Arithmetical proofs of strong normalization results for the symmetric $\lambda \mu$-calculus

TL;DR

This paper provides arithmetical proofs demonstrating strong normalization for the symmetric $ ext{λ} ext{μ}$-calculus, including new results for untyped $ ext{μ} ext{μ'}$-reduction and simplified proofs for typed $eta ext{μ} ext{μ'}$-reduction.

Contribution

It introduces arithmetical proofs for strong normalization in the symmetric $ ext{λ} ext{μ}$-calculus, notably proving $ ext{μ} ext{μ'}$-reduction is strongly normalizing, and simplifies existing proofs for typed cases.

Findings

$ ext{μ} ext{μ'}$-reduction is strongly normalizing for untyped calculus

Strong normalization of $eta ext{μ} ext{μ'}$-reduction for typed calculus

Previous proofs relied on complex reducibility candidates, now replaced by arithmetical methods

Abstract

The symmetric $\lambda \mu$-calculus is the $\lambda \mu$-calculus introduced by Parigot in which the reduction rule $\m'$, which is the symmetric of $\mu$, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the $\mu\mu'$-reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the $\beta\mu\mu'$-reduction for the typed calculus: this was already known but the previous proofs use candidates of reducibility where the interpretation of a type was defined as the fix point of some increasing operator and thus, were highly non arithmetical.