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**Harvard**

Nuyts, A., Vezzosi, A. och Devriese, D. (2017) *Parametric quantifiers for dependent type theory*.

** BibTeX **

@conference{

Nuyts2017,

author={Nuyts, Andreas and Vezzosi, Andrea and Devriese, Dominique},

title={Parametric quantifiers for dependent type theory},

booktitle={Proceedings of the ACM on Programming Languages},

pages={32:1--32:29},

abstract={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 Reynoldsâs 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-LÃ¶f 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 Î to generalize both the parametric quantifier â and the type former â which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers â and â alongside the existing Î and Î£, 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.},

year={2017},

keywords={Agda, Parametricity, cubical type theory, presheaf semantics, sized types},

}

** RefWorks **

RT Conference Proceedings

SR Electronic

ID 252073

A1 Nuyts, Andreas

A1 Vezzosi, Andrea

A1 Devriese, Dominique

T1 Parametric quantifiers for dependent type theory

YR 2017

T2 Proceedings of the ACM on Programming Languages

SP 321

AB 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 Reynoldsâs 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-LÃ¶f 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 Î to generalize both the parametric quantifier â and the type former â which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers â and â alongside the existing Î and Î£, 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.

LA eng

DO 10.1145/3110276

LK http://doi.acm.org/10.1145/3110276

LK http://publications.lib.chalmers.se/records/fulltext/252073/local_252073.pdf

OL 30