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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)))