Extending the Applicability Condition in the Formal System $\lambda\delta$

TL;DR

This paper introduces an extended version of the $\lambda ext{delta}$ formal system, providing machine-checked proofs of key properties like confluence, normalization, and validity preservation, enhancing its foundational robustness.

Contribution

It presents a revised $\lambda ext{delta}$ system with fully machine-checked proofs of important properties, including the first proof of the 'big tree' theorem for a calculus with $\Lambda_ extinfty$.

Findings

Proved confluence of reduction in the revised system.

Established strong normalization via the 'big tree' theorem.

Demonstrated preservation of validity during reduction.

Abstract

The formal system $\lambda\delta$ is a typed lambda calculus derived from $\Lambda_\infty$, aiming to support the foundations of Mathematics that require an underlying theory of expressions (for example the Minimal Type Theory). The system is developed in the context of the Hypertextual Electronic Library of Mathematics as a machine-checked digital specification, that is not the formal counterpart of previous informal material. The first version of the calculus appeared in 2006 and proved unsatisfactory for some reasons. In this article we present a revised version of the system and we prove three relevant desired properties: the confluence of reduction, the strong normalization of an extended form of reduction, known as the "big tree" theorem, and the preservation of validity by reduction. To our knowledge, we are presenting here the first fully machine-checked proof of the "big tree" theorem for a calculus that includes $\Lambda_\infty$.