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Proposal-set

Proposal-setp

Recognizer for proposal-set.

Signature
(proposal-setp x) → *

Definitions and Theorems

Function: proposal-setp

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

Theorem: booleanp-ofproposal-setp

(defthm booleanp-ofproposal-setp
  (booleanp (proposal-setp x)))

Theorem: setp-when-proposal-setp

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

Theorem: proposalp-of-head-when-proposal-setp

(defthm proposalp-of-head-when-proposal-setp
  (implies (proposal-setp x)
           (equal (proposalp (head x))
                  (not (emptyp x)))))

Theorem: proposal-setp-of-tail-when-proposal-setp

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

Theorem: proposal-setp-of-insert

(defthm proposal-setp-of-insert
  (equal (proposal-setp (insert a x))
         (and (proposalp a)
              (proposal-setp (sfix x)))))

Theorem: proposalp-when-in-proposal-setp-binds-free-x

(defthm proposalp-when-in-proposal-setp-binds-free-x
  (implies (and (in a x) (proposal-setp x))
           (proposalp a)))

Theorem: not-in-proposal-setp-when-not-proposalp

(defthm not-in-proposal-setp-when-not-proposalp
  (implies (and (proposal-setp x)
                (not (proposalp a)))
           (not (in a x))))

Theorem: proposal-setp-of-union

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

Theorem: proposal-setp-of-intersect

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

Theorem: proposal-setp-of-difference

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

Theorem: proposal-setp-of-delete

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