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Integer-range-listp

Integer-range-listp-basics

Basic theorems about integer-range-listp, generated by std::deflist.

Definitions and Theorems

Theorem: integer-range-listp-of-cons

(defthm integer-range-listp-of-cons
  (equal (integer-range-listp lower upper (cons a x))
         (and (integer-range-p lower upper a)
              (integer-range-listp lower upper x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-cdr-when-integer-range-listp

(defthm integer-range-listp-of-cdr-when-integer-range-listp
  (implies (integer-range-listp lower upper (double-rewrite x))
           (integer-range-listp lower upper (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-when-not-consp

(defthm integer-range-listp-when-not-consp
  (implies (not (consp x))
           (equal (integer-range-listp lower upper x)
                  (not x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-p-of-car-when-integer-range-listp

(defthm integer-range-p-of-car-when-integer-range-listp
  (implies (integer-range-listp lower upper x)
           (iff (integer-range-p lower upper (car x))
                (consp x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-integer-range-listp

(defthm true-listp-when-integer-range-listp
  (implies (integer-range-listp lower upper x)
           (true-listp x))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-list-fix

(defthm integer-range-listp-of-list-fix
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-sfix

(defthm integer-range-listp-of-sfix
  (iff (integer-range-listp lower upper (set::sfix x))
       (or (integer-range-listp lower upper x)
           (not (set::setp x))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-insert

(defthm integer-range-listp-of-insert
  (iff (integer-range-listp lower upper (set::insert a x))
       (and (integer-range-listp lower upper (set::sfix x))
            (integer-range-p lower upper a)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-delete

(defthm integer-range-listp-of-delete
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (set::delete k x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-mergesort

(defthm integer-range-listp-of-mergesort
  (iff (integer-range-listp lower upper (set::mergesort x))
       (integer-range-listp lower upper (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-union

(defthm integer-range-listp-of-union
  (iff (integer-range-listp lower upper (set::union x y))
       (and (integer-range-listp lower upper (set::sfix x))
            (integer-range-listp lower upper (set::sfix y))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-intersect-1

(defthm integer-range-listp-of-intersect-1
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-intersect-2

(defthm integer-range-listp-of-intersect-2
  (implies (integer-range-listp lower upper y)
           (integer-range-listp lower upper (set::intersect x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-difference

(defthm integer-range-listp-of-difference
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (set::difference x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-duplicated-members

(defthm integer-range-listp-of-duplicated-members
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (duplicated-members x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-rev

(defthm integer-range-listp-of-rev
  (equal (integer-range-listp lower upper (rev x))
         (integer-range-listp lower upper (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-append

(defthm integer-range-listp-of-append
  (equal (integer-range-listp lower upper (append a b))
         (and (integer-range-listp lower upper (list-fix a))
              (integer-range-listp lower upper b)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-rcons

(defthm integer-range-listp-of-rcons
  (iff (integer-range-listp lower upper (rcons a x))
       (and (integer-range-p lower upper a)
            (integer-range-listp lower upper (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-p-when-member-equal-of-integer-range-listp

(defthm integer-range-p-when-member-equal-of-integer-range-listp
  (and (implies (and (member-equal a x)
                     (integer-range-listp lower upper x))
                (integer-range-p lower upper a))
       (implies (and (integer-range-listp lower upper x)
                     (member-equal a x))
                (integer-range-p lower upper a)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-when-subsetp-equal

(defthm integer-range-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal x y)
                     (integer-range-listp lower upper y))
                (equal (integer-range-listp lower upper x)
                       (true-listp x)))
       (implies (and (integer-range-listp lower upper y)
                     (subsetp-equal x y))
                (equal (integer-range-listp lower upper x)
                       (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-set-difference-equal

(defthm integer-range-listp-of-set-difference-equal
  (implies
       (integer-range-listp lower upper x)
       (integer-range-listp lower upper (set-difference-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-intersection-equal-1

(defthm integer-range-listp-of-intersection-equal-1
  (implies
       (integer-range-listp lower upper (double-rewrite x))
       (integer-range-listp lower upper (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-intersection-equal-2

(defthm integer-range-listp-of-intersection-equal-2
  (implies
       (integer-range-listp lower upper (double-rewrite y))
       (integer-range-listp lower upper (intersection-equal x y)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-union-equal

(defthm integer-range-listp-of-union-equal
  (equal (integer-range-listp lower upper (union-equal x y))
         (and (integer-range-listp lower upper (list-fix x))
              (integer-range-listp lower upper (double-rewrite y))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-take

(defthm integer-range-listp-of-take
  (implies (integer-range-listp lower upper (double-rewrite x))
           (iff (integer-range-listp lower upper (take n x))
                (or (integer-range-p lower upper nil)
                    (<= (nfix n) (len x)))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-repeat

(defthm integer-range-listp-of-repeat
  (iff (integer-range-listp lower upper (repeat n x))
       (or (integer-range-p lower upper x)
           (zp n)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-p-of-nth-when-integer-range-listp

(defthm integer-range-p-of-nth-when-integer-range-listp
  (implies (integer-range-listp lower upper x)
           (iff (integer-range-p lower upper (nth n x))
                (< (nfix n) (len x))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-update-nth

(defthm integer-range-listp-of-update-nth
  (implies (integer-range-listp lower upper (double-rewrite x))
           (iff (integer-range-listp lower upper (update-nth n y x))
                (and (integer-range-p lower upper y)
                     (or (<= (nfix n) (len x))
                         (integer-range-p lower upper nil)))))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-butlast

(defthm integer-range-listp-of-butlast
  (implies (integer-range-listp lower upper (double-rewrite x))
           (integer-range-listp lower upper (butlast x n)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-nthcdr

(defthm integer-range-listp-of-nthcdr
  (implies (integer-range-listp lower upper (double-rewrite x))
           (integer-range-listp lower upper (nthcdr n x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-last

(defthm integer-range-listp-of-last
  (implies (integer-range-listp lower upper (double-rewrite x))
           (integer-range-listp lower upper (last x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-remove

(defthm integer-range-listp-of-remove
  (implies (integer-range-listp lower upper x)
           (integer-range-listp lower upper (remove a x)))
  :rule-classes ((:rewrite)))

Theorem: integer-range-listp-of-revappend

(defthm integer-range-listp-of-revappend
  (equal (integer-range-listp lower upper (revappend x y))
         (and (integer-range-listp lower upper (list-fix x))
              (integer-range-listp lower upper y)))
  :rule-classes ((:rewrite)))