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Std/lists/member

Basic-member-lemmas

Very basic lemmas about member.

Definitions and Theorems

Theorem: member-when-atom

(defthm member-when-atom
  (implies (atom x) (not (member a x))))

Theorem: member-of-cons

(defthm member-of-cons
  (equal (member a (cons b x))
         (if (equal a b)
             (cons b x)
           (member a x))))

Theorem: member-of-car

(defthm member-of-car
  (equal (member (car x) x)
         (if (consp x) x nil)))

Theorem: member-of-append

(defthm member-of-append
  (iff (member a (append x y))
       (or (member a x) (member a y))))

Theorem: member-of-set-difference-equal

(defthm member-of-set-difference-equal
  (iff (member a (set-difference-equal x y))
       (and (member a x) (not (member a y)))))

Theorem: member-of-intersection-equal

(defthm member-of-intersection-equal
  (iff (member a (intersection-equal x y))
       (and (member a x) (member a y))))

Theorem: member-of-remove

(defthm member-of-remove
  (iff (member a (remove b x))
       (and (member a x) (not (equal a b)))))

Theorem: acl2-count-when-member

(defthm acl2-count-when-member
  (implies (member a x)
           (< (acl2-count a) (acl2-count x)))
  :rule-classes ((:rewrite) (:linear)))

Theorem: member-self

(defthm member-self (not (member x x)))

Theorem: element-p-when-member-equal-of-element-list-not-negated

(defthm element-p-when-member-equal-of-element-list-not-negated
  (and (implies (and (member-equal a x)
                     (element-list-p x))
                (element-p a))
       (implies (and (element-list-p x)
                     (member-equal a x))
                (element-p a)))
  :rule-classes :rewrite)

Theorem: element-p-when-member-equal-of-element-list-negated

(defthm element-p-when-member-equal-of-element-list-negated
  (and (implies (and (member-equal a x)
                     (element-list-p x))
                (not (non-element-p a)))
       (implies (and (element-list-p x)
                     (member-equal a x))
                (not (non-element-p a))))
  :rule-classes :rewrite)

Theorem: member-of-element-xformer-in-elementlist-projection

(defthm member-of-element-xformer-in-elementlist-projection
  (implies (member k x)
           (member (element-xformer k)
                   (elementlist-projection x)))
  :rule-classes :rewrite)

Theorem: member-in-elementlist-mapappend

(defthm member-in-elementlist-mapappend
  (implies (and (member k (element-listxformer j))
                (member j x))
           (member k (elementlist-mapappend x)))
  :rule-classes :rewrite)