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Type-set

Type-setp

Recognizer for type-set.

Signature
(type-setp x) → *

Definitions and Theorems

Function: type-setp

(defun type-setp (x)
  (declare (xargs :guard t))
  (if (atom x)
      (null x)
    (and (typep (car x))
         (or (null (cdr x))
             (and (consp (cdr x))
                  (acl2::fast-<< (car x) (cadr x))
                  (type-setp (cdr x)))))))

Theorem: booleanp-oftype-setp

(defthm booleanp-oftype-setp
  (booleanp (type-setp x)))

Theorem: setp-when-type-setp

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

Theorem: typep-of-head-when-type-setp

(defthm typep-of-head-when-type-setp
  (implies (type-setp x)
           (equal (typep (head x))
                  (not (emptyp x)))))

Theorem: type-setp-of-tail-when-type-setp

(defthm type-setp-of-tail-when-type-setp
  (implies (type-setp x)
           (type-setp (tail x))))

Theorem: type-setp-of-insert

(defthm type-setp-of-insert
  (equal (type-setp (insert a x))
         (and (typep a) (type-setp (sfix x)))))

Theorem: typep-when-in-type-setp-binds-free-x

(defthm typep-when-in-type-setp-binds-free-x
  (implies (and (in a x) (type-setp x))
           (typep a)))

Theorem: not-in-type-setp-when-not-typep

(defthm not-in-type-setp-when-not-typep
  (implies (and (type-setp x) (not (typep a)))
           (not (in a x))))

Theorem: type-setp-of-union

(defthm type-setp-of-union
  (equal (type-setp (union x y))
         (and (type-setp (sfix x))
              (type-setp (sfix y)))))

Theorem: type-setp-of-intersect

(defthm type-setp-of-intersect
  (implies (or (type-setp x) (type-setp y))
           (type-setp (intersect x y))))

Theorem: type-setp-of-difference

(defthm type-setp-of-difference
  (implies (type-setp x)
           (type-setp (difference x y))))

Theorem: type-setp-of-delete

(defthm type-setp-of-delete
  (implies (type-setp x)
           (type-setp (delete a x))))