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• Std/lists/abstract

Element-list-p

Generic typed list recognizer function.

Definitions and Theorems

Function: element-list-p

(defun element-list-p (x)
  (if (atom x)
      (element-list-final-cdr-p x)
    (and (element-p (car x))
         (element-list-p (cdr x)))))

Theorem: element-list-p-of-cons

(defthm element-list-p-of-cons
  (equal (element-list-p (cons a x))
         (and (element-p a) (element-list-p x)))
  :rule-classes :rewrite)

Theorem: element-list-p-of-cdr-when-element-list-p

(defthm element-list-p-of-cdr-when-element-list-p
  (implies (element-list-p (double-rewrite x))
           (element-list-p (cdr x)))
  :rule-classes :rewrite)

Theorem: element-list-p-when-not-consp-non-true-list

(defthm element-list-p-when-not-consp-non-true-list
  (implies (and (element-list-final-cdr-p t)
                (not (consp x)))
           (element-list-p x))
  :rule-classes :rewrite)

Theorem: element-list-p-when-not-consp-true-list

(defthm element-list-p-when-not-consp-true-list
  (implies (and (not (element-list-final-cdr-p t))
                (not (consp x)))
           (equal (element-list-p x) (not x)))
  :rule-classes :rewrite)

Theorem: element-p-of-car-when-element-list-p-when-element-p-nil

(defthm element-p-of-car-when-element-list-p-when-element-p-nil
  (implies (and (element-p nil) (element-list-p x))
           (element-p (car x)))
  :rule-classes :rewrite)

Theorem: element-p-of-car-when-element-list-p-when-not-element-p-nil-and-not-negated

(defthm
 element-p-of-car-when-element-list-p-when-not-element-p-nil-and-not-negated
 (implies (and (not (element-p nil))
               (element-list-p x))
          (iff (element-p (car x)) (consp x)))
 :rule-classes :rewrite)

Theorem: element-p-of-car-when-element-list-p-when-not-element-p-nil-and-negated

(defthm
 element-p-of-car-when-element-list-p-when-not-element-p-nil-and-negated
 (implies (and (not (element-p nil))
               (element-list-p x))
          (iff (non-element-p (car x))
               (not (consp x))))
 :rule-classes :rewrite)

Theorem: element-p-of-car-when-element-list-p-when-unknown-nil

(defthm element-p-of-car-when-element-list-p-when-unknown-nil
  (implies (element-list-p x)
           (iff (element-p (car x))
                (or (consp x) (element-p nil))))
  :rule-classes :rewrite)

Theorem: element-p-of-car-when-element-list-p-when-unknown-nil-negated

(defthm
      element-p-of-car-when-element-list-p-when-unknown-nil-negated
  (implies (element-list-p x)
           (iff (non-element-p (car x))
                (and (not (consp x))
                     (non-element-p nil))))
  :rule-classes :rewrite)

Theorem: true-listp-when-element-list-p-rewrite

(defthm true-listp-when-element-list-p-rewrite
  (implies (and (element-list-p x)
                (not (element-list-final-cdr-p t)))
           (true-listp x))
  :rule-classes :rewrite)

Theorem: true-listp-when-element-list-p-compound-recognizer

(defthm true-listp-when-element-list-p-compound-recognizer
  (implies (and (element-list-p x)
                (not (element-list-final-cdr-p t)))
           (true-listp x))
  :rule-classes nil)

Theorem: element-list-p-of-append-non-true-list

(defthm element-list-p-of-append-non-true-list
  (implies (element-list-final-cdr-p t)
           (equal (element-list-p (append a b))
                  (and (element-list-p a)
                       (element-list-p b))))
  :rule-classes :rewrite)