Parametric quantifiers for dependent type theory
Paper i proceeding, 2017

Polymorphic type systems such as System F enjoy the parametricity property: polymorphic functions cannot inspect their type argument and will therefore apply the same algorithm to any type they are instantiated on. This idea is formalized mathematically in Reynolds’s theory of relational parametricity, which allows the metatheoretical derivation of parametricity theorems about all values of a given type. Although predicative System F embeds into dependent type systems such as Martin-Löf Type Theory (MLTT), parametricity does not carry over as easily. The identity extension lemma, which is crucial if we want to prove theorems involving equality, has only been shown to hold for small types, excluding the universe. We attribute this to the fact that MLTT uses a single type former Πto generalize both the parametric quantifier ∀ and the type former → which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers ∀ and ∃ alongside the existing Πand Σ, and provide relation type formers for proving parametricity theorems internally. We show internally the existence of initial algebras and final co-algebras of indexed functors both by Church encoding and, for a large class of functors, by using sized types. We prove soundness of our type system by enhancing existing iterated reflexive graph (cubical set) models of dependently typed parametricity by distinguishing between edges that express relatedness of objects (bridges) and edges that express equality (paths). The parametric functions are those that map bridges to paths. We implement an extension to the Agda proof assistant that type-checks proofs in our type system.

presheaf semantics

cubical type theory

Agda

sized types

Parametricity

Författare

Andreas Nuyts

Andrea Vezzosi

Chalmers, Data- och informationsteknik, Datavetenskap

Dominique Devriese

Proceedings of the ACM on Programming Languages

2475-1421 (ISSN)

Vol. 1 32:1--32:29-

Fundament

Grundläggande vetenskaper

Ämneskategorier

Datavetenskap (datalogi)

DOI

10.1145/3110276