Related papers: From type theory to setoids and back
We present Tores, a core language for encoding metatheoretic proofs. The novel features we introduce are well-founded Mendler-style (co)recursion over indexed data types and a form of recursion over objects in the index language to build…
We investigate existentially closed models (of a quite arbitrary theory) equipped which an action of a fixed group G. We embed these structures in a monster model D of some well-rounded theory and describe them as PAC substructures of D.…
We present a formalization of a version of Abadi and Plotkin's logic for parametricity for a polymorphic dual intuitionistic/linear type theory with fixed points, and show, following Plotkin's suggestions, that it can be used to define a…
We introduce a novel notion of pasting shapes for iterated Segal spaces which classify particular arrangements of composing cells in d-uple Segal spaces. Using this formalism, we then continue to prove a pasting theorem for these iterated…
We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler…
Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical,…
Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad…
In functional programming languages, generalized algebraic data types (GADTs) are very useful as the unnecessary pattern matching over them can be ruled out by the failure of unification of type arguments. In dependent type systems, this is…
Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can…
We propose a new type-theoretic approach to SLD-resolution and Horn-clause logic programming. It views Horn formulas as types, and derivations for a given query as a construction of the inhabitant (a proof-term) for the type given by the…
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image…
Dependently-typed proof assistants furnish expressive foundations for mechanised mathematics and verified software. However, automation for these systems has been either modest in scope or complex in implementation. We aim to improve the…
Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To…
This is the fourth in a series of papers extending Martin-L\"of's meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of…
In functional programming, datatypes a la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof…
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
For each n\geq 1 we introduce two new Segal-type models of n-types of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…