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• Primitives

Setp

(setp x) recognizes well-formed ordered sets.

A proper ordered set is a true-listp whose elements are fully ordered under <<. Note that this implicitly means that sets have no duplicate elements.

Testing setp is necessarily somewhat slow: we have to check that the elements are in order. Its cost is linear in the size of n.

Definitions and Theorems

Function: setp

(defun setp (x)
       (declare (xargs :guard t))
       (if (atom x)
           (null x)
           (or (null (cdr x))
               (and (consp (cdr x))
                    (fast-<< (car x) (cadr x))
                    (setp (cdr x))))))

Theorem: setp-type

(defthm setp-type
        (or (equal (setp x) t)
            (equal (setp x) nil))
        :rule-classes :type-prescription)

Theorem: sets-are-true-lists

(defthm sets-are-true-lists
        (implies (setp x) (true-listp x)))

Theorem: sets-are-true-lists-compound-recognizer

(defthm sets-are-true-lists-compound-recognizer
        (implies (setp x) (true-listp x))
        :rule-classes :compound-recognizer)

Theorem: sets-are-true-lists-cheap

(defthm sets-are-true-lists-cheap
        (implies (setp x) (true-listp x))
        :rule-classes ((:rewrite :backchain-limit-lst (1))))