  In this paper we examine the Paxos protocol and demonstrate how the discrete
numbering of ballots can be leveraged to weaken the conditions for learning.
Specifically, we define the notion of consecutive ballots and use this to
define Consecutive Quorums. Consecutive Quorums weakens the learning criterion
such that a learner does not need matching $accept$ messages sent in the $same
\; ballot$ from a majority of acceptors to learn a value. We prove that this
modification preserves the original safety and liveness guarantees of Paxos. We
define $Consecutive \; Paxos$ which encapsulates the properties of discrete
consecutive ballots. To establish the correctness of these results, we, in
addition to a paper proof, formally verify the correctness of a State Machine
Replication Library built on top of an optimized version of Multi-Paxos
modified to reflect $Consecutive \; Paxos$.
