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

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 calculiLKandLJenjoy 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 ofLJ, a system we callCLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs inCLJwith fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae inCLJ.