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• Mode-set

Mode-setp

Recognizer for mode-set.

Signature

(mode-setp x) → *

Definitions and Theorems

Function: mode-setp

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

Theorem: booleanp-ofmode-setp

(defthm booleanp-ofmode-setp
  (booleanp (mode-setp x)))

Theorem: setp-when-mode-setp

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

Theorem: modep-of-head-when-mode-setp

(defthm modep-of-head-when-mode-setp
  (implies (mode-setp x)
           (equal (modep (head x))
                  (not (emptyp x)))))

Theorem: mode-setp-of-tail-when-mode-setp

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

Theorem: mode-setp-of-insert

(defthm mode-setp-of-insert
  (equal (mode-setp (insert a x))
         (and (modep a) (mode-setp (sfix x)))))

Theorem: modep-when-in-mode-setp-binds-free-x

(defthm modep-when-in-mode-setp-binds-free-x
  (implies (and (in a x) (mode-setp x))
           (modep a)))

Theorem: not-in-mode-setp-when-not-modep

(defthm not-in-mode-setp-when-not-modep
  (implies (and (mode-setp x) (not (modep a)))
           (not (in a x))))

Theorem: mode-setp-of-union

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

Theorem: mode-setp-of-intersect

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

Theorem: mode-setp-of-difference

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

Theorem: mode-setp-of-delete

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