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Rulename-set

Rulename-setp

Recognizer for rulename-set.

Signature
(rulename-setp x) → *

Definitions and Theorems

Function: rulename-setp

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

Theorem: booleanp-ofrulename-setp

(defthm booleanp-ofrulename-setp
  (booleanp (rulename-setp x)))

Theorem: setp-when-rulename-setp

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

Theorem: rulenamep-of-head-when-rulename-setp

(defthm rulenamep-of-head-when-rulename-setp
  (implies (rulename-setp x)
           (equal (rulenamep (head x))
                  (not (emptyp x)))))

Theorem: rulename-setp-of-tail-when-rulename-setp

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

Theorem: rulename-setp-of-insert

(defthm rulename-setp-of-insert
  (equal (rulename-setp (insert a x))
         (and (rulenamep a)
              (rulename-setp (sfix x)))))

Theorem: rulenamep-when-in-rulename-setp-binds-free-x

(defthm rulenamep-when-in-rulename-setp-binds-free-x
  (implies (and (in a x) (rulename-setp x))
           (rulenamep a)))

Theorem: not-in-rulename-setp-when-not-rulenamep

(defthm not-in-rulename-setp-when-not-rulenamep
  (implies (and (rulename-setp x)
                (not (rulenamep a)))
           (not (in a x))))

Theorem: rulename-setp-of-union

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

Theorem: rulename-setp-of-intersect

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

Theorem: rulename-setp-of-difference

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

Theorem: rulename-setp-of-delete

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