Gradualizing the Calculus of Inductive Constructions

TL;DR

This paper explores gradual variations of the Calculus of Inductive Constructions, analyzing tradeoffs and proposing a parametrized framework with formal metatheory to enable more flexible dependently-typed programming and proof development.

Contribution

It introduces a parametrized Gradual CIC framework with three variations, formalizes their metatheory, and studies their gradual guarantees and semantic models.

Findings

Identifies a tradeoff between graduality and normalization in CIC.

Develops a bidirectional elaboration to CastCIC for better understanding.

Establishes normalization and gradual guarantees for the proposed variations.

Abstract

We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys. Beyond this Fire Triangle of Graduality, we explore the gradualization of CIC with three different compromises, each relaxing one edge of the Fire Triangle. We develop a parametrized presentation of Gradual CIC (GCIC) that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of GCIC to a dependently-typed cast calculus, CastCIC, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model of CastCIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We study the static and dynamic gradual guarantees as well as the stronger notion of graduality with embedding-projection pairs formulated by New and Ahmed, using appropriate semantic model constructions. This work informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.