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• Std/lists
• Intersectp

Intersectp-witness

(intersectp-witness x y) finds a some element that is a member of both x and y, if one exists.

This function is useful for basic pick-a-point style reasoning about set intersection.

Definitions and Theorems

Function: intersectp-witness

(defun intersectp-witness (x y)
  (if (atom x)
      nil
    (if (member (car x) y)
        (car x)
      (intersectp-witness (cdr x) y))))

Theorem: intersectp-witness-correct

(defthm intersectp-witness-correct
  (let ((a (intersectp-witness x y)))
    (equal (intersectp x y)
           (and (member a x) (member a y) t))))

Theorem: intersectp-witness-rw

(defthm intersectp-witness-rw
 (implies (rewriting-negative-literal (cons 'intersectp-equal
                                            (cons x (cons y 'nil))))
          (let ((a (intersectp-witness x y)))
            (equal (intersectp x y)
                   (and (member a x) (member a y) t)))))