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• Std/osets

Difference

(difference x y) removes all members of Y from X.

The logical definition is very simple, and the essential correctness property is given by difference-in.

The execution uses a better, O(n) algorithm to remove the elements by exploiting the set order.

Definitions and Theorems

Function: difference

(defun difference (x y)
  (declare (xargs :guard (and (setp x) (setp y))))
  (mbe :logic (cond ((emptyp x) (sfix x))
                    ((in (head x) y)
                     (difference (tail x) y))
                    (t (insert (head x)
                               (difference (tail x) y))))
       :exec (fast-difference x y nil)))

Theorem: difference-set

(defthm difference-set
  (setp (difference x y)))

Theorem: difference-sfix-x

(defthm difference-sfix-x
  (equal (difference (sfix x) y)
         (difference x y)))

Theorem: difference-sfix-y

(defthm difference-sfix-y
  (equal (difference x (sfix y))
         (difference x y)))

Theorem: difference-emptyp-x

(defthm difference-emptyp-x
  (implies (emptyp x)
           (equal (difference x y) (sfix x))))

Theorem: difference-emptyp-y

(defthm difference-emptyp-y
  (implies (emptyp y)
           (equal (difference x y) (sfix x))))

Theorem: difference-in

(defthm difference-in
  (equal (in a (difference x y))
         (and (in a x) (not (in a y)))))

Theorem: difference-subset-x

(defthm difference-subset-x
  (subset (difference x y) x))

Theorem: subset-difference

(defthm subset-difference
  (equal (emptyp (difference x y))
         (subset x y)))

Theorem: difference-insert-x

(defthm difference-insert-x
  (equal (difference (insert a x) y)
         (if (in a y)
             (difference x y)
           (insert a (difference x y)))))

Theorem: difference-preserves-subset

(defthm difference-preserves-subset
  (implies (subset x y)
           (subset (difference x z)
                   (difference y z))))