Factorization and Normalization, Essentially

TL;DR

This paper introduces simplified proof techniques for factorization and normalization theorems in lambda-calculi, applicable to various evaluation strategies, including non-deterministic and less classic reductions.

Contribution

It presents a more powerful, simpler proof method for factorization theorems and introduces the concept of essential systems to derive normalization results.

Findings

Technique works for head reduction and beyond

Applicable to non-deterministic weak call-by-value

Proven effective on four case studies

Abstract

Lambda-calculi come with no fixed evaluation strategy. Different strategies may then be considered, and it is important that they satisfy some abstract rewriting property, such as factorization or normalization theorems. In this paper we provide simple proof techniques for these theorems. Our starting point is a revisitation of Takahashi's technique to prove factorization for head reduction. Our technique is both simpler and more powerful, as it works in cases where Takahishi's does not. We then pair factorization with two other abstract properties, defining \emph{essential systems}, and show that normalization follows. Concretely, we apply the technique to four case studies, two classic ones, head and the leftmost-outermost reductions, and two less classic ones, non-deterministic weak call-by-value and least-level reductions.