  Cut-elimination is the bedrock of proof theory. It is the algorithm that
eliminates cuts from a sequent calculus proof that leads to cut-free calculi
and applications. Cut-elimination applies to many logics irrespective of their
semantics. Such is its influence that whenever cut-elimination is not provable
in a sequent calculus the invariable response has been a move to a richer proof
system to regain it. In this paper we investigate a radically different
approach to the latter: adapting age-old cut-elimination to restrict the shape
of the cut-formulas when elimination is not possible. We tackle the "first
level" above cut-free: analytic cuts. Our methodology is applied to the sequent
calculi for bi-intuitionistic logic and S5 where analytic cuts are already
known to be required. This marks the first steps in a theory of
cut-restriction.
