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• Expression-const/var-listp

Expression-const/var-listp-basics

Basic theorems about expression-const/var-listp, generated by std::deflist.

Definitions and Theorems

Theorem: expression-const/var-listp-of-cons

(defthm expression-const/var-listp-of-cons
  (equal (expression-const/var-listp (cons acl2::a acl2::x))
         (and (or (expression-case acl2::a :const)
                  (expression-case acl2::a :var))
              (expression-const/var-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-cdr-when-expression-const/var-listp

(defthm
  expression-const/var-listp-of-cdr-when-expression-const/var-listp
  (implies (expression-const/var-listp (double-rewrite acl2::x))
           (expression-const/var-listp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-when-not-consp

(defthm expression-const/var-listp-when-not-consp
  (implies (not (consp acl2::x))
           (expression-const/var-listp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-append

(defthm expression-const/var-listp-of-append
  (equal (expression-const/var-listp (append acl2::a acl2::b))
         (and (expression-const/var-listp acl2::a)
              (expression-const/var-listp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-list-fix

(defthm expression-const/var-listp-of-list-fix
  (equal (expression-const/var-listp (list-fix acl2::x))
         (expression-const/var-listp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-sfix

(defthm expression-const/var-listp-of-sfix
  (iff (expression-const/var-listp (sfix acl2::x))
       (or (expression-const/var-listp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-insert

(defthm expression-const/var-listp-of-insert
  (iff (expression-const/var-listp (insert acl2::a acl2::x))
       (and (expression-const/var-listp (sfix acl2::x))
            (or (expression-case acl2::a :const)
                (expression-case acl2::a :var))))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-delete

(defthm expression-const/var-listp-of-delete
  (implies (expression-const/var-listp acl2::x)
           (expression-const/var-listp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-mergesort

(defthm expression-const/var-listp-of-mergesort
  (iff (expression-const/var-listp (mergesort acl2::x))
       (expression-const/var-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-union

(defthm expression-const/var-listp-of-union
  (iff (expression-const/var-listp (union acl2::x acl2::y))
       (and (expression-const/var-listp (sfix acl2::x))
            (expression-const/var-listp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-intersect-1

(defthm expression-const/var-listp-of-intersect-1
  (implies (expression-const/var-listp acl2::x)
           (expression-const/var-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-intersect-2

(defthm expression-const/var-listp-of-intersect-2
  (implies (expression-const/var-listp acl2::y)
           (expression-const/var-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-difference

(defthm expression-const/var-listp-of-difference
 (implies (expression-const/var-listp acl2::x)
          (expression-const/var-listp (difference acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-duplicated-members

(defthm expression-const/var-listp-of-duplicated-members
 (implies (expression-const/var-listp acl2::x)
          (expression-const/var-listp (duplicated-members acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-rev

(defthm expression-const/var-listp-of-rev
  (equal (expression-const/var-listp (rev acl2::x))
         (expression-const/var-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-rcons

(defthm expression-const/var-listp-of-rcons
  (iff (expression-const/var-listp (rcons acl2::a acl2::x))
       (and (or (expression-case acl2::a :const)
                (expression-case acl2::a :var))
            (expression-const/var-listp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-when-subsetp-equal

(defthm expression-const/var-listp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (expression-const/var-listp acl2::y))
                (expression-const/var-listp acl2::x))
       (implies (and (expression-const/var-listp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (expression-const/var-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-set-equiv-congruence

(defthm expression-const/var-listp-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (expression-const/var-listp acl2::x)
                  (expression-const/var-listp acl2::y)))
  :rule-classes :congruence)

Theorem: expression-const/var-listp-of-set-difference-equal

(defthm expression-const/var-listp-of-set-difference-equal
  (implies (expression-const/var-listp acl2::x)
           (expression-const/var-listp
                (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-intersection-equal-1

(defthm expression-const/var-listp-of-intersection-equal-1
 (implies
  (expression-const/var-listp (double-rewrite acl2::x))
  (expression-const/var-listp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-intersection-equal-2

(defthm expression-const/var-listp-of-intersection-equal-2
 (implies
  (expression-const/var-listp (double-rewrite acl2::y))
  (expression-const/var-listp (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-union-equal

(defthm expression-const/var-listp-of-union-equal
 (equal (expression-const/var-listp (union-equal acl2::x acl2::y))
        (and (expression-const/var-listp (list-fix acl2::x))
             (expression-const/var-listp (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-take

(defthm expression-const/var-listp-of-take
  (implies (expression-const/var-listp (double-rewrite acl2::x))
           (iff (expression-const/var-listp (take acl2::n acl2::x))
                (or (or (expression-case nil :const)
                        (expression-case nil :var))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-repeat

(defthm expression-const/var-listp-of-repeat
  (iff (expression-const/var-listp (repeat acl2::n acl2::x))
       (or (or (expression-case acl2::x :const)
               (expression-case acl2::x :var))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-update-nth

(defthm expression-const/var-listp-of-update-nth
 (implies
  (expression-const/var-listp (double-rewrite acl2::x))
  (iff
   (expression-const/var-listp (update-nth acl2::n acl2::y acl2::x))
   (and (or (expression-case acl2::y :const)
            (expression-case acl2::y :var))
        (or (<= (nfix acl2::n) (len acl2::x))
            (or (expression-case nil :const)
                (expression-case nil :var))))))
 :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-butlast

(defthm expression-const/var-listp-of-butlast
  (implies (expression-const/var-listp (double-rewrite acl2::x))
           (expression-const/var-listp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-nthcdr

(defthm expression-const/var-listp-of-nthcdr
  (implies (expression-const/var-listp (double-rewrite acl2::x))
           (expression-const/var-listp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-last

(defthm expression-const/var-listp-of-last
  (implies (expression-const/var-listp (double-rewrite acl2::x))
           (expression-const/var-listp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-remove

(defthm expression-const/var-listp-of-remove
  (implies (expression-const/var-listp acl2::x)
           (expression-const/var-listp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: expression-const/var-listp-of-revappend

(defthm expression-const/var-listp-of-revappend
  (equal (expression-const/var-listp (revappend acl2::x acl2::y))
         (and (expression-const/var-listp (list-fix acl2::x))
              (expression-const/var-listp acl2::y)))
  :rule-classes ((:rewrite)))