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• Set-equiv

Set-unequal-witness

(set-unequal-witness x y) finds a member of x that is not a member of y, or vice versa.

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

Definitions and Theorems

Function: set-unequal-witness

(defun set-unequal-witness (x y)
  (cond ((not (subsetp x y))
         (subsetp-witness x y))
        ((not (subsetp y x))
         (subsetp-witness y x))))

Theorem: set-unequal-witness-correct

(defthm set-unequal-witness-correct
  (equal (set-equiv x y)
         (iff (member (set-unequal-witness x y) x)
              (member (set-unequal-witness x y) y))))

Theorem: set-unequal-witness-rw

(defthm set-unequal-witness-rw
 (implies (rewriting-positive-literal (cons 'set-equiv
                                            (cons x (cons y 'nil))))
          (equal (set-equiv x y)
                 (iff (member (set-unequal-witness x y) x)
                      (member (set-unequal-witness x y) y)))))