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• Qualified-ident-option-set

Qualified-ident-option-setp

Recognizer for qualified-ident-option-set.

Signature

(qualified-ident-option-setp x) → *

Definitions and Theorems

Function: qualified-ident-option-setp

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

Theorem: booleanp-ofqualified-ident-option-setp

(defthm booleanp-ofqualified-ident-option-setp
  (booleanp (qualified-ident-option-setp x)))

Theorem: setp-when-qualified-ident-option-setp

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

Theorem: qualified-ident-optionp-of-head-when-qualified-ident-option-setp

(defthm
   qualified-ident-optionp-of-head-when-qualified-ident-option-setp
  (implies (qualified-ident-option-setp x)
           (qualified-ident-optionp (head x))))

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

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

Theorem: qualified-ident-option-setp-of-insert

(defthm qualified-ident-option-setp-of-insert
  (equal (qualified-ident-option-setp (insert a x))
         (and (qualified-ident-optionp a)
              (qualified-ident-option-setp (sfix x)))))

Theorem: qualified-ident-optionp-when-in-qualified-ident-option-setp-binds-free-x

(defthm
 qualified-ident-optionp-when-in-qualified-ident-option-setp-binds-free-x
 (implies (and (in a x)
               (qualified-ident-option-setp x))
          (qualified-ident-optionp a)))

Theorem: not-in-qualified-ident-option-setp-when-not-qualified-ident-optionp

(defthm
 not-in-qualified-ident-option-setp-when-not-qualified-ident-optionp
 (implies (and (qualified-ident-option-setp x)
               (not (qualified-ident-optionp a)))
          (not (in a x))))

Theorem: qualified-ident-option-setp-of-union

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

Theorem: qualified-ident-option-setp-of-intersect

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

Theorem: qualified-ident-option-setp-of-difference

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

Theorem: qualified-ident-option-setp-of-delete

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