scholarly journals Implementation of Two Layers Type Theory in Dedukti and Application to Cubical Type Theory

2021 ◽  
Vol 332 ◽  
pp. 54-67
Author(s):  
Bruno Barras ◽  
Valentin Maestracci
Keyword(s):  
Type Theory ◽  
Cubical Type Theory

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

scholarly journals Guarded Cubical Type Theory

2018 ◽  
Vol 63 (2) ◽  
pp. 211-253 ◽  
Author(s):  
Lars Birkedal ◽  
Aleš Bizjak ◽  
Ranald Clouston ◽  
Hans Bugge Grathwohl ◽  
Bas Spitters ◽  
...  
Keyword(s):  
Type Theory ◽  
Cubical Type Theory

scholarly journals Internal Parametricity for Cubical Type Theory

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Evan Cavallo ◽  
Robert Harper
Keyword(s):  
Type Theory ◽  
Formal Interface ◽  
Inductive Types ◽  
Cubical Type Theory

We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, observe how cubical equality regularizes parametric type theory, and examine the similarities and discrepancies between cubical and parametric type theory, which are closely related. We also abstract a formal interface to the computational interpretation and show that this also has a presheaf model.

scholarly journals Canonicity for Cubical Type Theory

2018 ◽  
Vol 63 (2) ◽  
pp. 173-210 ◽  
Author(s):  
Simon Huber
Keyword(s):  
Type Theory ◽  
Cubical Type Theory

Type Theory and Formal Proof

2009 ◽  
Author(s):  
Rob Nederpelt ◽  
Herman Geuvers
Keyword(s):  
Type Theory ◽  
Formal Proof
Sign in / Sign up

Export Citation Format

Share Document