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Subsetp-witness

(subsetp-witness x y) finds an element of x that is not a member of y, if one exists.

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

Definitions and Theorems

Function: subsetp-witness

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

Theorem: subsetp-witness-correct

(defthm subsetp-witness-correct
  (let ((a (subsetp-witness x y)))
    (iff (subsetp x y)
         (implies (member a x) (member a y)))))

Theorem: subsetp-witness-rw

(defthm subsetp-witness-rw
 (implies (rewriting-positive-literal (cons 'subsetp-equal
                                            (cons x (cons y 'nil))))
          (let ((a (subsetp-witness x y)))
            (iff (subsetp x y)
                 (implies (member a x) (member a y))))))