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• Function-set

Function-setp

Recognizer for function-set.

Signature

(function-setp x) → *

Definitions and Theorems

Function: function-setp

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

Theorem: booleanp-offunction-setp

(defthm booleanp-offunction-setp
  (booleanp (function-setp x)))

Theorem: setp-when-function-setp

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

Theorem: functionp-of-head-when-function-setp

(defthm functionp-of-head-when-function-setp
  (implies (function-setp x)
           (equal (functionp (head x))
                  (not (emptyp x)))))

Theorem: function-setp-of-tail-when-function-setp

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

Theorem: function-setp-of-insert

(defthm function-setp-of-insert
  (equal (function-setp (insert a x))
         (and (functionp a)
              (function-setp (sfix x)))))

Theorem: functionp-when-in-function-setp-binds-free-x

(defthm functionp-when-in-function-setp-binds-free-x
  (implies (and (in a x) (function-setp x))
           (functionp a)))

Theorem: not-in-function-setp-when-not-functionp

(defthm not-in-function-setp-when-not-functionp
  (implies (and (function-setp x)
                (not (functionp a)))
           (not (in a x))))

Theorem: function-setp-of-union

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

Theorem: function-setp-of-intersect

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

Theorem: function-setp-of-difference

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

Theorem: function-setp-of-delete

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