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Character-set

Character-setp

Recognizer for character-set.

Signature
(character-setp x) → *

Definitions and Theorems

Function: character-setp

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

Theorem: booleanp-ofcharacter-setp

(defthm booleanp-ofcharacter-setp
  (booleanp (character-setp x)))

Theorem: setp-when-character-setp

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

Theorem: characterp-of-head-when-character-setp

(defthm characterp-of-head-when-character-setp
  (implies (character-setp x)
           (equal (characterp (set::head x))
                  (not (set::emptyp x)))))

Theorem: character-setp-of-tail-when-character-setp

(defthm character-setp-of-tail-when-character-setp
  (implies (character-setp x)
           (character-setp (set::tail x))))

Theorem: character-setp-of-insert

(defthm character-setp-of-insert
  (equal (character-setp (set::insert a x))
         (and (characterp a)
              (character-setp (set::sfix x)))))

Theorem: characterp-when-in-character-setp-binds-free-x

(defthm characterp-when-in-character-setp-binds-free-x
  (implies (and (set::in a x) (character-setp x))
           (characterp a)))

Theorem: not-in-character-setp-when-not-characterp

(defthm not-in-character-setp-when-not-characterp
  (implies (and (character-setp x)
                (not (characterp a)))
           (not (set::in a x))))

Theorem: character-setp-of-union

(defthm character-setp-of-union
  (equal (character-setp (set::union x y))
         (and (character-setp (set::sfix x))
              (character-setp (set::sfix y)))))

Theorem: character-setp-of-intersect

(defthm character-setp-of-intersect
  (implies (or (character-setp x)
               (character-setp y))
           (character-setp (set::intersect x y))))

Theorem: character-setp-of-difference

(defthm character-setp-of-difference
  (implies (character-setp x)
           (character-setp (set::difference x y))))

Theorem: character-setp-of-delete

(defthm character-setp-of-delete
  (implies (character-setp x)
           (character-setp (set::delete a x))))