  In sequent calculi, cut elimination is a property that guarantees that any
provable formula can be proven analytically. For example, Gentzen's classical
and intuitionistic calculi LK and LJ enjoy cut elimination. The property is
less studied in coinductive extensions of sequent calculi. In this paper, we
use coinductive Horn clause theories to show that cut is not eliminable in a
coinductive extension of LJ, a system we call CLJ. We derive two further
practical results from this study. We show that CoLP by Gupta et al. gives rise
to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a
novel method of coinductive theory exploration that provides several heuristics
for discovery of cut formulae in CLJ.
