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• Pseudo-event-landmark-listp

Pseudo-event-landmark-listp-basics

Basic theorems about pseudo-event-landmark-listp, generated by std::deflist.

Definitions and Theorems

Theorem: pseudo-event-landmark-listp-of-cons

(defthm pseudo-event-landmark-listp-of-cons
        (equal (pseudo-event-landmark-listp (cons a x))
               (and (pseudo-event-landmarkp a)
                    (pseudo-event-landmark-listp x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-cdr-when-pseudo-event-landmark-listp

(defthm
 pseudo-event-landmark-listp-of-cdr-when-pseudo-event-landmark-listp
 (implies (pseudo-event-landmark-listp (double-rewrite x))
          (pseudo-event-landmark-listp (cdr x)))
 :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-when-not-consp

(defthm pseudo-event-landmark-listp-when-not-consp
        (implies (not (consp x))
                 (equal (pseudo-event-landmark-listp x)
                        (not x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmarkp-of-car-when-pseudo-event-landmark-listp

(defthm
     pseudo-event-landmarkp-of-car-when-pseudo-event-landmark-listp
     (implies (pseudo-event-landmark-listp x)
              (iff (pseudo-event-landmarkp (car x))
                   (consp x)))
     :rule-classes ((:rewrite)))

Theorem: true-listp-when-pseudo-event-landmark-listp

(defthm true-listp-when-pseudo-event-landmark-listp
        (implies (pseudo-event-landmark-listp x)
                 (true-listp x))
        :rule-classes :compound-recognizer)

Theorem: pseudo-event-landmark-listp-of-list-fix

(defthm pseudo-event-landmark-listp-of-list-fix
        (implies (pseudo-event-landmark-listp x)
                 (pseudo-event-landmark-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-sfix

(defthm pseudo-event-landmark-listp-of-sfix
        (iff (pseudo-event-landmark-listp (set::sfix x))
             (or (pseudo-event-landmark-listp x)
                 (not (set::setp x))))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-insert

(defthm pseudo-event-landmark-listp-of-insert
        (iff (pseudo-event-landmark-listp (set::insert a x))
             (and (pseudo-event-landmark-listp (set::sfix x))
                  (pseudo-event-landmarkp a)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-delete

(defthm pseudo-event-landmark-listp-of-delete
        (implies (pseudo-event-landmark-listp x)
                 (pseudo-event-landmark-listp (set::delete k x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-mergesort

(defthm pseudo-event-landmark-listp-of-mergesort
        (iff (pseudo-event-landmark-listp (set::mergesort x))
             (pseudo-event-landmark-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-union

(defthm pseudo-event-landmark-listp-of-union
        (iff (pseudo-event-landmark-listp (set::union x y))
             (and (pseudo-event-landmark-listp (set::sfix x))
                  (pseudo-event-landmark-listp (set::sfix y))))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-intersect-1

(defthm pseudo-event-landmark-listp-of-intersect-1
        (implies (pseudo-event-landmark-listp x)
                 (pseudo-event-landmark-listp (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-intersect-2

(defthm pseudo-event-landmark-listp-of-intersect-2
        (implies (pseudo-event-landmark-listp y)
                 (pseudo-event-landmark-listp (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-difference

(defthm
     pseudo-event-landmark-listp-of-difference
     (implies (pseudo-event-landmark-listp x)
              (pseudo-event-landmark-listp (set::difference x y)))
     :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-duplicated-members

(defthm
     pseudo-event-landmark-listp-of-duplicated-members
     (implies (pseudo-event-landmark-listp x)
              (pseudo-event-landmark-listp (duplicated-members x)))
     :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-rev

(defthm pseudo-event-landmark-listp-of-rev
        (equal (pseudo-event-landmark-listp (rev x))
               (pseudo-event-landmark-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-append

(defthm pseudo-event-landmark-listp-of-append
        (equal (pseudo-event-landmark-listp (append a b))
               (and (pseudo-event-landmark-listp (list-fix a))
                    (pseudo-event-landmark-listp b)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-rcons

(defthm pseudo-event-landmark-listp-of-rcons
        (iff (pseudo-event-landmark-listp (rcons a x))
             (and (pseudo-event-landmarkp a)
                  (pseudo-event-landmark-listp (list-fix x))))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmarkp-when-member-equal-of-pseudo-event-landmark-listp

(defthm
 pseudo-event-landmarkp-when-member-equal-of-pseudo-event-landmark-listp
 (and (implies (and (member-equal a x)
                    (pseudo-event-landmark-listp x))
               (pseudo-event-landmarkp a))
      (implies (and (pseudo-event-landmark-listp x)
                    (member-equal a x))
               (pseudo-event-landmarkp a)))
 :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-when-subsetp-equal

(defthm pseudo-event-landmark-listp-when-subsetp-equal
        (and (implies (and (subsetp-equal x y)
                           (pseudo-event-landmark-listp y))
                      (equal (pseudo-event-landmark-listp x)
                             (true-listp x)))
             (implies (and (pseudo-event-landmark-listp y)
                           (subsetp-equal x y))
                      (equal (pseudo-event-landmark-listp x)
                             (true-listp x))))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-set-difference-equal

(defthm
  pseudo-event-landmark-listp-of-set-difference-equal
  (implies (pseudo-event-landmark-listp x)
           (pseudo-event-landmark-listp (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-intersection-equal-1

(defthm
    pseudo-event-landmark-listp-of-intersection-equal-1
    (implies (pseudo-event-landmark-listp (double-rewrite x))
             (pseudo-event-landmark-listp (intersection-equal x y)))
    :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-intersection-equal-2

(defthm
    pseudo-event-landmark-listp-of-intersection-equal-2
    (implies (pseudo-event-landmark-listp (double-rewrite y))
             (pseudo-event-landmark-listp (intersection-equal x y)))
    :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-union-equal

(defthm
     pseudo-event-landmark-listp-of-union-equal
     (equal (pseudo-event-landmark-listp (union-equal x y))
            (and (pseudo-event-landmark-listp (list-fix x))
                 (pseudo-event-landmark-listp (double-rewrite y))))
     :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-take

(defthm pseudo-event-landmark-listp-of-take
        (implies (pseudo-event-landmark-listp (double-rewrite x))
                 (iff (pseudo-event-landmark-listp (take n x))
                      (or (pseudo-event-landmarkp nil)
                          (<= (nfix n) (len x)))))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-repeat

(defthm pseudo-event-landmark-listp-of-repeat
        (iff (pseudo-event-landmark-listp (repeat n x))
             (or (pseudo-event-landmarkp x) (zp n)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmarkp-of-nth-when-pseudo-event-landmark-listp

(defthm
     pseudo-event-landmarkp-of-nth-when-pseudo-event-landmark-listp
     (implies (pseudo-event-landmark-listp x)
              (iff (pseudo-event-landmarkp (nth n x))
                   (< (nfix n) (len x))))
     :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-update-nth

(defthm
     pseudo-event-landmark-listp-of-update-nth
     (implies (pseudo-event-landmark-listp (double-rewrite x))
              (iff (pseudo-event-landmark-listp (update-nth n y x))
                   (and (pseudo-event-landmarkp y)
                        (or (<= (nfix n) (len x))
                            (pseudo-event-landmarkp nil)))))
     :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-butlast

(defthm pseudo-event-landmark-listp-of-butlast
        (implies (pseudo-event-landmark-listp (double-rewrite x))
                 (pseudo-event-landmark-listp (butlast x n)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-nthcdr

(defthm pseudo-event-landmark-listp-of-nthcdr
        (implies (pseudo-event-landmark-listp (double-rewrite x))
                 (pseudo-event-landmark-listp (nthcdr n x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-last

(defthm pseudo-event-landmark-listp-of-last
        (implies (pseudo-event-landmark-listp (double-rewrite x))
                 (pseudo-event-landmark-listp (last x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-remove

(defthm pseudo-event-landmark-listp-of-remove
        (implies (pseudo-event-landmark-listp x)
                 (pseudo-event-landmark-listp (remove a x)))
        :rule-classes ((:rewrite)))

Theorem: pseudo-event-landmark-listp-of-revappend

(defthm pseudo-event-landmark-listp-of-revappend
        (equal (pseudo-event-landmark-listp (revappend x y))
               (and (pseudo-event-landmark-listp (list-fix x))
                    (pseudo-event-landmark-listp y)))
        :rule-classes ((:rewrite)))