Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study

TL;DR

This paper formalizes a normalization proof for Leivant's Predicative System F using hereditary substitutions and the Equations package in Coq, advancing the understanding of type quantification and reflection in dependently-typed languages.

Contribution

It introduces a novel hereditary substitution method for Leivant's original system, formalized in Coq with the Equations package, enabling clearer algorithmic understanding and reflection techniques.

Findings

Normalization proof for Leivant's system achieved

Hereditary substitution explicitly defined and verified

Enlarged class of languages for reflection methods

Abstract

This paper presents a case study of formalizing a normalization proof for Leivant's Predicative System F using the Equations package. Leivant's Predicative System F is a stratified version of System F, where type quantification is annotated with kinds representing universe levels. A weaker variant of this system was studied by Stump & Eades, employing the hereditary substitution method to show normalization. We improve on this result by showing normalization for Leivant's original system using hereditary substitutions and a novel multiset ordering on types. Our development is done in the Coq proof assistant using the Equations package, which provides an interface to define dependently-typed programs with well-founded recursion and full dependent pattern- matching. Equations allows us to define explicitly the hereditary substitution function, clarifying its algorithmic behavior in presence of term and type substitutions. From this definition, consistency can easily be derived. The algorithmic nature of our development is crucial to reflect languages with type quantification, enlarging the class of languages on which reflection methods can be used in the proof assistant.