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• Proof-utilities

Disjoint-p

Signature

(disjoint-p x y) → *

Arguments

x — Guard (true-listp x).

y — Guard (true-listp y).

disjoint-p returns t if true-listp-satisfying inputs x and y have no element in common. Otherwise, nil is returned.

Definitions and Theorems

Function: disjoint-p

(defun disjoint-p (x y)
       (declare (xargs :guard (and (true-listp x) (true-listp y))))
       (let ((__function__ 'disjoint-p))
            (declare (ignorable __function__))
            (if (atom x)
                t
                (if (member-p (car x) y)
                    nil (disjoint-p (cdr x) y)))))

Theorem: disjoint-p-x-x

(defthm disjoint-p-x-x
        (implies (consp x)
                 (equal (disjoint-p x x) nil)))

Theorem: disjoint-p-nil-1

(defthm disjoint-p-nil-1
        (equal (disjoint-p nil y) t))

Theorem: disjoint-p-nil-2

(defthm disjoint-p-nil-2
        (equal (disjoint-p x nil) t))

Theorem: disjoint-p-cdr-1

(defthm disjoint-p-cdr-1
        (implies (disjoint-p x y)
                 (disjoint-p (cdr x) y))
        :rule-classes ((:rewrite :backchain-limit-lst (0))))

Theorem: disjoint-p-cdr-2

(defthm disjoint-p-cdr-2
        (implies (disjoint-p x y)
                 (disjoint-p x (cdr y)))
        :rule-classes ((:rewrite :backchain-limit-lst (0))))

Theorem: disjoint-p-cons-1

(defthm disjoint-p-cons-1
        (equal (disjoint-p (cons e x) a)
               (and (disjoint-p x a)
                    (equal (member-p e a) nil))))

Theorem: disjoint-p-cons-2

(defthm disjoint-p-cons-2
        (equal (disjoint-p a (cons e x))
               (and (disjoint-p a x)
                    (equal (member-p e a) nil))))

Theorem: disjoint-p-commutative

(defthm disjoint-p-commutative
        (equal (disjoint-p a b)
               (disjoint-p b a))
        :rule-classes ((:rewrite :loop-stopper ((a b)))))

Theorem: member-p-when-not-disjoint-p

(defthm member-p-when-not-disjoint-p
        (implies (and (member-p e x) (member-p e y))
                 (equal (disjoint-p x y) nil)))

Theorem: not-member-p-when-disjoint-p

(defthm not-member-p-when-disjoint-p
        (implies (and (disjoint-p x y) (member-p e x))
                 (equal (member-p e y) nil)))

Theorem: disjoint-p-append-1

(defthm disjoint-p-append-1
        (implies (true-listp a)
                 (equal (disjoint-p (append a b) c)
                        (and (disjoint-p a c)
                             (disjoint-p b c)))))

Theorem: disjoint-p-append-2

(defthm disjoint-p-append-2
        (implies (true-listp b)
                 (equal (disjoint-p a (append b c))
                        (and (disjoint-p a b)
                             (disjoint-p a c)))))