• Top
  • Documentation
  • Books
  • Recursion-and-induction
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
    • Std/lists
    • Std/alists
    • Obags
    • Std/util
    • Std/strings
    • Std/io
    • Std/osets
    • Std/system
    • Std/basic
    • Std/typed-lists
      • Std/typed-lists/character-listp
      • Std/typed-lists/symbol-listp
      • Std/typed-lists/boolean-listp
      • Std/typed-lists/string-listp
      • Std/typed-lists/eqlable-listp
      • Theorems-about-true-list-lists
      • Std/typed-lists/atom-listp
      • Unsigned-byte-listp
      • Cons-listp
      • Cons-list-listp
      • Signed-byte-listp
      • String-or-symbol-listp
    • Std/bitsets
    • Std/testing
    • Std/typed-alists
    • Std/stobjs
    • Std-extensions
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Testing-utilities
  • Math

• Std/typed-lists
• Atom-listp

Std/typed-lists/atom-listp

Lemmas about atom-listp available in the std/typed-lists library.

Most of these are generated automatically with std::deflist.

Definitions and Theorems

Theorem: atom-listp-of-cons

(defthm atom-listp-of-cons
        (equal (atom-listp (cons a x))
               (and (not (consp a)) (atom-listp x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-cdr-when-atom-listp

(defthm atom-listp-of-cdr-when-atom-listp
        (implies (atom-listp (double-rewrite x))
                 (atom-listp (cdr x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-when-not-consp

(defthm atom-listp-when-not-consp
        (implies (not (consp x))
                 (equal (atom-listp x) (not x)))
        :rule-classes ((:rewrite)))

Theorem: consp-of-car-when-atom-listp

(defthm consp-of-car-when-atom-listp
        (implies (atom-listp x)
                 (not (consp (car x))))
        :rule-classes ((:rewrite :backchain-limit-lst 0)))

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

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

Theorem: atom-listp-of-list-fix

(defthm atom-listp-of-list-fix
        (implies (atom-listp x)
                 (atom-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-sfix

(defthm atom-listp-of-sfix
        (iff (atom-listp (set::sfix x))
             (or (atom-listp x) (not (set::setp x))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-insert

(defthm atom-listp-of-insert
        (iff (atom-listp (set::insert a x))
             (and (atom-listp (set::sfix x))
                  (not (consp a))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-delete

(defthm atom-listp-of-delete
        (implies (atom-listp x)
                 (atom-listp (set::delete k x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-mergesort

(defthm atom-listp-of-mergesort
        (iff (atom-listp (set::mergesort x))
             (atom-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-union

(defthm atom-listp-of-union
        (iff (atom-listp (set::union x y))
             (and (atom-listp (set::sfix x))
                  (atom-listp (set::sfix y))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-intersect-1

(defthm atom-listp-of-intersect-1
        (implies (atom-listp x)
                 (atom-listp (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-intersect-2

(defthm atom-listp-of-intersect-2
        (implies (atom-listp y)
                 (atom-listp (set::intersect x y)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-difference

(defthm atom-listp-of-difference
        (implies (atom-listp x)
                 (atom-listp (set::difference x y)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-duplicated-members

(defthm atom-listp-of-duplicated-members
        (implies (atom-listp x)
                 (atom-listp (duplicated-members x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-rev

(defthm atom-listp-of-rev
        (equal (atom-listp (rev x))
               (atom-listp (list-fix x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-append

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

Theorem: atom-listp-of-rcons

(defthm atom-listp-of-rcons
        (iff (atom-listp (rcons a x))
             (and (not (consp a))
                  (atom-listp (list-fix x))))
        :rule-classes ((:rewrite)))

Theorem: consp-when-member-equal-of-atom-listp

(defthm consp-when-member-equal-of-atom-listp
        (and (implies (and (member-equal a x) (atom-listp x))
                      (not (consp a)))
             (implies (and (atom-listp x) (member-equal a x))
                      (not (consp a))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-when-subsetp-equal

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

Theorem: atom-listp-of-set-difference-equal

(defthm atom-listp-of-set-difference-equal
        (implies (atom-listp x)
                 (atom-listp (set-difference-equal x y)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-intersection-equal-1

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

Theorem: atom-listp-of-intersection-equal-2

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

Theorem: atom-listp-of-union-equal

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

Theorem: atom-listp-of-take

(defthm atom-listp-of-take
        (implies (atom-listp (double-rewrite x))
                 (iff (atom-listp (take n x))
                      (or (not (consp nil))
                          (<= (nfix n) (len x)))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-repeat

(defthm atom-listp-of-repeat
        (iff (atom-listp (repeat n x))
             (or (not (consp x)) (zp n)))
        :rule-classes ((:rewrite)))

Theorem: consp-of-nth-when-atom-listp

(defthm consp-of-nth-when-atom-listp
        (implies (atom-listp x)
                 (not (consp (nth n x))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-update-nth

(defthm atom-listp-of-update-nth
        (implies (atom-listp (double-rewrite x))
                 (iff (atom-listp (update-nth n y x))
                      (and (not (consp y))
                           (or (<= (nfix n) (len x))
                               (not (consp nil))))))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-butlast

(defthm atom-listp-of-butlast
        (implies (atom-listp (double-rewrite x))
                 (atom-listp (butlast x n)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-nthcdr

(defthm atom-listp-of-nthcdr
        (implies (atom-listp (double-rewrite x))
                 (atom-listp (nthcdr n x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-last

(defthm atom-listp-of-last
        (implies (atom-listp (double-rewrite x))
                 (atom-listp (last x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-remove

(defthm atom-listp-of-remove
        (implies (atom-listp x)
                 (atom-listp (remove a x)))
        :rule-classes ((:rewrite)))

Theorem: atom-listp-of-revappend

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

Theorem: atom-listp-of-remove-equal

(defthm atom-listp-of-remove-equal
        (implies (atom-listp x)
                 (atom-listp (remove-equal a x))))

Theorem: atom-listp-of-make-list-ac

(defthm atom-listp-of-make-list-ac
        (equal (atom-listp (make-list-ac n x ac))
               (and (atom-listp ac)
                    (or (atom x) (zp n)))))