Proof terms for infinitary rewriting, progress report

TL;DR

This paper extends proof terms to transfinite reductions, enabling formal analysis of infinitary rewriting and establishing permutation equivalence and compression properties for such reductions.

Contribution

It introduces a novel framework of proof terms for transfinite reductions, facilitating their formal analysis and proving key properties like permutation equivalence and compression.

Findings

Proof terms can faithfully represent any transfinite reduction.

Permutation equivalence is established via adapted equational logic.

Compression property is proven using proof terms.

Abstract

We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. We show that any transfinite reduction can be faithfully represented as an infinitary proof term, which is unique up to, infinitary, associativity. Our main use of proof terms is in a definition of permutation equivalence for transfinite reductions, on the basis of permutation equations. This definition involves a variant of equational logic, adapted for dealing with infinite objects. A proof of the compression property via proof terms is presented, which establishes permutation equivalence between the original and the compressed reductions.