Multi-level Contextual Type Theory

TL;DR

This paper generalizes contextual type theory to n levels, introducing a uniform framework with variables indexed by levels, enabling reasoning about higher-order unification and proof holes with a decidable type system.

Contribution

It extends contextual type theory to arbitrary levels, unifying various variable types into a single indexed notion and providing a decidable type system for beta-eta-normal forms.

Findings

Unified variable indexing for multiple levels

Decidable bi-directional type system established

Formal system supports higher-order reasoning

Abstract

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.