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Alternationp

Alternationp-basics

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

Definitions and Theorems

Theorem: alternationp-of-cons

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

Theorem: alternationp-of-cdr-when-alternationp

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

Theorem: alternationp-when-not-consp

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

Theorem: concatenationp-of-car-when-alternationp

(defthm concatenationp-of-car-when-alternationp
  (implies (alternationp acl2::x)
           (concatenationp (car acl2::x)))
  :rule-classes ((:rewrite)))

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

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

Theorem: alternationp-of-list-fix

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

Theorem: alternationp-of-rev

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

Theorem: alternationp-of-append

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

Theorem: alternationp-of-last

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

Theorem: concatenationp-of-nth-when-alternationp

(defthm concatenationp-of-nth-when-alternationp
  (implies (alternationp acl2::x)
           (concatenationp (nth acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: alternationp-of-nthcdr

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

Theorem: alternationp-of-remove

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

Theorem: alternationp-of-repeat

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

Theorem: alternationp-of-revappend

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

Theorem: alternationp-of-rcons

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

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

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

Theorem: alternationp-when-subsetp-equal

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

Theorem: alternationp-of-set-difference-equal

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

Theorem: alternationp-of-intersection-equal-1

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

Theorem: alternationp-of-intersection-equal-2

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

Theorem: alternationp-of-union-equal

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

Theorem: alternationp-of-take

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

Theorem: alternationp-of-update-nth

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