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Ident-set

Ident-setp

Recognizer for ident-set.

Signature
(ident-setp x) → *

Definitions and Theorems

Function: ident-setp

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

Theorem: booleanp-ofident-setp

(defthm booleanp-ofident-setp
  (booleanp (ident-setp x)))

Theorem: setp-when-ident-setp

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

Theorem: identp-of-head-when-ident-setp

(defthm identp-of-head-when-ident-setp
  (implies (ident-setp x)
           (equal (identp (head x))
                  (not (emptyp x)))))

Theorem: ident-setp-of-tail-when-ident-setp

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

Theorem: ident-setp-of-insert

(defthm ident-setp-of-insert
  (equal (ident-setp (insert a x))
         (and (identp a) (ident-setp (sfix x)))))

Theorem: identp-when-in-ident-setp-binds-free-x

(defthm identp-when-in-ident-setp-binds-free-x
  (implies (and (in a x) (ident-setp x))
           (identp a)))

Theorem: not-in-ident-setp-when-not-identp

(defthm not-in-ident-setp-when-not-identp
  (implies (and (ident-setp x) (not (identp a)))
           (not (in a x))))

Theorem: ident-setp-of-union

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

Theorem: ident-setp-of-intersect

(defthm ident-setp-of-intersect
  (implies (and (ident-setp x) (ident-setp y))
           (ident-setp (intersect x y))))

Theorem: ident-setp-of-difference

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

Theorem: ident-setp-of-delete

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