Related papers: Impredicativity in Linear Dependent Type Theory
Logical frameworks can be used to translate proofs from a proof system to another one. For this purpose, we should be able to encode the theory of the proof system in the logical framework. The Lambda Pi calculus modulo theory is one of…
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…
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…
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a…
We present a type theory dealing with non-linear, "ordinary" dependent types (which we will call cartesian) and linear types, where both constructs may depend on terms of the former. In the interplay between these, we find new type formers…
This is the second in a series of papers extending Martin-L\"{o}f's meaning explanation of dependent type theory to account for higher-dimensional types. We build on the cubical realizability framework for simple types developed in Part I,…
We describe a realizability framework for classical first-order logic in which realizers live in (a model of) typed {\lambda}{\mu}-calculus. This allows a direct interpretation of classical proofs, avoiding the usual negative translation to…
This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems…
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…
We present a type theory combining both linearity and dependency by stratifying typing rules into a level for logics and a level for programs. The distinction between logics and programs decouples their semantics, allowing the type system…
The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the…
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…
In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of…
This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of…
We present a novel dependent linear type theory in which the multiplicity of some variable-i.e., the number of times the variable can be used in a program-can depend on other variables. This allows us to give precise resource annotations to…
We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We define an extension of lambda-calculus with dependents types that enables us to encode transparent and opaque probabilistic programs and prove a strong normalisation result for it by a reducibility technique. While transparent…
Bernardy et al. [2018] proposed a linear type system $\lambda^q_\to$ as a core type system of Linear Haskell. In the system, linearity is represented by annotated arrow types $A \to_m B$, where $m$ denotes the multiplicity of the argument.…
This paper continues the series of papers that develop a new approach to syntax and semantics of dependent type theories. Here we study the interpretation of the rules of the identity types in the intensional Martin-Lof type theories on the…