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Subset-p

Signature

(subset-p x y) → *

Arguments

x — Guard (true-listp x).

y — Guard (true-listp y).

Definitions and Theorems

Function: subset-p

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

Theorem: subset-p-cdr-x

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

Theorem: subset-p-cdr-y

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

Theorem: subset-p-cons

(defthm subset-p-cons
        (implies (subset-p x y)
                 (subset-p (cons e x) (cons e y)))
        :rule-classes ((:rewrite :backchain-limit-lst (0))))

Theorem: subset-p-reflexive

(defthm subset-p-reflexive
        (equal (subset-p x x) t))

Theorem: subset-p-transitive

(defthm subset-p-transitive
        (implies (and (subset-p x y) (subset-p y z))
                 (subset-p x z)))

Theorem: subset-p-of-append-1

(defthm subset-p-of-append-1
        (equal (subset-p (append a b) x)
               (and (subset-p a x) (subset-p b x))))

Theorem: subset-p-of-append-2

(defthm subset-p-of-append-2
        (implies (or (subset-p a x) (subset-p a y))
                 (subset-p a (append x y))))

Theorem: subset-p-and-append-both

(defthm subset-p-and-append-both
        (implies (subset-p a b)
                 (subset-p (append e a) (append e b))))

Theorem: subset-p-of-nil

(defthm subset-p-of-nil
        (equal (subset-p x nil) (atom x)))

Theorem: subset-p-cons-2

(defthm subset-p-cons-2
        (implies (subset-p x y)
                 (subset-p x (cons e y))))

Theorem: member-p-of-subset-is-member-p-of-superset

(defthm member-p-of-subset-is-member-p-of-superset
        (implies (and (subset-p x y) (member-p e x))
                 (member-p e y)))

Theorem: not-member-p-of-superset-is-not-member-p-of-subset

(defthm not-member-p-of-superset-is-not-member-p-of-subset
        (implies (and (equal (member-p e y) nil)
                      (subset-p x y))
                 (equal (member-p e x) nil)))

Theorem: subset-p-and-remove-duplicates-equal-1

(defthm subset-p-and-remove-duplicates-equal-1
        (implies (subset-p x y)
                 (subset-p (remove-duplicates-equal x)
                           y)))

Theorem: subset-p-and-remove-duplicates-equal-2

(defthm subset-p-and-remove-duplicates-equal-2
        (implies (subset-p x y)
                 (subset-p x (remove-duplicates-equal y))))

Theorem: subset-p-and-remove-duplicates-equal-both

(defthm subset-p-and-remove-duplicates-equal-both
        (implies (subset-p x y)
                 (subset-p (remove-duplicates-equal x)
                           (remove-duplicates-equal y))))