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Tree-set

Tree-setp

Recognizer for tree-set.

Signature
(tree-setp x) → *

Definitions and Theorems

Function: tree-setp

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

Theorem: booleanp-oftree-setp

(defthm booleanp-oftree-setp
  (booleanp (tree-setp x)))

Theorem: setp-when-tree-setp

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

Theorem: treep-of-head-when-tree-setp

(defthm treep-of-head-when-tree-setp
  (implies (tree-setp x)
           (equal (treep (head x))
                  (not (emptyp x)))))

Theorem: tree-setp-of-tail-when-tree-setp

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

Theorem: tree-setp-of-insert

(defthm tree-setp-of-insert
  (equal (tree-setp (insert a x))
         (and (treep a) (tree-setp (sfix x)))))

Theorem: treep-when-in-tree-setp-binds-free-x

(defthm treep-when-in-tree-setp-binds-free-x
  (implies (and (in a x) (tree-setp x))
           (treep a)))

Theorem: not-in-tree-setp-when-not-treep

(defthm not-in-tree-setp-when-not-treep
  (implies (and (tree-setp x) (not (treep a)))
           (not (in a x))))

Theorem: tree-setp-of-union

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

Theorem: tree-setp-of-intersect

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

Theorem: tree-setp-of-difference

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

Theorem: tree-setp-of-delete

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