  In his seminal paper of 2004, Ulrich Junker proposed the QuickXPlain
algorithm, which provides a divide-and-conquer computation strategy to find
within a given set an irreducible subset with a particular (monotone) property.
Beside its original application in the domain of constraint satisfaction
problems, the algorithm has since then found widespread adoption in areas as
different as model-based diagnosis, recommender systems, verification, or the
Semantic Web. This popularity is due to the frequent occurrence of the problem
of finding irreducible subsets on the one hand, and to QuickXPlain's general
applicability and favorable computational complexity on the other hand.
  However, although (we regularly experience) people are having a hard time
understanding QuickXPlain and seeing why it works correctly, a proof of
correctness of the algorithm has never been published. This is what we account
for in this work, by explaining QuickXPlain in a novel tried and tested way and
by presenting an intelligible formal proof of it. Apart from showing the
correctness of the algorithm and excluding the later detection of errors (proof
and trust effect), the added value of the availability of a formal proof is,
e.g., (i) that the workings of the algorithm often become completely clear only
after studying, verifying and comprehending the proof (didactic effect), (ii)
the shown proof methodology can be used as a guidance for proving other
recursive algorithms (transfer effect), and (iii) the possibility of providing
"gapless" correctness proofs of systems that rely on (results computed by)
QuickXPlain, such as numerous model-based debuggers (completeness effect).
