DeepAI

# Cubical Syntax for Reflection-Free Extensional Equality

We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-Löf's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel extension of the logical families or categorical gluing argument inspired by Coquand and Shulman: every closed element of boolean type is derivably equal to either 'true' or 'false'.

• 19 publications
• 7 publications
• 12 publications
03/03/2020

### A Cubical Language for Bishop Sets

We present XTT, a version of Cartesian cubical type theory specialized f...
07/17/2019

### Typal Heterogeneous Equality Types

The usual homogeneous form of equality type in Martin-Löf Type Theory co...
05/02/2022

02/06/2023

### Algebraic Semantics of Datalog with Equality

We discuss the syntax and semantics of relational Horn logic (RHL) and p...
01/17/2019

### Path Spaces of Higher Inductive Types in Homotopy Type Theory

The study of equality types is central to homotopy type theory. Characte...
10/02/2020

### On the Nielsen-Schreier Theorem in Homotopy Type Theory

We give a formulation of the Nielsen-Schreier theorem (subgroups of free...
02/10/2023

### Strictly Associative and Unital ∞-Categories as a Generalized Algebraic Theory

We present the first definition of strictly associative and unital ∞-cat...