Compression for Coinductive Rewriting and the Cut-Elimination of Non-Wellfounded Proofs

Abstract

We introduce a generic presentation of "syntactic objects built by mixed induction and coinduction" encompassing all standard kinds of infinitary terms, as well as derivation trees in non-wellfounded proof systems. We then define a coinductive notion of infinitary rewriting of such objects, which is equivalent to the original presentation of infinitary rewriting relying on metric convergence and ordinal-indexed sequences of rewriting steps. This provides a unified coinductive presentation of e.g. first-order infinitary rewriting, infinitary λ-calculi, and cut-elimination in non-wellfounded proofs. We then formulate and study the coinductive counterpart of compression, i.e. the property of an infinitary rewriting system such that all rewriting sequences of any ordinal length can be "compressed" to equivalent sequences of length at most ω(which ensures that they can be finitely approximated). We characterise compression in our generic setting for coinductive rewriting, "factorising" the part of the proof that can be performed at this level of generality. Our proof is fully coinductive, avoiding any detour via rewriting sequences. Finally we focus on the non-wellfounded proof system \muMALL\infty for multiplicative-additive linear logic with fixed points, and we put our results to work in order to prove that compression holds for cut-elimination in this setting, which is a key lemma of several extensions of cut-elimination to similar systems.