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    • Dag-omni-paths

    Dag-omni-paths-rounds-p

    Check if dag-omni-paths-rounds-p holds for all rounds that are at least two rounds after the given certificate.

    This provides an alternative definition of dag-omni-paths-p as a quantification over the rounds of dag-omni-paths-round-p, which is convenient for our proof by induction, as mentioned above. We prove that it is indeed equivalent to dag-omni-paths-p, which is conceptually easy but takes a few steps to deal with the quantifiers.

    Definitions and Theorems

    Theorem: dag-omni-paths-rounds-p-necc

    (defthm dag-omni-paths-rounds-p-necc
      (implies (dag-omni-paths-rounds-p cert dag)
               (implies (and (posp round)
                             (>= round (+ 2 (certificate->round cert))))
                        (dag-omni-paths-round-p round cert dag))))

    Theorem: booleanp-of-dag-omni-paths-rounds-p

    (defthm booleanp-of-dag-omni-paths-rounds-p
      (b* ((yes/no (dag-omni-paths-rounds-p cert dag)))
        (booleanp yes/no))
      :rule-classes :rewrite)

    Theorem: dag-omni-paths-p-alt-def

    (defthm dag-omni-paths-p-alt-def
      (equal (dag-omni-paths-p cert dag)
             (dag-omni-paths-rounds-p cert dag)))