "extra-operations")
;; UBDD-DEPS: Produce a list of Ts and NILs indicating which variables the input
;; BDD depends on.
(defn or-lists (x y)
(if (atom x)
y
(if (atom y) x
(cons (or (car x) (car y))
(or-lists (cdr x) (cdr y))))))
(defthm nth-of-or-lists
(equal (nth n (or-lists x y))
(or (nth n x) (nth n y))))
(in-theory (disable or-lists))
(defn ubdd-deps (x)
(if (atom x)
nil
(cons (not (hons-equal (car x) (cdr x)))
(or-lists (ubdd-deps (car x)) (ubdd-deps (cdr x))))))
(memoize 'ubdd-deps :condition '(consp x))
(in-theory (disable ubdd-deps))
(local (defun eval-ubdd-deps-ind (x n env)
(if (atom x)
(list n env)
(if (car env)
(eval-ubdd-deps-ind (car x) (1- n) (cdr env))
(eval-ubdd-deps-ind (cdr x) (1- n) (cdr env))))))
(defthm eval-bdd-of-update-when-not-dependent
(implies (not (nth n (ubdd-deps x)))
(equal (eval-bdd x (update-nth n v env))
(eval-bdd x env)))
:hints (("goal" :induct (eval-ubdd-deps-ind x n env)
:in-theory (enable (:i eval-bdd))
:expand ((:free (env) (eval-bdd x env))
(ubdd-deps x)
(:free (a b) (nth n (cons a b)))))))
(local (defun bdd-deps-ind (x n)
(if (zp n)
x
(list (bdd-deps-ind (car x) (1- n))
(bdd-deps-ind (cdr x) (1- n))))))
(local (defun 2bdd-deps-ind (x y n)
(if (zp n)
(list x y)
(list (2bdd-deps-ind (car x) (car y) (1- n))
(2bdd-deps-ind (cdr x) (cdr y) (1- n))))))
(local (defun q-ite-deps-ind (x y z n)
(declare (xargs :measure (nfix n)))
(if (zp n)
(list x y z)
(list (q-ite-deps-ind (car x)
(if (hons-equal x y) t (qcar y))
(if (hons-equal x z) nil (qcar z))
(1- n))
(q-ite-deps-ind (cdr x)
(if (hons-equal x y) t (qcdr y))
(if (hons-equal x z) nil (qcdr z))
(1- n))))))
(local (in-theory (disable equal-of-booleans-rewrite)))
(local (defthm nth-ubdd-deps-of-qcons
(equal (nth n (ubdd-deps (qcons x y)))
(if (zp n)
(not (equal x y))
(nth (1- n) (or-lists (ubdd-deps x) (ubdd-deps y)))))
:hints(("Goal" :in-theory (enable ubdd-deps)))))
(local (defthm nth-ubdd-deps-of-cons
(equal (nth n (ubdd-deps (cons x y)))
(if (zp n)
(not (equal x y))
(nth (1- n) (or-lists (ubdd-deps x) (ubdd-deps y)))))
:hints(("Goal" :in-theory (enable ubdd-deps)))))
(local (defthm nth-of-ubdd-deps-when-zp
(implies (and (syntaxp (symbolp x))
(zp n))
(equal (nth n (ubdd-deps x))
(not (equal (car x) (cdr x)))))
:hints(("Goal" :in-theory (enable ubdd-deps)))
:rule-classes ((:rewrite :backchain-limit-lst 0))))
(local (defthm nth-of-ubdd-deps-when-not-zp
(implies (and (syntaxp (symbolp x))
(not (zp n)))
(equal (nth n (ubdd-deps x))
(nth (1- n) (or-lists
(ubdd-deps (car x))
(ubdd-deps (cdr x))))))
:hints(("Goal" :in-theory (enable ubdd-deps)))
:rule-classes ((:rewrite :backchain-limit-lst 0))))
(local (in-theory (disable qcons nth)))
(local (defthm nth-of-nil
(not (nth n nil))
:hints(("Goal" :in-theory (enable nth)))))
(local (defthm or-lists-identical
(equal (or-lists x x)
x)
:hints(("Goal" :in-theory (enable or-lists)))))
(defthm ubdd-deps-of-qv
(equal (ubdd-deps (qv n))
(update-nth n t nil))
:hints(("Goal" :in-theory (enable (:i qv))
:induct (qv n)
:expand ((qv n)
(:free (x y) (ubdd-deps (cons x y)))))))
;; BOZO may also prove that if (nth n (ubdd-deps x)) and (ubddp x), then there
;; exists some env, v under which
;; (eval-bdd x env) != (eval-bdd x (update-nth n v env)).
;; Stronger theorems may be proved provided X is ubddp. At the moment I'll
;; just prove simple conservative stuff.
(defthm q-not-no-new-deps
(implies (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps (q-not x)))))
:hints (("goal" :induct (bdd-deps-ind x n))))
(defthm q-and-no-new-deps
(implies (and (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps y))))
(not (nth n (ubdd-deps (q-binary-and x y)))))
:hints (("goal" :induct (2bdd-deps-ind x y n)
:in-theory (disable (force))
:expand ((q-binary-and x y)))))
(defthm q-or-no-new-deps
(implies (and (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps y))))
(not (nth n (ubdd-deps (q-binary-or x y)))))
:hints (("goal" :induct (2bdd-deps-ind x y n)
:in-theory (disable (force))
:expand ((q-binary-or x y)))))
(defthm q-xor-no-new-deps
(implies (and (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps y))))
(not (nth n (ubdd-deps (q-binary-xor x y)))))
:hints (("goal" :induct (2bdd-deps-ind x y n)
:in-theory (disable (force))
:expand ((q-binary-xor x y)))))
(defthm q-iff-no-new-deps
(implies (and (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps y))))
(not (nth n (ubdd-deps (q-binary-iff x y)))))
:hints (("goal" :induct (2bdd-deps-ind x y n)
:in-theory (disable (force))
:expand ((q-binary-iff x y)))))
(defthm ubdd-deps-of-atom
(implies (not (consp x))
(equal (ubdd-deps x) nil))
:hints(("Goal" :in-theory (enable ubdd-deps)))
:rule-classes ((:rewrite :backchain-limit-lst 0)))
; Added by Matt K., 9/28/2013, to get around ACL2(hp) error such as:
; Error: Not owner of hash table #<HASH-TABLE :TEST EQL size 3/62 #x30200BB9920D>
; David Rager points out (email, 9/28/2013) that "memoization is known
; not to be thread-safe"; Jared Davis says this too. (Perhaps this will be
; addressed in the future.)
(local (unmemoize 'ubdd-deps))
(defthm q-ite-no-new-deps
(implies (and (not (nth n (ubdd-deps x)))
(not (nth n (ubdd-deps y)))
(not (nth n (ubdd-deps z))))
(not (nth n (ubdd-deps (q-ite-fn x y z)))))
:hints (("goal" :induct (q-ite-deps-ind x y z n)
:in-theory (disable (force))
:expand ((q-ite-fn x y z)
(:free (a b) (nth n (cons a b)))))))
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