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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/
; param.lisp -- bdd parameterization
(in-package "ACL2")
(include-book "extra-operations")
(include-book "misc/hons-help2" :dir :system)
(include-book "tools/rulesets" :dir :system)
(local (include-book "std/lists/take" :dir :system))
(local (include-book "arithmetic/top" :dir :system))
(local (in-theory (disable* default-car default-cdr ;; blp-implies-t
default-+-2 default-+-1 default-<-2 default-<-1
(:ruleset canonicalize-to-q-ite)
equal-by-eval-bdds
)))
(local (in-theory (enable eval-bdd eval-bdd-list ubddp ubdd-listp
q-compose q-compose-list)))
;; -------------------------------------------------------------------
;; Function Definitions
;; -------------------------------------------------------------------
;; find-diff: given two different BDDs, finds a variable assignment
;; that makes their evaulations unequal.
(defn find-diff (a b)
(declare (xargs :measure (+ (acl2-count a) (acl2-count b))))
(if (and (atom a) (atom b))
nil
(if (hqual (qcdr a) (qcdr b))
(cons t (find-diff (qcar a) (qcar b)))
(cons nil (find-diff (qcdr a) (qcdr b))))))
(defn find-diff-of-length (a b n)
(declare (xargs :measure (nfix n)))
(if (zp (nfix n))
nil
(if (and (atom a) (atom b))
(cons nil (find-diff-of-length a b (1- n)))
(if (hqual (qcdr a) (qcdr b))
(cons t (find-diff-of-length (qcar a) (qcar b) (1- n)))
(cons nil (find-diff-of-length (qcdr a) (qcdr b) (1- n)))))))
; Regarless of whether
; x = f(a,b,c), l = g(p,q), h(p,q), i(p,q),
; q-compose(f(a,b,c), (g(p,q), h(p,q), i(p,q)) )
; = f(g(p,q),h(p,q),i(p,q))
(defun head-tail-bind-fn (bindings body)
(if (atom bindings)
body
(let ((b (car bindings)))
`(mv-let (,(car b) ,(cadr b))
(if (consp ,(caddr b))
(mv (car ,(caddr b)) (cdr ,(caddr b)))
(mv nil nil))
(declare (ignorable ,(car b) ,(cadr b)))
,(head-tail-bind-fn (cdr bindings) body)))))
(defmacro head-tail-bind (bindings body)
(head-tail-bind-fn bindings body))
;; Makes a list each element of which is (q-ite x yi zi), where yi and
;; zi are the corresponding elements in lists ys and zs.
(defn q-ite-list (x ys zs)
(declare (xargs :measure (+ (acl2-count ys)
(acl2-count zs))))
(if (and (atom ys) (atom zs))
nil
(head-tail-bind
((z1 zr zs)
(y1 yr ys))
(cons (q-ite x y1 z1)
(q-ite-list x yr zr)))))
;; QCONSes together corresponding elements of lists L1 and L2.
(defn q-zipper (l1 l2)
(declare (xargs :measure (+ (len l1) (len l2))))
(if (and (atom l1) (atom l2))
nil
(head-tail-bind
((l11 l1r l1)
(l21 l2r l2))
(cons (qcons l11 l21)
(q-zipper l1r l2r)))))
(in-theory (disable q-zipper))
;; Makes a list of N bdd variables starting at START and increasing by
;; BY at each step.
(defn qv-list (start by n)
(declare (xargs :measure (nfix n)))
(if (zp (nfix n))
nil
(cons (qv (nfix start))
(qv-list (ec-call (binary-+ by start)) by (1- n)))))
;; Fixes each element in list L to a Boolean.
(defn bfix-list (l)
(if (atom l)
nil
(cons (if (car l) t nil)
(bfix-list (cdr l)))))
;; Given a BDD X representing some predicate of n variables, makes a
;; list of n BDDs for which 1) any evaluation of the n outputs under
;; the same variable assignment produces a variable assignment that
;; satisfies X, and 2) any variable assignment that satisfies X can be
;; produced by the n BDDs.
;; More precisely, NVARS is the number of variables "used" by X; for
;; 1) above to hold it should be at least (max-depth x) but it can be
;; more. 2) only holds for variable assignments of length NVARS.
(defn q-param (x nvars)
(cond ((zp (nfix nvars)) nil)
((atom x) (qv-list 0 1 nvars))
((not (car x))
(cons nil (q-param (cdr x) (1- nvars))))
((not (cdr x))
(cons t (q-param (car x) (1- nvars))))
(t (cons (qv 0)
(q-zipper
(q-param (car x) (1- nvars))
(q-param (cdr x) (1- nvars)))))))
;; Given a BDD X, find an "inverse" for q-param
(defn q-param-inv (x nvars)
(cond ((zp (nfix nvars)) nil)
((atom x) (qv-list 0 1 nvars))
((not (car x))
(let ((x (q-param-inv (cdr x) (1- nvars))))
(q-zipper x x)))
((not (cdr x))
(let ((x (q-param-inv (car x) (1- nvars))))
(q-zipper x x)))
(t (cons (qv 0)
(q-zipper (q-param-inv (car x) (1- nvars))
(q-param-inv (cdr x) (1- nvars)))))))
(defn max-max-depth (x)
(if (atom x)
0
(max (max-depth (car x))
(max-max-depth (cdr x)))))
(defn extend-list (a b)
(declare (xargs :measure (+ (len a) (len b))))
(if (and (atom a) (atom b))
a
(head-tail-bind
((a1 ar a)
(b1 br b))
(cons a1 (extend-list ar br)))))
(defun ebmde-ind (x l l2)
(if (atom x)
(list l l2)
(list (ebmde-ind (car x) (cdr l) (cdr l2))
(ebmde-ind (cdr x) (cdr l) (cdr l2)))))
(defthm eval-bdd-max-depth-extend
(implies (<= (max-depth x) (len l))
(equal (eval-bdd x (extend-list l l2))
(eval-bdd x l)))
:hints (("goal" :induct (ebmde-ind x l l2)
:in-theory (e/d (default-car default-cdr max-depth)
()))))
(defthm eval-bdd-list-max-max-depth-extend
(implies (<= (max-max-depth x) (len l))
(equal (eval-bdd-list x (extend-list l l2))
(eval-bdd-list x l)))
:hints (("goal" :in-theory (disable ))))
(defthm len-eval-bdd-list
(equal (len (eval-bdd-list x l))
(len x)))
;; UBDDP forward-chaining rules
(defthm ubddp-cons-forward-chain
(implies (and (ubddp x) (consp x))
(and (ubddp (car x)) (ubddp (cdr x))))
:rule-classes :forward-chaining)
(defthm ubddp-atom-forward-chain
(implies (and (ubddp x) (not (consp x)))
(or (equal x nil) (equal x t)))
:rule-classes :forward-chaining)
;; -------------------------------------------------------------------
(defthm car-nthcdr
(equal (car (nthcdr n l))
(nth n l)))
(defthm nth-implies-consp-nthcdr
(implies (nth n l)
(consp (nthcdr n l))))
(defthm cdr-nthcdr
(equal (nthcdr n (cdr l))
(cdr (nthcdr n l))))
;; FIND-DIFF always produces a variable assignment which
;; differentiates two BDDs, if they are different:
(defthm eval-bdd-diff
(implies (and (ubddp a)
(ubddp b)
(not (equal a b)))
(not (equal (eval-bdd a (find-diff a b))
(eval-bdd b (find-diff a b)))))
:hints (("Goal" :induct (find-diff a b))))
;; FIND-DIFF-OF-LENGTH always produces a variable assignment which
;; differentiates two BDDs, if they are different:
(defthm eval-bdd-diff-of-length
(implies (and (ubddp a)
(ubddp b)
(not (equal a b))
(<= (max-depth a) (nfix n))
(<= (max-depth b) (nfix n)))
(not (equal (eval-bdd a (find-diff-of-length a b n))
(eval-bdd b (find-diff-of-length a b n)))))
:hints (("goal" :in-theory (enable max-depth))))
(defthm find-diff-of-length-len
(equal (len (find-diff-of-length a b n))
(nfix n)))
;; This is an indirect (but easier than a direct, inductive proof) way
;; of proving identities about BDD operations. To prove a generic
;; equation of two expressions that produce BDDs, we prove that the
;; eval-bdds of the two expressions are equal under all variable
;; assignments, then use eval-bdd-diff (above) to show that the
;; expressions themselves are equal.
;; (defthm eval-bdd-q-ite-of-q-ite
;; (implies (and (ubddp a)
;; (ubddp b) (ubddp c)
;; (ubddp d) (ubddp e))
;; (equal (eval-bdd (q-ite (q-ite a b c) d e) vals)
;; (eval-bdd (q-ite a
;; (q-ite b d e)
;; (q-ite c d e))
;; vals))))
;; (defthm q-ite-of-q-ite
;; (implies (and (ubddp a)
;; (ubddp b) (ubddp c)
;; (ubddp d) (ubddp e))
;; (equal (q-ite (q-ite a b c) d e)
;; (q-ite a
;; (q-ite b d e)
;; (q-ite c d e)))))
(defthm q-and-equals-t
(implies (equal (q-and x y) t)
(and (implies (ubddp x) (equal x t))
(implies (ubddp y) (equal y t))))
:hints (("goal" :in-theory (enable q-and)))
:rule-classes :forward-chaining)
(defthmd ubddp-q-not-t-implies-not
(implies (and (ubddp x)
x)
(not (equal (q-not x) t)))
:hints (("goal" :in-theory (enable q-not))))
(defthmd ubddp-q-not-nil-implies-t
(implies (and (ubddp x)
(not (q-not x)))
(equal (equal x t) t))
:hints (("goal" :in-theory (enable q-not))))
(defthm ubdd-listp-q-ite-list
(implies (and (ubddp x) (ubdd-listp ys) (ubdd-listp zs))
(ubdd-listp (q-ite-list x ys zs)))
:hints (("goal" :induct (q-ite-list x ys zs))))
;; (defthm q-compose-ubddp
;; (implies (and (ubddp x) (ubdd-listp l))
;; (ubddp (q-compose x l))))
;; Show that Q-COMPOSE really performs function composition.
(defthm q-compose-to-eval-bdd
(implies (and (ubddp x) (ubdd-listp l))
(equal (eval-bdd (q-compose x l) vals)
(eval-bdd x (eval-bdd-list l vals)))))
; (add-bdd-fn-pat q-compose)
(defthm q-ite-list-correct
(implies (and (ubddp x) (ubdd-listp ys) (ubdd-listp zs)
(equal (len ys) (len zs)))
(equal (eval-bdd-list (q-ite-list x ys zs) vals)
(if (eval-bdd x vals)
(eval-bdd-list ys vals)
(eval-bdd-list zs vals)))))
;; (if a (f b) (f c)) == (f (if a b c))
(defthm eval-bdd-q-compose-commutes-with-q-ite
(implies (and (ubddp x) (ubddp y)
(ubdd-listp l1) (ubdd-listp l2)
(equal (len l1) (len l2)))
(equal (eval-bdd (q-ite x
(q-compose y l1)
(q-compose y l2))
vals)
(eval-bdd (q-compose y (q-ite-list x l1 l2))
vals)))
:hints (("goal" :in-theory (disable q-compose))))
(defthm q-compose-commutes-with-q-ite
(implies (and (ubddp x) (ubddp y)
(ubdd-listp l1) (ubdd-listp l2)
(equal (len l1) (len l2)))
(equal (q-ite x
(q-compose y l1)
(q-compose y l2))
(q-compose y (q-ite-list x l1 l2))))
:hints(("Goal" :in-theory (enable equal-by-eval-bdds))))
;; (defthm ubdd-listp-q-compose-list
;; (implies (and (ubdd-listp ys) (ubdd-listp zs))
;; (ubdd-listp (q-compose-list ys zs))))
(defthm q-compose-list-correct
(implies (and (ubdd-listp ys) (ubdd-listp zs))
(equal (eval-bdd-list (q-compose-list ys zs) vals)
(eval-bdd-list ys (eval-bdd-list zs vals)))))
(defthm eval-bdd-q-compose-associative
(implies (and (ubddp x) (ubdd-listp ys) (ubdd-listp zs))
(equal (eval-bdd (q-compose (q-compose x ys) zs) vals)
(eval-bdd (q-compose x (q-compose-list ys zs))
vals))))
(defthm q-compose-associative
(implies (and (ubddp x) (ubdd-listp ys) (ubdd-listp zs))
(equal (q-compose (q-compose x ys) zs)
(q-compose x (q-compose-list ys zs))))
:hints(("Goal" :in-theory (enable equal-by-eval-bdds))))
(defthm eval-bdd-depth
(implies (<= (max-depth x) (len l))
(equal (eval-bdd x (append l l2))
(eval-bdd x l)))
:hints (("goal" :in-theory (enable max-depth))))
(in-theory (enable q-zipper))
(defthm ubdd-listp-q-zipper
(implies (and (ubdd-listp l1) (ubdd-listp l2))
(ubdd-listp (q-zipper l1 l2))))
(defthm len-q-zipper
(equal (len (q-zipper l1 l2))
(max (len l1) (len l2))))
(defun nth-q-zipper-ind (n a b)
(if (zp n)
(cons a b)
(head-tail-bind
((a1 ar a)
(b1 br b))
(nth-q-zipper-ind (1- n) ar br))))
(defthm nth-q-zipper
(implies (and (<= (nfix n) (len a))
(<= (nfix n) (len b)))
(equal (nth n (q-zipper a b))
(qcons (nth n a) (nth n b))))
:hints (("goal" :induct (nth-q-zipper-ind n a b))))
(in-theory (disable q-zipper))
(defun nth-qv-list-ind (m start by n)
(if (zp m)
(list start by n)
(nth-qv-list-ind (1- m) (+ start by) by (1- n))))
(defthm nth-qv-list
(implies (and (integerp n)
(integerp m)
(<= 0 m)
(< m n)
(integerp start)
(<= 0 start)
(integerp by)
(<= 0 by))
(equal (nth m (qv-list start by n))
(qv (+ start (* m by)))))
:hints (("goal" :induct (nth-qv-list-ind m start by n)
:in-theory (enable qv zp nfix))
(and (case-match id (((0 1) . &) t))
'(:expand (qv-list start by n)))))
(defthm eval-bdd-extend-list
(equal (eval-bdd-list (extend-list a b) vars)
(extend-list (eval-bdd-list a vars) b)))
(defthm eval-q-zipper
(equal (eval-bdd-list
(q-zipper l1 l2)
vars)
(if (car vars)
(eval-bdd-list (extend-list l1 l2) (cdr vars))
(eval-bdd-list (extend-list l2 l1) (cdr vars))))
:hints (("goal" :in-theory (enable q-zipper))))
(defthm ubddp-qv-all
(ubddp (qv n)))
(defthm ubdd-listp-qv-list
(ubdd-listp (qv-list start by n)))
(defthm len-qv-list
(equal (len (qv-list start by n))
(nfix n)))
(defthm max-max-depth-q-zipper-upper-bound
(implies (and (<= (max-max-depth x) c)
(<= (max-max-depth y) c))
(<= (max-max-depth (q-zipper x y))
(+ 1 c)))
:hints (("goal" :in-theory (enable q-zipper max-depth)))
:rule-classes (:rewrite :linear))
(defthm max-depth-qv
(equal (max-depth (qv n))
(1+ (nfix n)))
:hints (("goal" :in-theory (enable max-depth qv))))
(encapsulate nil
(local (include-book "arithmetic/top-with-meta" :dir :system))
(defthm max-max-depth-qv-list
(implies (and (natp start) (natp by) (posp n))
(equal (max-max-depth (qv-list start by n))
(+ 1 start (* by (1- n)))))
:hints (("goal" :in-theory (enable max-depth natp posp))
("Subgoal *1/5" :expand (qv-list start by 1))))
(defthm max-max-depth-make-list-ac
(equal (max-max-depth (make-list-ac n nil acc))
(max-max-depth acc)))
(defthm len-q-param-inv
(implies (<= (max-depth x) (nfix nvars))
(<= (max-max-depth (q-param x nvars))
(len (q-param-inv x nvars))))
:hints (("goal" :in-theory (enable max-depth))
("Subgoal *1/16"
:use ((:instance max-max-depth-q-zipper-upper-bound
(x (q-param (car x) (1- nvars)))
(y (q-param (cdr x) (1- nvars)))
(c (len (q-param-inv (car x) (+ -1 nvars)))))
(:instance max-max-depth-q-zipper-upper-bound
(x (q-param (car x) (1- nvars)))
(y (q-param (cdr x) (1- nvars)))
(c (len (q-param-inv (cdr x) (+ -1 nvars))))))))
:rule-classes :linear))
(defthm max-max-depth-q-param
(<= (max-max-depth (q-param pred n))
(nfix n))
:rule-classes (:rewrite :linear))
(defthm max-max-depth-q-param-inv
(<= (max-max-depth (q-param-inv pred n))
(nfix n))
:rule-classes (:rewrite :linear))
(defthm not-eval-bdd-qv-nil
(not (eval-bdd (qv n) nil)))
(defthm eval-bdd-qv1
(equal (eval-bdd (qv n) vars)
(if (nth n vars) t nil)))
(local
(encapsulate nil
(local (include-book "arithmetic/top" :dir :system))
(defthm take-bfix-list
(equal (take (len vars) (bfix-list vars))
(bfix-list vars)))))
(defthm bfix-list-bfix-list
(equal (bfix-list (bfix-list x))
(bfix-list x)))
(defthm eval-qv-list
(implies (natp start)
(equal (eval-bdd-list (qv-list start 1 n)
vars)
(bfix-list
(take n (nthcdr start vars)))))
:hints(("Goal" :in-theory (enable take))))
(defthm ubdd-listp-make-list-ac
(equal (ubdd-listp (make-list-ac n nil acc))
(ubdd-listp acc)))
(defthm ubdd-listp-q-param
(ubdd-listp (q-param x depth)))
(defthm len-of-make-list-ac
(equal (len (make-list-ac n val acc))
(+ (nfix n) (len acc))))
(defthm len-q-param
(equal (len (q-param x depth))
(nfix depth)))
(defun q-param-eval-bdd-ind (x depth vars)
(if (atom x)
(cons depth vars)
(list (q-param-eval-bdd-ind (car x) (1- depth) (cdr vars))
(q-param-eval-bdd-ind (cdr x) (1- depth) (cdr vars))
(q-param-eval-bdd-ind (car x) (1- depth) vars)
(q-param-eval-bdd-ind (cdr x) (1- depth) vars))))
(defthm eval-q-param
(implies (and (natp depth)
(<= (max-depth x) depth)
(ubddp x)
x)
(equal (eval-bdd x (eval-bdd-list (q-param x depth) vars))
t))
:hints (("Goal" :induct (q-param-eval-bdd-ind x depth vars)
:in-theory (enable max-depth))))
(local
(defthm bfix-list-list-fix
(equal (bfix-list (list-fix x))
(bfix-list x))))
;; (defthm eval-q-param-inv
;; (implies (and (equal (eval-bdd x vars) t)
;; (<= (max-depth x) (len vars)))
;; (equal (eval-bdd-list (q-param x (len vars))
;; (q-param-inv x vars))
;; (bfix-list vars)))
;; :hints (("goal" :induct (q-param-inv x vars))))
(defn eqpib-ind (x vars)
(cond ((atom vars) x)
((atom x) vars)
((not (car x))
(eqpib-ind (cdr x) (cdr vars)))
((not (cdr x))
(eqpib-ind (car x) (cdr vars)))
(t (list (eqpib-ind (car x) (cdr vars))
(eqpib-ind (cdr x) (cdr vars))))))
(defthm extend-list-len
(implies (<= (len b) (len a))
(equal (extend-list a b) a)))
(defthm ubdd-listp-q-param-inv
(ubdd-listp (q-param-inv x nvars)))
(defthm eval-q-param-inv
(implies (and (equal (eval-bdd x vars) t)
(<= (max-depth x) (len vars)))
(equal (eval-bdd-list (q-param x (len vars))
(eval-bdd-list (q-param-inv x (len vars))
vars))
(bfix-list vars)))
:hints (("goal" :induct (eqpib-ind x vars)
:in-theory (e/d (max-depth)))))
(defthm len-q-param-inv-upper-bound
(<= (len (q-param-inv x nvars))
(nfix nvars))
:rule-classes :linear)
(defthm len-bfix-list
(equal (len (bfix-list x))
(len x)))
(defthm compose-q-param-inv
(implies (and (equal (eval-bdd x vars) t)
(<= (max-depth x) (len vars)))
(equal (eval-bdd-list (q-compose-list
(q-param x (len vars))
(q-param-inv x (len vars)))
vars)
(bfix-list vars))))
(defthm bfix-list-boolean-list
(and (boolean-listp (bfix-list x))
(implies (boolean-listp x)
(equal (bfix-list x) x))))
;; -------------------------------------------------------------------
;; Theorem 1 from Aagard-Jones-Seger 1999:
;; part 1:
(defthm forall-y-p-of-param-of-y-is-true
(implies (and p
(ubddp p)
(integerp n)
(<= (max-depth p) n))
(equal (eval-bdd p
(eval-bdd-list (q-param p n)
y))
t)))
;; part 2:
(defthm exists-y-such-that-x-is-param-of-y
(implies (and (equal (eval-bdd p x) t)
(<= (max-depth p) (len x))
(boolean-listp x))
(let ((y (eval-bdd-list (q-param-inv p (len x)) x)))
(equal (eval-bdd-list (q-param p (len x))
y)
x))))
;; -------------------------------------------------------------------
;; part 2 using compose instead of eval:
(defthm x-is-param-of-y-compose-eval
(implies (and (equal (eval-bdd p x) t)
(<= (max-depth p) (len x))
(boolean-listp x))
(equal (eval-bdd-list (q-compose-list (q-param p (len x))
(q-param-inv p (len x)))
x)
x)))
(memoize 'q-param :condition '(consp x))
(memoize 'q-param-inv :condition '(consp x))
(memoize 'q-compose :condition '(consp x))
;; From-param-space pulls a bdd y in param(p) space back into the original
;; space. To prove it correct we also define to-param-space.
(defn from-param-space (p y)
(cond ((atom y) (if y p nil))
((atom p) (if p y nil))
((eq (car p) nil)
(let ((x (from-param-space (cdr p) y)))
(qcons nil x)))
((eq (cdr p) nil)
(let ((x (from-param-space (car p) y)))
(qcons x nil)))
(t (qcons (from-param-space (car p) (car y))
(from-param-space (cdr p) (cdr y))))))
(memoize 'from-param-space :condition '(or (consp p) (consp y)))
(defn to-param-space (p y)
(cond ((atom p) y)
((atom y) y)
((eq (car p) nil)
(to-param-space (cdr p) (cdr y)))
((eq (cdr p) nil)
(to-param-space (car p) (car y)))
(t (qcons (to-param-space (car p) (car y))
(to-param-space (cdr p) (cdr y))))))
;; [Jared]: tweaking this to only memoize when car&cdr are non-nil, to slightly
;; reduce memo table overhead.
(memoize 'to-param-space :condition '(and (consp p)
(consp y)
(car p)
(cdr p)))
(defthm ubddp-to-param-space
(implies (ubddp x)
(ubddp (to-param-space p x))))
(defun param-env (p env)
(declare (xargs :guard t))
(cond ((atom p) env)
((atom env) nil)
((eq (car p) nil) (param-env (cdr p) (cdr env)))
((eq (cdr p) nil) (param-env (car p) (cdr env)))
((car env) (cons t (param-env (car p) (cdr env))))
(t (cons nil (param-env (cdr p) (cdr env))))))
(defthm param-env-to-param-space
(implies (eval-bdd p env)
(equal (eval-bdd (to-param-space p x)
(param-env p env))
(eval-bdd x env)))
:hints(("Goal" :in-theory (enable eval-bdd to-param-space))))
(defun unparam-env (p env)
(declare (xargs :guard t))
(cond ((atom p) env)
((eq (car p) nil) (cons nil (unparam-env (cdr p) env)))
((eq (cdr p) nil) (cons t (unparam-env (car p) env)))
(t (mv-let (car cdr) (if (consp env)
(mv (car env) (cdr env))
(mv nil nil))
(cons car (unparam-env (if car (car p) (cdr p))
cdr))))))
(encapsulate
nil
(local (defun ind (x p env)
(if (consp x)
(if (consp p)
(if (car p)
(if (cdr p)
(if (car env)
(ind (car x) (car p) (cdr env))
(ind (cdr x) (cdr p) (cdr env)))
(ind x (car p) env))
(ind x (cdr p) env))
x)
env)))
(defthm eval-param-env-of-unparam-env
(equal (eval-bdd x (param-env p (unparam-env p env)))
(eval-bdd x env))
:hints (("goal" :induct (ind x p env)
:in-theory (enable default-car default-cdr))
(and stable-under-simplificationp
'(:expand ((:free (env) (eval-bdd x env))))))))
(defthm eval-with-unparam-env
(implies (and p (ubddp p))
(eval-bdd p (unparam-env p env))))
(defun unparam-env-ind (x p env)
(cond ((atom p) (list x env))
((eq (car p) nil) (unparam-env-ind (cdr x) (cdr p) env))
((eq (cdr p) nil) (unparam-env-ind (car x) (car p) env))
(t (unparam-env-ind (if (car env) (car x) (cdr x))
(if (car env) (car p) (cdr p))
(cdr env)))))
(defthmd unparam-env-to-param-space
(equal (eval-bdd (to-param-space p x) env)
(eval-bdd x (unparam-env p env)))
:hints (("goal" :induct (unparam-env-ind x p env)
:expand ((:free (env) (eval-bdd x env))
(:free (env) (unparam-env p env)))
:in-theory (enable default-car default-cdr))))
(defthm unparam-env-of-param-env
(implies (eval-bdd p env)
(equal (eval-bdd x (unparam-env p (param-env p env)))
(eval-bdd x env)))
:hints (("goal" :use ((:instance param-env-to-param-space)
(:instance unparam-env-to-param-space
(env (param-env p env)))))))
(defn to-param-space-list (p list)
(if (atom list)
nil
(cons (to-param-space p (car list))
(to-param-space-list p (cdr list)))))
(defthm eval-bdd-lst-to-param-space-lst
(implies (eval-bdd p env)
(equal (eval-bdd-list (to-param-space-list p list)
(param-env p env))
(eval-bdd-list list env))))
(defun q-param-to-param-ind (p m n)
(if (atom p)
(cons m n)
(cons (q-param-to-param-ind (car p) (1- m) (1- n))
(q-param-to-param-ind (cdr p) (1- m) (1- n)))))
(encapsulate nil
(local (include-book "arithmetic/top-with-meta" :dir :system))
(defthm nth-make-list-ac-nil
(equal (nth m (make-list-ac n nil acc))
(and (<= (nfix n) (nfix m))
(nth (- (nfix m) (nfix n)) acc)))
:hints (("goal" :induct (make-list-ac n nil acc)))))
;; TO-PARAM-SPACE of (QV m) is the same as the mth element of Q-PARAM.
(defthm q-param-to-param
(implies (and (integerp n)
(integerp m)
(<= 0 m)
(< m n)
(<= (max-depth p) n))
(equal (nth m (q-param p n))
(to-param-space p (qv m))))
:hints (("goal" :induct (q-param-to-param-ind p m n)
:in-theory (enable max-depth qv))))
(encapsulate nil
(local (in-theory (disable equal-by-eval-bdds)))
(local (defthm q-not-equal-t
(implies (ubddp x)
(equal (equal (q-not x) t)
(equal x nil)))
:hints (("goal" :in-theory (enable q-not)))))
(local (defthm not-q-not-x
(implies (ubddp x)
(iff (q-not x)
(not (equal x t))))
:hints (("goal" :in-theory (enable q-not)))))
(local (in-theory (disable q-not-under-iff)))
;; NOT commutes over to-param-space:
(defthm q-not-to-param
(implies (and p (ubddp p) (ubddp x))
(equal (q-not (to-param-space p x))
(to-param-space p (q-not x))))
:hints (("goal" :in-theory
(e/d (q-not) (equal-by-eval-bdds)))
("Subgoal *1/9"
:in-theory (e/d (q-not ubddp-q-not-t-implies-not
ubddp-q-not-nil-implies-t))))))
(encapsulate
()
(local (defthm not-consp-not-nil-implies-t
(implies (and (not (consp (to-param-space p x)))
(ubddp x)
(to-param-space p x))
(equal (to-param-space p x) t))
:rule-classes :forward-chaining))
(defun q-and-to-param-ind (p x y)
(if (atom p)
(cons x y)
(cons (q-and-to-param-ind (car p) (qcar x) (qcar y))
(q-and-to-param-ind (cdr p) (qcdr x) (qcdr y)))))
;; AND commutes over to-param-space:
(defthm q-and-to-param
(implies (and (ubddp x) (ubddp y))
(equal (q-and (to-param-space p x)
(to-param-space p y))
(to-param-space p (q-and x y))))
:hints (("goal" :induct (q-and-to-param-ind p x y)
;; BOZO: Why do we have to disable force?
:in-theory (e/d (q-and) (ubddp (force)))))))
(defthm q-ite-in-terms-of-and-and-not
(implies (and (ubddp x) (ubddp y) (ubddp z))
(equal (q-ite x y z)
(q-not (q-and
(q-not (q-and x y))
(q-not (q-and (q-not x) z))))))
:hints(("Goal" :in-theory (enable equal-by-eval-bdds)))
:rule-classes nil)
;; Using the AND and NOT thms we can prove that all basic BDD operations
;; commute over to-param-space:
(defthm q-ite-to-param
(implies (and p (ubddp p) (ubddp x) (ubddp y) (ubddp z))
(equal (q-ite (to-param-space p x)
(to-param-space p y)
(to-param-space p z))
(to-param-space p (q-ite x y z))))
:hints (("goal"
:use ((:instance q-ite-in-terms-of-and-and-not)
(:instance q-ite-in-terms-of-and-and-not
(x (to-param-space p x))
(y (to-param-space p y))
(z (to-param-space p z)))))))
;; Finally, from-param-space of to-param-space reduces to q-and with the predicate.
(defthm to-from-param-space
(implies (and (ubddp x) (ubddp p))
(equal (from-param-space p (to-param-space p x))
(q-and p x)))
:hints (("goal" :induct (to-param-space p x)
:in-theory (e/d (q-and) (ubddp)))))
(defthm eval-bdd-of-qcons
(equal (eval-bdd (qcons x y) env)
(if (car env)
(eval-bdd x (cdr env))
(eval-bdd y (cdr env)))))
(defun to-from-ind (p x env)
(if (consp p)
(cond ((eq (car p) nil)
(and (not (car env))
(to-from-ind (cdr p) (cdr x) (cdr env))))
((eq (cdr p) nil)
(and (car env)
(to-from-ind (car p) (car x) (cdr env))))
(t (list (to-from-ind (car p) (car x) (cdr env))
(to-from-ind (cdr p) (cdr x) (cdr env)))))
(list p x env)))
(defthm from-param-space-of-qcons
(implies (and (car p) (cdr p))
(equal (from-param-space p (qcons x y))
(if (atom (qcons x y))
(and x p)
(qcons (from-param-space (car p) x)
(from-param-space (cdr p) y)))))
:otf-flg t)
(defthm consp-qcons
(equal (consp (qcons x y))
(not (and (booleanp x)
(equal x y)))))
(defthm to-from-param-space-eval
(equal (eval-bdd (from-param-space p (to-param-space p x)) env)
(and (eval-bdd p env)
(eval-bdd x env)))
:hints (("goal" :induct (to-from-ind p x env)
:in-theory (disable qcons))))
(defthm to-param-space-self
(implies (and (ubddp p) p)
(equal (to-param-space p p) t))
:hints(("Goal" :in-theory (enable ubddp))))
(defthm from-to-param-space
(implies (and (ubddp x) (ubddp p) p)
(equal (to-param-space p (from-param-space p x))
x))
:hints (("goal" :induct (from-param-space p x)
:in-theory (e/d (q-and) (ubddp)))))
(defun param-env-ind (x p env)
(cond ((atom p) (list x env))
((eq (car p) nil) (param-env-ind x (cdr p) (cdr env)))
((eq (cdr p) nil) (param-env-ind x (car p) (cdr env)))
((car env) (cons t (param-env-ind (car x) (car p) (cdr env))))
(t (cons nil (param-env-ind (cdr x) (cdr p) (cdr env))))))
(defthm eval-of-from-param-space
(implies (eval-bdd p env)
(equal (eval-bdd (from-param-space p x)
env)
(eval-bdd x (param-env p env))))
:hints (("goal" :induct (param-env-ind x p env)
:expand ((from-param-space p x)
(unparam-env p env)
(:free (env) (eval-bdd x env))
(eval-bdd p env)))))
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