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• Std/lists
• Set-difference$

Std/lists/set-difference

Lemmas about set-difference$ available in the std/lists library.

Definitions and Theorems

Theorem: set-difference-equal-when-atom

(defthm set-difference-equal-when-atom
        (implies (atom x)
                 (equal (set-difference-equal x y) nil)))

Theorem: set-difference-equal-of-cons

(defthm set-difference-equal-of-cons
        (equal (set-difference-equal (cons a x) y)
               (if (member-equal a y)
                   (set-difference-equal x y)
                   (cons a (set-difference-equal x y)))))

Theorem: set-difference-equal-when-subsetp-equal

(defthm set-difference-equal-when-subsetp-equal
        (implies (subsetp-equal x y)
                 (equal (set-difference-equal x y) nil)))

Theorem: set-difference-equal-of-self

(defthm set-difference-equal-of-self
        (equal (set-difference-equal x x) nil))

Theorem: empty-intersect-with-difference-of-self

(defthm empty-intersect-with-difference-of-self
        (not (intersectp-equal a (set-difference-equal b a))))

Theorem: no-duplicatesp-of-set-difference-equal

(defthm
     no-duplicatesp-of-set-difference-equal
     (implies (no-duplicatesp-equal l1)
              (no-duplicatesp-equal (set-difference-equal l1 l2))))