A direct proof of the confluence of combinatory strong reduction

TL;DR

This paper provides a direct proof of the confluence property for combinatory strong reduction, along with straightforward proofs of standardization and strong normalization for simply typed terms, avoiding lambda-calculus methods.

Contribution

It introduces a novel, direct proof approach for confluence in combinatory strong reduction, simplifying existing proofs and extending understanding.

Findings

Proof of confluence without lambda-calculus

Standardization theorem for combinatory reduction

Strong normalization of simply typed terms

Abstract

I give a proof of the confluence of combinatory strong reduction that does not use the one of lambda-calculus. I also give simple and direct proofs of a standardization theorem for this reduction and the strong normalization of simply typed terms.