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Quorum intersection.
This is a fairly ubiquitous concept in BFT systems.
This concept can be generally described like this:
certain decisions are made (by correct validators)
only if they are supported by a quorum of validators;
contradictory decisions cannot be made because,
because each would be supported by a quorum,
but the intersection of the two quora would contain enough validators
that there is at least a correct one in the intersection,
which would not have supported both decisions.
This is the case if the system is fault-tolerant,
i.e.
In AleoBFT this concept applies to certificate non-equivocation. By requiring a quorum of signatures, where each signature supports the certificate (in the sense of `supporting' mentioned above), we ensure that two incompatible certificates, i.e. two different certificates with the same author and round, cannot exist because they would have to be both signed by at least one correct validator in the intersection of the quora.
Here we introduce a function that picks a correct validator (if any) from the intersection of two sets of (addresses of) validators from a common committee. We prove that, if the committee is fault-tolerant, and each set of addresses forms a quorum, the function indeed picks a correct validator. This is then used (elsewhere) to prove certificate non-equivocation.