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• Concatenationp

Concatenationp-basics

Basic theorems about concatenationp, generated by std::deflist.

Definitions and Theorems

Theorem: concatenationp-of-cons

(defthm concatenationp-of-cons
  (equal (concatenationp (cons acl2::a acl2::x))
         (and (repetitionp acl2::a)
              (concatenationp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-cdr-when-concatenationp

(defthm concatenationp-of-cdr-when-concatenationp
  (implies (concatenationp (double-rewrite acl2::x))
           (concatenationp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-when-not-consp

(defthm concatenationp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (concatenationp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetitionp-of-car-when-concatenationp

(defthm repetitionp-of-car-when-concatenationp
  (implies (concatenationp acl2::x)
           (iff (repetitionp (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-concatenationp-compound-recognizer

(defthm true-listp-when-concatenationp-compound-recognizer
  (implies (concatenationp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: concatenationp-of-list-fix

(defthm concatenationp-of-list-fix
  (implies (concatenationp acl2::x)
           (concatenationp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-rev

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

Theorem: concatenationp-of-append

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

Theorem: concatenationp-of-last

(defthm concatenationp-of-last
  (implies (concatenationp (double-rewrite acl2::x))
           (concatenationp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: repetitionp-of-nth-when-concatenationp

(defthm repetitionp-of-nth-when-concatenationp
  (implies (concatenationp acl2::x)
           (iff (repetitionp (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-nthcdr

(defthm concatenationp-of-nthcdr
  (implies (concatenationp (double-rewrite acl2::x))
           (concatenationp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-remove

(defthm concatenationp-of-remove
  (implies (concatenationp acl2::x)
           (concatenationp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-repeat

(defthm concatenationp-of-repeat
  (iff (concatenationp (repeat acl2::n acl2::x))
       (or (repetitionp acl2::x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-revappend

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

Theorem: concatenationp-of-rcons

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

Theorem: repetitionp-when-member-equal-of-concatenationp

(defthm repetitionp-when-member-equal-of-concatenationp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (concatenationp acl2::x))
                (repetitionp acl2::a))
       (implies (and (concatenationp acl2::x)
                     (member-equal acl2::a acl2::x))
                (repetitionp acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-when-subsetp-equal

(defthm concatenationp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (concatenationp acl2::y))
                (equal (concatenationp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (concatenationp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (concatenationp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: concatenationp-of-set-difference-equal

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

Theorem: concatenationp-of-intersection-equal-1

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

Theorem: concatenationp-of-intersection-equal-2

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

Theorem: concatenationp-of-union-equal

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

Theorem: concatenationp-of-take

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

Theorem: concatenationp-of-update-nth

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