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-L"of 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 $\Pi$ to generalize both the parametric quantifier $\forall$ and the type former $\to$ which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers $\forall$ and $\exists$ alongside the existing $\Pi$ and $\Sigma$, 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.
Wed 6 SepDisplayed time zone: Belfast change
10:30 - 12:00
|A Specification for Dependent Types in Haskell|
Stephanie Weirich University of Pennsylvania, USA, Antoine Voizard University of Pennsylvania, USA, Pedro Henrique Azevedo de Amorim Ecole Polytechnique, n.n. / University of Campinas, Brazil, Richard A. Eisenberg Bryn Mawr College, USADOI
|Parametric Quantifiers for Dependent Type Theory|
Andreas Nuyts KU Leuven, Belgium, Andrea Vezzosi Chalmers University of Technology, Sweden, Dominique Devriese KU Leuven, BelgiumDOI
|Normalization by Evaluation for Sized Dependent Types|
Andreas Abel University of Gothenburg, Sweden, Andrea Vezzosi Chalmers University of Technology, Sweden, Theo Winterhalter ENS Paris-Saclay, FranceDOI
|A Metaprogramming Framework for Formal Verification|
Gabriel Ebner Vienna University of Technology, Austria, Sebastian Ullrich KIT, Germany, Jared Roesch University of Washington, USA, Jeremy Avigad Carnegie Mellon University, USA, Leonardo de Moura Microsoft Research, n.n.DOI