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    • Successor-predecessor-intersection

    Not-empty-successor-predecessor-intersection

    Non-empty intersection of successors and predecessors

    This lifts not-empty-successor-predecessor-author-intersection from the authors, over which stake is calculated, to the certificates, which are the ones whose non-empty intersection we need to show. With reference to successor-predecessor-intersection, here cert1 is A and cert2 is C; we show that the successors of A and the predecessors of C have a non-empty intersection. The key theorem used in the proof is certs-same-round-unequiv-intersect-when-authors-intersect.

    Definitions and Theorems

    Theorem: not-empty-successor-predecessor-intersection

    (defthm not-empty-successor-predecessor-intersection
     (implies
          (and (certificate-setp dag1)
               (certificate-setp dag2)
               (certificate-set-unequivocalp dag1)
               (certificate-set-unequivocalp dag2)
               (certificate-sets-unequivocalp dag1 dag2)
               (equal (certificate->round cert2)
                      (+ 2 (certificate->round cert1)))
               (subset (cert-set->author-set (successors cert1 dag1))
                       (committee-members commtt))
               (subset (cert-set->author-set (predecessors cert2 dag2))
                       (committee-members commtt))
               (> (committee-members-stake
                       (cert-set->author-set (successors cert1 dag1))
                       commtt)
                  (committee-max-faulty-stake commtt))
               (>= (committee-members-stake
                        (cert-set->author-set (predecessors cert2 dag2))
                        commtt)
                   (committee-quorum-stake commtt)))
          (not (emptyp (intersect (successors cert1 dag1)
                                  (predecessors cert2 dag2))))))