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• Good-pseudo-term-listp

Good-pseudo-term-listp-basics

Basic theorems about good-pseudo-term-listp, generated by std::deflist.

Definitions and Theorems

Theorem: good-pseudo-term-listp-of-cons

(defthm good-pseudo-term-listp-of-cons
  (equal (good-pseudo-term-listp (cons a x))
         (and (good-pseudo-termp a)
              (good-pseudo-term-listp x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-cdr-when-good-pseudo-term-listp

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

Theorem: good-pseudo-term-listp-when-not-consp

(defthm good-pseudo-term-listp-when-not-consp
  (implies (not (consp x))
           (good-pseudo-term-listp x))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-termp-of-car-when-good-pseudo-term-listp

(defthm good-pseudo-termp-of-car-when-good-pseudo-term-listp
  (implies (good-pseudo-term-listp x)
           (iff (good-pseudo-termp (car x))
                (or (consp x) (good-pseudo-termp nil))))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-append

(defthm good-pseudo-term-listp-of-append
  (equal (good-pseudo-term-listp (append a b))
         (and (good-pseudo-term-listp a)
              (good-pseudo-term-listp b)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-list-fix

(defthm good-pseudo-term-listp-of-list-fix
  (equal (good-pseudo-term-listp (list-fix x))
         (good-pseudo-term-listp x))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-sfix

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

Theorem: good-pseudo-term-listp-of-insert

(defthm good-pseudo-term-listp-of-insert
  (iff (good-pseudo-term-listp (set::insert a x))
       (and (good-pseudo-term-listp (set::sfix x))
            (good-pseudo-termp a)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-delete

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

Theorem: good-pseudo-term-listp-of-mergesort

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

Theorem: good-pseudo-term-listp-of-union

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

Theorem: good-pseudo-term-listp-of-intersect-1

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

Theorem: good-pseudo-term-listp-of-intersect-2

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

Theorem: good-pseudo-term-listp-of-difference

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

Theorem: good-pseudo-term-listp-of-duplicated-members

(defthm good-pseudo-term-listp-of-duplicated-members
  (implies (good-pseudo-term-listp x)
           (good-pseudo-term-listp (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-rev

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

Theorem: good-pseudo-term-listp-of-rcons

(defthm good-pseudo-term-listp-of-rcons
  (iff (good-pseudo-term-listp (rcons a x))
       (and (good-pseudo-termp a)
            (good-pseudo-term-listp (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-termp-when-member-equal-of-good-pseudo-term-listp

(defthm
      good-pseudo-termp-when-member-equal-of-good-pseudo-term-listp
  (and (implies (and (member-equal a x)
                     (good-pseudo-term-listp x))
                (good-pseudo-termp a))
       (implies (and (good-pseudo-term-listp x)
                     (member-equal a x))
                (good-pseudo-termp a)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-when-subsetp-equal

(defthm good-pseudo-term-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (good-pseudo-term-listp y))
                (good-pseudo-term-listp x))
       (implies (and (good-pseudo-term-listp y)
                     (subsetp-equal x y))
                (good-pseudo-term-listp x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-set-equiv-congruence

(defthm good-pseudo-term-listp-set-equiv-congruence
  (implies (set-equiv x y)
           (equal (good-pseudo-term-listp x)
                  (good-pseudo-term-listp y)))
  :rule-classes :congruence)

Theorem: good-pseudo-term-listp-of-set-difference-equal

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

Theorem: good-pseudo-term-listp-of-intersection-equal-1

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

Theorem: good-pseudo-term-listp-of-intersection-equal-2

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

Theorem: good-pseudo-term-listp-of-union-equal

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

Theorem: good-pseudo-term-listp-of-take

(defthm good-pseudo-term-listp-of-take
  (implies (good-pseudo-term-listp (double-rewrite x))
           (iff (good-pseudo-term-listp (take n x))
                (or (good-pseudo-termp nil)
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-repeat

(defthm good-pseudo-term-listp-of-repeat
  (iff (good-pseudo-term-listp (repeat n x))
       (or (good-pseudo-termp x) (zp n)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-termp-of-nth-when-good-pseudo-term-listp

(defthm good-pseudo-termp-of-nth-when-good-pseudo-term-listp
  (implies (and (good-pseudo-term-listp x)
                (< (nfix n) (len x)))
           (good-pseudo-termp (nth n x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-update-nth

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

Theorem: good-pseudo-term-listp-of-butlast

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

Theorem: good-pseudo-term-listp-of-nthcdr

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

Theorem: good-pseudo-term-listp-of-last

(defthm good-pseudo-term-listp-of-last
  (implies (good-pseudo-term-listp (double-rewrite x))
           (good-pseudo-term-listp (last x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-remove

(defthm good-pseudo-term-listp-of-remove
  (implies (good-pseudo-term-listp x)
           (good-pseudo-term-listp (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: good-pseudo-term-listp-of-revappend

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