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Nat-set

Nat-setp

Recognizer for nat-set.

Signature
(nat-setp x) → *

Definitions and Theorems

Function: nat-setp

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

Theorem: booleanp-ofnat-setp

(defthm booleanp-ofnat-setp
  (booleanp (nat-setp x)))

Theorem: setp-when-nat-setp

(defthm setp-when-nat-setp
  (implies (nat-setp x) (set::setp x))
  :rule-classes (:rewrite))

Theorem: natp-of-head-when-nat-setp

(defthm natp-of-head-when-nat-setp
  (implies (nat-setp x)
           (equal (natp (set::head x))
                  (not (set::emptyp x)))))

Theorem: nat-setp-of-tail-when-nat-setp

(defthm nat-setp-of-tail-when-nat-setp
  (implies (nat-setp x)
           (nat-setp (set::tail x))))

Theorem: nat-setp-of-insert

(defthm nat-setp-of-insert
  (equal (nat-setp (set::insert a x))
         (and (natp a)
              (nat-setp (set::sfix x)))))

Theorem: natp-when-in-nat-setp-binds-free-x

(defthm natp-when-in-nat-setp-binds-free-x
  (implies (and (set::in a x) (nat-setp x))
           (natp a)))

Theorem: not-in-nat-setp-when-not-natp

(defthm not-in-nat-setp-when-not-natp
  (implies (and (nat-setp x) (not (natp a)))
           (not (set::in a x))))

Theorem: nat-setp-of-union

(defthm nat-setp-of-union
  (equal (nat-setp (set::union x y))
         (and (nat-setp (set::sfix x))
              (nat-setp (set::sfix y)))))

Theorem: nat-setp-of-intersect

(defthm nat-setp-of-intersect
  (implies (or (nat-setp x) (nat-setp y))
           (nat-setp (set::intersect x y))))

Theorem: nat-setp-of-difference

(defthm nat-setp-of-difference
  (implies (nat-setp x)
           (nat-setp (set::difference x y))))

Theorem: nat-setp-of-delete

(defthm nat-setp-of-delete
  (implies (nat-setp x)
           (nat-setp (set::delete a x))))