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Proof-utilities

Member-p

Signature
(member-p e x) → *
Arguments: x — Guard (true-listp x).

Definitions and Theorems

Function: member-p

(defun member-p (e x)
  (declare (xargs :guard (true-listp x)))
  (let ((__function__ 'member-p))
    (declare (ignorable __function__))
    (if (atom x)
        nil
      (if (equal e (car x))
          t
        (member-p e (cdr x))))))

Theorem: member-p-of-not-a-consp

(defthm member-p-of-not-a-consp
  (implies (not (consp x))
           (equal (member-p e x) nil)))

Theorem: member-p-of-nil

(defthm member-p-of-nil
  (equal (member-p e nil) nil))

Theorem: member-p-cons

(defthm member-p-cons
  (equal (member-p e1 (cons e2 x))
         (if (equal e1 e2) t (member-p e1 x))))

Theorem: member-p-append

(defthm member-p-append
  (equal (member-p e (append x y))
         (or (member-p e x) (member-p e y))))

Theorem: member-p-cdr

(defthm member-p-cdr
  (implies (member-p e (cdr x))
           (member-p e x))
  :rule-classes ((:rewrite :backchain-limit-lst (0))))

Theorem: member-p-iff-member-equal

(defthm member-p-iff-member-equal
  (iff (member-p e x) (member-equal e x)))