Revisiting Cut-Elimination: One Difficult Proof Is Really a Proof

Abstract

Powerful proof techniques, such as logical relation arguments, have been developed for establishing the strong normalisation property of term- rewriting systems. The first author used such a logical relation argument to establish strong normalising for a cut-elimination procedure in classical logic. He presented a rather complicated, but informal, proof establishing this property. The difficulties in this proof arise from a quite subtle substitution operation, which implements proof transformation that permute cuts over other inference rules. We have formalised this proof in the theorem prover Isabelle/HOL using the Nominal Datatype Package, closely following the informal proof given by the first author in his PhD-thesis. In the process, we identified and resolved a gap in one central lemma and a number of smaller problems in others. We also needed to make one informal definition rigorous. We thus show that the original proof is indeed a proof and that present automated proving technology is adequate for formalising such difficult proofs.