Compression for Coinductive Infinitary Rewriting: A Generic Approach, with Applications to Cut-Elimination for Non-Wellfounded Proofs

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 notion of coinductive 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 extension of cut-elimination to similar systems.

💡 Research Summary

The paper presents a unified, fully coinductive framework for infinitary rewriting that simultaneously covers infinitary terms (first‑order, λ‑calculus, higher‑order) and non‑well‑founded proof trees such as those arising in fixed‑point extensions of linear logic. The authors start by describing “syntactic objects built by mixed induction and coinduction”: finite syntax is generated inductively (µ‑algebra) while infinite objects are obtained coinductively (ν‑coalgebra). They show that the metric completion traditionally used to define infinitary terms is isomorphic to this coinductive presentation, thereby establishing a clean correspondence between topological and coalgebraic views.

Next, they define a coinductive notion of rewriting. Instead of ordinal‑indexed, strongly convergent sequences, they introduce a family of relations “→γ∞” (rewriting steps occurring at depth ≤ γ) together with auxiliary “⇁γ∞” that lifts rewriting under constructors. The main technical theorem (Theorem 3/4) proves that the union of all →γ∞ (for γ < ω₁) coincides with the usual infinitary rewriting relation →∞. This result holds for first‑order term rewriting, infinitary λ‑calculus, and, more generally, any system fitting their mixed‑induction/coinduction schema.

The core contribution is a coinductive characterisation of the compression property. Compression states that any infinitary rewriting sequence of arbitrary ordinal length can be “compressed” to an equivalent sequence of length at most ω, guaranteeing that finite approximations of the limit are produced in finitely many steps. The authors factorise the proof of compression into three generic conditions: (i) a depth‑decreasing (ordinal) measure on rewriting steps, (ii) a global commutation property ensuring that independent steps can be reordered, and (iii) a strong convergence condition encoded by the ∞‑label on steps. They give a fully coinductive proof that, under these conditions, any →γ∞ derivation can be transformed into an ω‑length derivation without leaving the coinductive setting. This factorisation isolates the part of the argument that is universal, leaving only system‑specific checks for the three conditions.

To demonstrate the power of the framework, the authors instantiate it for the non‑well‑founded proof system µMALL∞, a linear logic with multiplicative‑additive connectives and least/greatest fixed points. Cut‑elimination in µMALL∞ is naturally expressed as an infinitary rewriting process on proof trees. By checking that µMALL∞’s rewrite rules satisfy the three generic compression conditions, they prove that cut‑elimination enjoys compression: any potentially transfinite cut‑reduction can be simulated by an ω‑step reduction, yielding a finite‑time construction of a cut‑free proof. This result is a key lemma for the broader cut‑elimination theory of µLL∞ and its embeddings of many other logics.

In the concluding section the authors discuss how their generic coinductive approach subsumes earlier compression lemmas for left‑linear first‑order systems, infinitary λ‑calculi, and fully‑extended higher‑order rewriting. They argue that the same methodology can be applied to other infinitary or cyclic proof systems, to coinductive type theories, and to program semantics involving infinite data structures. The paper thus provides both a conceptual unification of infinitary rewriting and a practical toolkit for establishing compression in a wide variety of settings.