A Deforestation of Reducts: Refocusing

TL;DR

This paper provides a formal proof, using Coq, of the refocusing transformation in small-step semantics, connecting reduction-based and reduction-free normalization methods, enhancing the rigor of prior informal proofs.

Contribution

It offers the first formal, Coq-based proof of refocusing, clarifying its correctness and formalizing an important concept in semantics.

Findings

Formal proof of refocusing using Coq

Clarification of the correctness of the transformation

Alignment with original simple idea of refocusing

Abstract

In a small-step semantics with a deterministic reduction strategy, refocusing is a transformation that connects a reduction-based normalization function (i.e., a normalization function that enumerates the successive terms in a reduction sequence -- the successive reducts) and a reduction-free normalization function (i.e., a normalization function that does not construct any reduct because all the reducts are deforested). This transformation was introduced by Nielsen and the author in the early 2000's with an informal correctness proof. Since then, it has been used in a variety of settings, starting with Biernacka and the author's syntactic correspondence between calculi and abstract machines, and several formal proofs of it have been put forward. This article presents a simple, if overdue, formal proof of refocusing that uses the Coq Proof Assistant and is aligned with the simplicity of the original idea.