Abstract

Inductive-inductive types are a joint generalization of mutual inductive types and indexed inductive types. In extensional type theory, inductive-inductive types can be constructed from inductive types, and this construction has been conjectured to work in intensional type theory as well. In this paper, we show that the existing construction requires Uniqueness of Identity Proofs, and present a new construction (which we conjecture generalizes) of one particular inductive-inductive type in cubical type theory, which is compatible with homotopy type theory.

Highlights

  • Inductive-inductive types allow for the mutual inductive definition of a type and a family over that type

  • We first show that this construction does not work in intensional type theory without assuming Uniqueness of Identity Proofs (UIP), which is incompatible with the Univalence axiom of Homotopy Type Theory [18]

  • We give an alternate construction in cubical type theory [6], which is compatible with Univalence

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Summary

Introduction

Inductive-inductive types allow for the mutual inductive definition of a type and a family over that type. Such definitions have been used for example by Danielsson [9] and Chapman [5] to define intrinsically typed syntax of a dependent type theory, and Agda supports such definitions natively These types have been studied extensively in Nordvall Forsberg [15]. 2, we show that, in intensional type theory, if the types constructed by Nordvall Forsberg satisfy the simple elimination rules, UIP holds (formalized in both Coq and Agda). 3, we give the construction of a particular inductive-inductive type with simple elimination rules in cubical type theory (formalized in cubical Agda)

Syntax and Conventions
Running Example of an Inductive-Inductive Definition
Deriving UIP
Unique Goodness Implies UIP
Simple Elimination Rules Imply Unique Goodness
Simple Elimination Rules for Nordvall Forsberg’s Construction only if UIP
Constructing an Inductive-Inductive Type in Cubical Type Theory
Pre-syntax
Goodness Algebras
Niceness
Successor Goodness Algebra
Related Work
Conclusions and Future Work
The Coq Development Team
18. The Univalent Foundations Program: Homotopy Type Theory
Full Text
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