Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi

TL;DR

This paper proves strong normalisation for an intuitionistic sequent calculus by embedding it into simply-typed lambda-calculus using a continuation-passing style translation that preserves reduction steps.

Contribution

It introduces a novel continuation-passing style embedding that strictly simulates reductions, establishing strong normalisation for the calculus and its extensions.

Findings

Embedding preserves reduction steps and proves strong normalisation.

Method applies to higher-order calculi.

First proof of strong normalisation via reduction-preserving embedding for these systems.

Abstract

The intuitionistic fragment of the call-by-name version of Curien and Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed lambda-calculus. Our embedding is a continuation-and-garbage-passing style translation, the inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's \lambda\_mu-calculus. The embedding strictly simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need "units of garbage" to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. The results obtained extend to second and higher-order calculi.