Complete Bidirectional Typing for the Calculus of Inductive Constructions

TL;DR

This paper introduces a bidirectional type system for the Calculus of Inductive Constructions, featuring a new constrained inference judgement that enhances understanding and proof strategies for CIC's properties.

Contribution

It proposes a novel bidirectional type system with constrained inference for CIC, formally proven to be complete and useful for proof and extension design.

Findings

Formal proof of completeness in Coq

Enhanced insights for CIC property proofs

Framework for designing CIC variations

Abstract

This article presents a bidirectional type system for the Calculus of Inductive Constructions (CIC). It introduces a new judgement intermediate between the usual inference and checking, dubbed constrained inference, to handle the presence of computation in types. The key property of the system is its completeness with respect to the usual undirected one, which has been formally proven in Coq as a part of the MetaCoq project. Although it plays an important role in an ongoing completeness proof for a realistic typing algorithm, the interest of bidirectionality is wider, as it gives insights and structure when trying to prove properties on CIC or design variations and extensions. In particular, we put forward constrained inference, an intermediate between the usual inference and checking judgements, to handle the presence of computation in types.