  We describe an algorithmic method of proof compression based on the
introduction of Pi_2-cuts into a cut-free LK-proof. The current approach is
based on an inversion of Gentzen s cut-elimination method and extends former
methods for introducing Pi_1-cuts. The Herbrand instances of a cut-free proof
pi of a sequent S are described by a grammar G which encodes substitutions
defined in the elimination of quantified cuts. We present an algorithm which,
given a grammar G, constructs a Pi_2-cut formula A and a proof phi of S with
one cut on A. It is shown that, by this algorithm, we can achieve an
exponential proof compression.
