scholarly journals Representing Continuous Functions between Greatest Fixed Points of Indexed Containers

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Pierre Hyvernat
Keyword(s):  
Fixed Points ◽  
Type Theory ◽  
Continuous Functions ◽  
Dependent Type Theory ◽  
Recursive Definitions ◽  
Computable Functions ◽  
Dependent Type

We describe a way to represent computable functions between coinductive types as particular transducers in type theory. This generalizes earlier work on functions between streams by P. Hancock to a much richer class of coinductive types. Those transducers can be defined in dependent type theory without any notion of equality but require inductive-recursive definitions. Most of the properties of these constructions only rely on a mild notion of equality (intensional equality) and can thus be formalized in the dependently typed language Agda.

scholarly journals Constructing Infinitary Quotient-Inductive Types

2020 ◽  
pp. 257-276
Author(s):  
Marcelo P. Fiore ◽  
Andrew M. Pitts ◽  
S. C. Steenkamp
Keyword(s):  
Type Theory ◽  
Theorem Prover ◽  
Dependent Type Theory ◽  
Inductive Definitions ◽  
Inductive Types ◽  
Recursive Definitions ◽  
Dependent Type

AbstractThis 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 inductive-inductive definitions involving strictly positive occurrences of Hofmann-style quotient types, and Abel’s size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem prover.

scholarly journals A relationally parametric model of dependent type theory

2014 ◽  
Vol 49 (1) ◽  
pp. 503-515 ◽  
Author(s):  
Robert Atkey ◽  
Neil Ghani ◽  
Patricia Johann
Keyword(s):  
Type Theory ◽  
Parametric Model ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Implementing a modal dependent type theory

2019 ◽  
Vol 3 (ICFP) ◽  
pp. 1-29 ◽  
Author(s):  
Daniel Gratzer ◽  
Jonathan Sterling ◽  
Lars Birkedal
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Guarded Dependent Type Theory with Coinductive Types

2016 ◽  
pp. 20-35 ◽  
Author(s):  
Aleš Bizjak ◽  
Hans Bugge Grathwohl ◽  
Ranald Clouston ◽  
Rasmus E. Møgelberg ◽  
Lars Birkedal
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Parametric quantifiers for dependent type theory

2017 ◽  
Vol 1 (ICFP) ◽  
pp. 1-29 ◽  
Author(s):  
Andreas Nuyts ◽  
Andrea Vezzosi ◽  
Dominique Devriese
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

Introduction to the Special Issue on Dependent Type Theory Meets Practical Programming

2004 ◽  
Vol 14 (1) ◽  
pp. 1-2
Author(s):  
GILLES BARTHE ◽  
PETER DYBJEN ◽  
PETER THIEMANN
Keyword(s):  
Programming Languages ◽  
Type Theory ◽  
Type System ◽  
Type Systems ◽  
Special Issue ◽  
Dependent Type Theory ◽  
Dependent Type

Modern programming languages rely on advanced type systems that detect errors at compile-time. While the benefits of type systems have long been recognized, there are some areas where the standard systems in programming languages are not expressive enough. Language designers usually trade expressiveness for decidability of the type system. Some interesting programs will always be rejected (despite their semantical soundness) or be assigned uninformative types.

scholarly journals Univalent categories and the Rezk completion

2015 ◽  
Vol 25 (5) ◽  
pp. 1010-1039 ◽  
Author(s):  
BENEDIKT AHRENS ◽  
KRZYSZTOF KAPULKIN ◽  
MICHAEL SHULMAN
Keyword(s):  
Category Theory ◽  
Type Theory ◽  
Dependent Type Theory ◽  
Equivalence Of Categories ◽  
Segal Spaces ◽  
Definition Of ◽  
Categorical Semantics ◽  
Univalent Foundations ◽  
Dependent Type

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.

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