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• Bfr-listp$

Bfr-listp$-basics

Basic theorems about bfr-listp$, generated by deflist.

Definitions and Theorems

Theorem: bfr-listp$-of-cons

(defthm bfr-listp$-of-cons
  (equal (bfr-listp$ (cons a x) bfrstate)
         (and (bfr-p a) (bfr-listp$ x bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-cdr-when-bfr-listp$

(defthm bfr-listp$-of-cdr-when-bfr-listp$
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (bfr-listp$ (cdr x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-when-not-consp

(defthm bfr-listp$-when-not-consp
  (implies (not (consp x))
           (bfr-listp$ x bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-p-of-car-when-bfr-listp$

(defthm bfr-p-of-car-when-bfr-listp$
  (implies (bfr-listp$ x bfrstate)
           (bfr-p (car x)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-append

(defthm bfr-listp$-of-append
  (equal (bfr-listp$ (append a b) bfrstate)
         (and (bfr-listp$ a bfrstate)
              (bfr-listp$ b bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-list-fix

(defthm bfr-listp$-of-list-fix
  (equal (bfr-listp$ (list-fix x) bfrstate)
         (bfr-listp$ x bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-sfix

(defthm bfr-listp$-of-sfix
  (iff (bfr-listp$ (set::sfix x) bfrstate)
       (or (bfr-listp$ x bfrstate)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-insert

(defthm bfr-listp$-of-insert
  (iff (bfr-listp$ (set::insert a x) bfrstate)
       (and (bfr-listp$ (set::sfix x) bfrstate)
            (bfr-p a)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-delete

(defthm bfr-listp$-of-delete
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (set::delete k x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-mergesort

(defthm bfr-listp$-of-mergesort
  (iff (bfr-listp$ (set::mergesort x) bfrstate)
       (bfr-listp$ (list-fix x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-union

(defthm bfr-listp$-of-union
  (iff (bfr-listp$ (set::union x y) bfrstate)
       (and (bfr-listp$ (set::sfix x) bfrstate)
            (bfr-listp$ (set::sfix y) bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-intersect-1

(defthm bfr-listp$-of-intersect-1
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (set::intersect x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-intersect-2

(defthm bfr-listp$-of-intersect-2
  (implies (bfr-listp$ y bfrstate)
           (bfr-listp$ (set::intersect x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-difference

(defthm bfr-listp$-of-difference
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (set::difference x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-duplicated-members

(defthm bfr-listp$-of-duplicated-members
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (acl2::duplicated-members x)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-rev

(defthm bfr-listp$-of-rev
  (equal (bfr-listp$ (rev x) bfrstate)
         (bfr-listp$ (list-fix x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-rcons

(defthm bfr-listp$-of-rcons
  (iff (bfr-listp$ (acl2::rcons a x) bfrstate)
       (and (bfr-p a)
            (bfr-listp$ (list-fix x) bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-p-when-member-equal-of-bfr-listp$

(defthm bfr-p-when-member-equal-of-bfr-listp$
  (and (implies (and (member-equal a x)
                     (bfr-listp$ x bfrstate))
                (bfr-p a))
       (implies (and (bfr-listp$ x bfrstate)
                     (member-equal a x))
                (bfr-p a)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-when-subsetp-equal

(defthm bfr-listp$-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (bfr-listp$ y bfrstate))
                (bfr-listp$ x bfrstate))
       (implies (and (bfr-listp$ y bfrstate)
                     (subsetp-equal x y))
                (bfr-listp$ x bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-set-equiv-congruence

(defthm bfr-listp$-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (bfr-listp$ x bfrstate)
                  (bfr-listp$ y bfrstate)))
  :rule-classes :congruence)

Theorem: bfr-listp$-of-set-difference-equal

(defthm bfr-listp$-of-set-difference-equal
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (set-difference-equal x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-intersection-equal-1

(defthm bfr-listp$-of-intersection-equal-1
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (bfr-listp$ (intersection-equal x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-intersection-equal-2

(defthm bfr-listp$-of-intersection-equal-2
  (implies (bfr-listp$ (double-rewrite y) bfrstate)
           (bfr-listp$ (intersection-equal x y)
                       bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-union-equal

(defthm bfr-listp$-of-union-equal
  (equal (bfr-listp$ (union-equal x y) bfrstate)
         (and (bfr-listp$ (list-fix x) bfrstate)
              (bfr-listp$ (double-rewrite y)
                          bfrstate)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-take

(defthm bfr-listp$-of-take
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (iff (bfr-listp$ (take n x) bfrstate)
                (or (bfr-p nil) (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-repeat

(defthm bfr-listp$-of-repeat
  (iff (bfr-listp$ (acl2::repeat n x) bfrstate)
       (or (bfr-p x) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: bfr-p-of-nth-when-bfr-listp$

(defthm bfr-p-of-nth-when-bfr-listp$
  (implies (bfr-listp$ x bfrstate)
           (bfr-p (nth n x)))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-update-nth

(defthm bfr-listp$-of-update-nth
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (iff (bfr-listp$ (update-nth n y x) bfrstate)
                (and (bfr-p y)
                     (or (<= (nfix n) (len x))
                         (bfr-p nil)))))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-butlast

(defthm bfr-listp$-of-butlast
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (bfr-listp$ (butlast x n) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-nthcdr

(defthm bfr-listp$-of-nthcdr
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (bfr-listp$ (nthcdr n x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-last

(defthm bfr-listp$-of-last
  (implies (bfr-listp$ (double-rewrite x) bfrstate)
           (bfr-listp$ (last x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-remove

(defthm bfr-listp$-of-remove
  (implies (bfr-listp$ x bfrstate)
           (bfr-listp$ (remove a x) bfrstate))
  :rule-classes ((:rewrite)))

Theorem: bfr-listp$-of-revappend

(defthm bfr-listp$-of-revappend
  (equal (bfr-listp$ (revappend x y) bfrstate)
         (and (bfr-listp$ (list-fix x) bfrstate)
              (bfr-listp$ y bfrstate)))
  :rule-classes ((:rewrite)))