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Bfr-updates

Bfr-updates-p

Signature
(bfr-updates-p x) → *

Definitions and Theorems

Function: bfr-updates-p

(defun bfr-updates-p (x)
       (declare (xargs :guard t))
       (let ((__function__ 'bfr-updates-p))
            (declare (ignorable __function__))
            (if (atom x)
                t
                (and (consp (car x))
                     (bfr-varname-p (caar x))
                     (bfr-updates-p (cdr x))))))

Theorem: bfr-updates-p-of-revappend

(defthm bfr-updates-p-of-revappend
        (equal (bfr-updates-p (revappend x y))
               (and (bfr-updates-p (list-fix x))
                    (bfr-updates-p y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-remove

(defthm bfr-updates-p-of-remove
        (implies (bfr-updates-p x)
                 (bfr-updates-p (remove a x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-last

(defthm bfr-updates-p-of-last
        (implies (bfr-updates-p (double-rewrite x))
                 (bfr-updates-p (last x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-nthcdr

(defthm bfr-updates-p-of-nthcdr
        (implies (bfr-updates-p (double-rewrite x))
                 (bfr-updates-p (nthcdr n x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-butlast

(defthm bfr-updates-p-of-butlast
        (implies (bfr-updates-p (double-rewrite x))
                 (bfr-updates-p (butlast x n)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-update-nth

(defthm bfr-updates-p-of-update-nth
        (implies (bfr-updates-p (double-rewrite x))
                 (iff (bfr-updates-p (update-nth n y x))
                      (and (and (consp y) (bfr-varname-p (car y)))
                           (or (<= (nfix n) (len x))
                               (and (consp nil)
                                    (bfr-varname-p (car nil)))))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-repeat

(defthm bfr-updates-p-of-repeat
        (iff (bfr-updates-p (acl2::repeat n x))
             (or (and (consp x) (bfr-varname-p (car x)))
                 (zp n)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-take

(defthm bfr-updates-p-of-take
        (implies (bfr-updates-p (double-rewrite x))
                 (iff (bfr-updates-p (take n x))
                      (or (and (consp nil)
                               (bfr-varname-p (car nil)))
                          (<= (nfix n) (len x)))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-union-equal

(defthm bfr-updates-p-of-union-equal
        (equal (bfr-updates-p (union-equal x y))
               (and (bfr-updates-p (list-fix x))
                    (bfr-updates-p (double-rewrite y))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-intersection-equal-2

(defthm bfr-updates-p-of-intersection-equal-2
        (implies (bfr-updates-p (double-rewrite y))
                 (bfr-updates-p (intersection-equal x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-intersection-equal-1

(defthm bfr-updates-p-of-intersection-equal-1
        (implies (bfr-updates-p (double-rewrite x))
                 (bfr-updates-p (intersection-equal x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-set-difference-equal

(defthm bfr-updates-p-of-set-difference-equal
        (implies (bfr-updates-p x)
                 (bfr-updates-p (set-difference-equal x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-set-equiv-congruence

(defthm bfr-updates-p-set-equiv-congruence
        (implies (acl2::set-equiv x y)
                 (equal (bfr-updates-p x)
                        (bfr-updates-p y)))
        :rule-classes :congruence)

Theorem: bfr-updates-p-when-subsetp-equal

(defthm bfr-updates-p-when-subsetp-equal
        (and (implies (and (subsetp-equal x y)
                           (bfr-updates-p y))
                      (bfr-updates-p x))
             (implies (and (bfr-updates-p y)
                           (subsetp-equal x y))
                      (bfr-updates-p x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-rcons

(defthm bfr-updates-p-of-rcons
        (iff (bfr-updates-p (acl2::rcons a x))
             (and (and (consp a) (bfr-varname-p (car a)))
                  (bfr-updates-p (list-fix x))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-rev

(defthm bfr-updates-p-of-rev
        (equal (bfr-updates-p (acl2::rev x))
               (bfr-updates-p (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-duplicated-members

(defthm bfr-updates-p-of-duplicated-members
        (implies (bfr-updates-p x)
                 (bfr-updates-p (acl2::duplicated-members x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-difference

(defthm bfr-updates-p-of-difference
        (implies (bfr-updates-p x)
                 (bfr-updates-p (set::difference x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-intersect-2

(defthm bfr-updates-p-of-intersect-2
        (implies (bfr-updates-p y)
                 (bfr-updates-p (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-intersect-1

(defthm bfr-updates-p-of-intersect-1
        (implies (bfr-updates-p x)
                 (bfr-updates-p (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-union

(defthm bfr-updates-p-of-union
        (iff (bfr-updates-p (set::union x y))
             (and (bfr-updates-p (set::sfix x))
                  (bfr-updates-p (set::sfix y))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-mergesort

(defthm bfr-updates-p-of-mergesort
        (iff (bfr-updates-p (set::mergesort x))
             (bfr-updates-p (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-delete

(defthm bfr-updates-p-of-delete
        (implies (bfr-updates-p x)
                 (bfr-updates-p (set::delete k x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-insert

(defthm bfr-updates-p-of-insert
        (iff (bfr-updates-p (set::insert a x))
             (and (bfr-updates-p (set::sfix x))
                  (and (consp a)
                       (bfr-varname-p (car a)))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-sfix

(defthm bfr-updates-p-of-sfix
        (iff (bfr-updates-p (set::sfix x))
             (or (bfr-updates-p x)
                 (not (set::setp x))))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-list-fix

(defthm bfr-updates-p-of-list-fix
        (equal (bfr-updates-p (list-fix x))
               (bfr-updates-p x))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-append

(defthm bfr-updates-p-of-append
        (equal (bfr-updates-p (append a b))
               (and (bfr-updates-p a)
                    (bfr-updates-p b)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-when-not-consp

(defthm bfr-updates-p-when-not-consp
        (implies (not (consp x))
                 (bfr-updates-p x))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-cdr-when-bfr-updates-p

(defthm bfr-updates-p-of-cdr-when-bfr-updates-p
        (implies (bfr-updates-p (double-rewrite x))
                 (bfr-updates-p (cdr x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-cons

(defthm bfr-updates-p-of-cons
        (equal (bfr-updates-p (cons a x))
               (and (and (consp a) (bfr-varname-p (car a)))
                    (bfr-updates-p x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-fast-alist-clean

(defthm bfr-updates-p-of-fast-alist-clean
        (implies (bfr-updates-p x)
                 (bfr-updates-p (fast-alist-clean x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-hons-shrink-alist

(defthm bfr-updates-p-of-hons-shrink-alist
        (implies (and (bfr-updates-p x)
                      (bfr-updates-p y))
                 (bfr-updates-p (hons-shrink-alist x y)))
        :rule-classes ((:rewrite)))

Theorem: bfr-updates-p-of-hons-acons

(defthm bfr-updates-p-of-hons-acons
        (equal (bfr-updates-p (hons-acons a n x))
               (and (bfr-varname-p a)
                    t (bfr-updates-p x)))
        :rule-classes ((:rewrite)))

Theorem: bfr-varname-p-of-caar-when-bfr-updates-p

(defthm bfr-varname-p-of-caar-when-bfr-updates-p
        (implies (bfr-updates-p x)
                 (iff (bfr-varname-p (caar x))
                      (or (consp x) (bfr-varname-p nil))))
        :rule-classes ((:rewrite)))