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• Repetition-list-wfp

Repetition-list-wfp-basics

Basic theorems about repetition-list-wfp, generated by std::deflist.

Definitions and Theorems

Theorem: repetition-list-wfp-of-cons

(defthm repetition-list-wfp-of-cons
  (equal (repetition-list-wfp (cons acl2::a acl2::x))
         (and (repetition-wfp acl2::a)
              (repetition-list-wfp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-cdr-when-repetition-list-wfp

(defthm repetition-list-wfp-of-cdr-when-repetition-list-wfp
  (implies (repetition-list-wfp (double-rewrite acl2::x))
           (repetition-list-wfp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-when-not-consp

(defthm repetition-list-wfp-when-not-consp
  (implies (not (consp acl2::x))
           (repetition-list-wfp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: repetition-wfp-of-car-when-repetition-list-wfp

(defthm repetition-wfp-of-car-when-repetition-list-wfp
  (implies (repetition-list-wfp acl2::x)
           (iff (repetition-wfp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-append

(defthm repetition-list-wfp-of-append
  (equal (repetition-list-wfp (append acl2::a acl2::b))
         (and (repetition-list-wfp acl2::a)
              (repetition-list-wfp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-list-fix

(defthm repetition-list-wfp-of-list-fix
  (equal (repetition-list-wfp (list-fix acl2::x))
         (repetition-list-wfp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-sfix

(defthm repetition-list-wfp-of-sfix
  (iff (repetition-list-wfp (sfix acl2::x))
       (or (repetition-list-wfp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-insert

(defthm repetition-list-wfp-of-insert
  (iff (repetition-list-wfp (insert acl2::a acl2::x))
       (and (repetition-list-wfp (sfix acl2::x))
            (repetition-wfp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-delete

(defthm repetition-list-wfp-of-delete
  (implies (repetition-list-wfp acl2::x)
           (repetition-list-wfp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-mergesort

(defthm repetition-list-wfp-of-mergesort
  (iff (repetition-list-wfp (mergesort acl2::x))
       (repetition-list-wfp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-union

(defthm repetition-list-wfp-of-union
  (iff (repetition-list-wfp (union acl2::x acl2::y))
       (and (repetition-list-wfp (sfix acl2::x))
            (repetition-list-wfp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-intersect-1

(defthm repetition-list-wfp-of-intersect-1
  (implies (repetition-list-wfp acl2::x)
           (repetition-list-wfp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-intersect-2

(defthm repetition-list-wfp-of-intersect-2
  (implies (repetition-list-wfp acl2::y)
           (repetition-list-wfp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-difference

(defthm repetition-list-wfp-of-difference
  (implies (repetition-list-wfp acl2::x)
           (repetition-list-wfp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-duplicated-members

(defthm repetition-list-wfp-of-duplicated-members
  (implies (repetition-list-wfp acl2::x)
           (repetition-list-wfp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-rev

(defthm repetition-list-wfp-of-rev
  (equal (repetition-list-wfp (rev acl2::x))
         (repetition-list-wfp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-rcons

(defthm repetition-list-wfp-of-rcons
  (iff (repetition-list-wfp (rcons acl2::a acl2::x))
       (and (repetition-wfp acl2::a)
            (repetition-list-wfp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: repetition-wfp-when-member-equal-of-repetition-list-wfp

(defthm repetition-wfp-when-member-equal-of-repetition-list-wfp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (repetition-list-wfp acl2::x))
                (repetition-wfp acl2::a))
       (implies (and (repetition-list-wfp acl2::x)
                     (member-equal acl2::a acl2::x))
                (repetition-wfp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-when-subsetp-equal

(defthm repetition-list-wfp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (repetition-list-wfp acl2::y))
                (repetition-list-wfp acl2::x))
       (implies (and (repetition-list-wfp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (repetition-list-wfp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-set-equiv-congruence

(defthm repetition-list-wfp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (repetition-list-wfp acl2::x)
                  (repetition-list-wfp acl2::y)))
  :rule-classes :congruence)

Theorem: repetition-list-wfp-of-set-difference-equal

(defthm repetition-list-wfp-of-set-difference-equal
  (implies
       (repetition-list-wfp acl2::x)
       (repetition-list-wfp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-intersection-equal-1

(defthm repetition-list-wfp-of-intersection-equal-1
  (implies
       (repetition-list-wfp (double-rewrite acl2::x))
       (repetition-list-wfp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-intersection-equal-2

(defthm repetition-list-wfp-of-intersection-equal-2
  (implies
       (repetition-list-wfp (double-rewrite acl2::y))
       (repetition-list-wfp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-union-equal

(defthm repetition-list-wfp-of-union-equal
  (equal (repetition-list-wfp (union-equal acl2::x acl2::y))
         (and (repetition-list-wfp (list-fix acl2::x))
              (repetition-list-wfp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-take

(defthm repetition-list-wfp-of-take
  (implies (repetition-list-wfp (double-rewrite acl2::x))
           (iff (repetition-list-wfp (take acl2::n acl2::x))
                (or (repetition-wfp nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-repeat

(defthm repetition-list-wfp-of-repeat
  (iff (repetition-list-wfp (repeat acl2::n acl2::x))
       (or (repetition-wfp acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: repetition-wfp-of-nth-when-repetition-list-wfp

(defthm repetition-wfp-of-nth-when-repetition-list-wfp
  (implies (repetition-list-wfp acl2::x)
           (iff (repetition-wfp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-update-nth

(defthm repetition-list-wfp-of-update-nth
 (implies
     (repetition-list-wfp (double-rewrite acl2::x))
     (iff (repetition-list-wfp (update-nth acl2::n acl2::y acl2::x))
          (and (repetition-wfp acl2::y)
               (or (<= (nfix acl2::n) (len acl2::x))
                   (repetition-wfp nil)))))
 :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-butlast

(defthm repetition-list-wfp-of-butlast
  (implies (repetition-list-wfp (double-rewrite acl2::x))
           (repetition-list-wfp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-nthcdr

(defthm repetition-list-wfp-of-nthcdr
  (implies (repetition-list-wfp (double-rewrite acl2::x))
           (repetition-list-wfp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-last

(defthm repetition-list-wfp-of-last
  (implies (repetition-list-wfp (double-rewrite acl2::x))
           (repetition-list-wfp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-remove

(defthm repetition-list-wfp-of-remove
  (implies (repetition-list-wfp acl2::x)
           (repetition-list-wfp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetition-list-wfp-of-revappend

(defthm repetition-list-wfp-of-revappend
  (equal (repetition-list-wfp (revappend acl2::x acl2::y))
         (and (repetition-list-wfp (list-fix acl2::x))
              (repetition-list-wfp acl2::y)))
  :rule-classes ((:rewrite)))