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Certificates

Certificate-with-author+round

Retrieve, from a set of certificates, a certificate with a given author and round.

Signature
(certificate-with-author+round author round certs) → cert?
Arguments: author — Guard (addressp author).; round — Guard (posp round).; certs — Guard (certificate-setp certs).
Returns: cert? — Type (certificate-optionp cert?).

If there is no certificate with the given author and round, nil is returned, for no certificate.

If there is a certificate with the given author and round, the first one found is returned, according to the total ordering of the set. However, when a certificate set is unequivocal, i.e. it has unique author-round combinations, the first certificate found is the only one.

Definitions and Theorems

Function: certificate-with-author+round

(defun certificate-with-author+round (author round certs)
  (declare (xargs :guard (and (addressp author)
                              (posp round)
                              (certificate-setp certs))))
  (let ((__function__ 'certificate-with-author+round))
    (declare (ignorable __function__))
    (b* (((when (emptyp certs)) nil)
         ((certificate cert) (head certs))
         ((when (and (equal author cert.author)
                     (equal round cert.round)))
          (certificate-fix cert)))
      (certificate-with-author+round author round (tail certs)))))

Theorem: certificate-optionp-of-certificate-with-author+round

(defthm certificate-optionp-of-certificate-with-author+round
  (b* ((cert? (certificate-with-author+round author round certs)))
    (certificate-optionp cert?))
  :rule-classes :rewrite)

Theorem: certificate->author-of-certificate-with-author+round

(defthm certificate->author-of-certificate-with-author+round
  (b* ((?cert? (certificate-with-author+round author round certs)))
    (implies cert?
             (equal (certificate->author cert?)
                    (address-fix author)))))

Theorem: certificate->round-of-certificate-with-author+round

(defthm certificate->round-of-certificate-with-author+round
  (b* ((?cert? (certificate-with-author+round author round certs)))
    (implies cert?
             (equal (certificate->round cert?)
                    (pos-fix round)))))

Theorem: certificate-with-author+round-element

(defthm certificate-with-author+round-element
  (implies (and (certificate-setp certs)
                (certificate-with-author+round author round certs))
           (in (certificate-with-author+round author round certs)
               certs)))

Theorem: certificate-with-author+round-when-element

(defthm certificate-with-author+round-when-element
  (implies (and (in cert certs)
                (equal (certificate->author cert)
                       author)
                (equal (certificate->round cert) round))
           (certificate-with-author+round author round certs)))

Theorem: certificate-with-author+round-when-subset

(defthm certificate-with-author+round-when-subset
  (implies (and (certificate-with-author+round author round certs0)
                (subset certs0 certs))
           (certificate-with-author+round author round certs)))

Theorem: certificate-with-author+round-of-insert-iff

(defthm certificate-with-author+round-of-insert-iff
 (iff
    (certificate-with-author+round author round (insert cert certs))
    (or (and (equal (certificate->author cert)
                    author)
             (equal (certificate->round cert) round))
        (certificate-with-author+round author round certs))))

Theorem: certificate-with-author+round-of-union-iff

(defthm certificate-with-author+round-of-union-iff
 (implies
    (certificate-setp certs2)
    (iff (certificate-with-author+round
              author round (union certs1 certs2))
         (or (certificate-with-author+round author round certs1)
             (certificate-with-author+round author round certs2)))))

Theorem: certificate-with-author+round-when-delete

(defthm certificate-with-author+round-when-delete
 (implies
    (certificate-with-author+round author round (delete cert certs))
    (certificate-with-author+round author round certs)))

Theorem: certificate-with-author+round-of-delete

(defthm certificate-with-author+round-of-delete
 (implies
  (and (certificate-with-author+round author round certs)
       (or (not (equal (certificate->author cert)
                       author))
           (not (equal (certificate->round cert)
                       round))))
  (certificate-with-author+round author round (delete cert certs))))