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In sequent calculi, cut elimination is a property that guarantees that anyprovable formula can be proven analytically. For example, Gentzen's classicaland intuitionistic calculi LK and LJ enjoy cut elimination. The property isless studied in coinductive extensions of sequent calculi. In this paper, weuse coinductive Horn clause theories to show that cut is not eliminable in acoinductive extension of LJ, a system we call CLJ. We derive two furtherpractical results from this study. We show that CoLP by Gupta et al. gives riseto cut-free proofs in CLJ with fixpoint terms, and we formulate and implement anovel method of coinductive theory exploration that provides several heuristicsfor discovery of cut formulae in CLJ.

The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them