A Modular Lean 4 Framework for Confluence and Strong Normalization of Lambda Calculi with Products and Sums

TL;DR

This paper introduces Metatheory, a modular Lean 4 library that mechanizes proofs of confluence and strong normalization for various lambda calculi and rewriting systems, ensuring rigorous formal verification.

Contribution

It provides the first comprehensive Lean 4 framework for confluence and normalization proofs across multiple lambda calculus variants and rewriting systems.

Findings

Successfully mechanized confluence proofs for six case studies.

Established strong normalization via logical relations.

Demonstrated the framework's applicability to diverse calculi.

Abstract

We present Metatheory, a comprehensive library for programming language foundations in Lean 4. The library features a modular framework for proving confluence of abstract rewriting systems using three classical proof techniques: the diamond property, Newmans lemma, and the Hindley-Rosen lemma. These are instantiated across six case studies including untyped lambda calculus, combinatory logic, term rewriting, simply typed lambda calculus, and STLC with products and sums. All theorems are fully mechanized with zero axioms or sorry statements. We provide complete proofs of de Bruijn substitution infrastructure and demonstrate strong normalization via logical relations. To our knowledge, this is the first comprehensive confluence and normalization framework for Lean 4.