Gradualizing the Calculus of Inductive Constructions

Abstract

Acknowledging the ordeal of a fully formal development in a proof assistant such as Coq, 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 that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of Gradual CIC to a dependently-typed cast calculus, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model into CIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We also study the stronger notion of graduality as embeddingprojection 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.

Approximate Normalization and Eager Equality Checking for Gradual Inductive Families

“…GrIf does not have to specify its dynamic semantics: this was precisely why we defined CastIf. Following Lennon-Bertrand et al [2020], we express the typing of GrIf terms in terms of CastIf types. To type applications of dependent functions, we elaborate GrIf terms into CastIf, which are substituted into the appropriate types.…”

“…Conversely, CIC has an untyped reduction semantics by which we can easily show a simulation of CastIf's reduction steps. The model uses a similar technique as Lennon-Bertrand et al [2020], but it serves as the foundation for our novel approach to equality ( §8).…”

“…El ℓ turns these descriptions of types into actual types, but because types may depend on values, the definition of the inductive type Code refers to the recursive function El ℓ that matches on Code values (thus the name inductive-recursive type). This technique is standard for dependent type introspection [Boulier et al 2017;Lennon-Bertrand et al 2020]. As a notational shorthand in Fig.…”

“…Finally, we add an rule that Γ ⊢ D t −→ ℧ C t ′ : C t ′ whenever t −→ ℧ T . So casts between types with incompatible indices produce ℧ immediately, unlike GCIC [Lennon-Bertrand et al 2020] where inconsistencies are only discovered during elimination.…”

“…We must also account for the possibility of dynamic type errors during type-checking. GrInd builds on two existing languages with different approaches to this challenge: GDTL [Eremondi et al 2019] and GCIC [Lennon-Bertrand et al 2020].…”