Signatures and Induction Principles for Higher Inductive-Inductive Types

TL;DR

This paper introduces a formal framework for higher inductive-inductive types (HIITs), providing definitions, induction principles, and a formalized implementation in Agda and Haskell, advancing the understanding of complex type constructions.

Contribution

It proposes a general definition of HIITs via the theory of signatures, and develops syntactic translations for induction and recursion, with formalization and implementation in Agda and Haskell.

Findings

Defined a formal theory of signatures for HIITs

Computed induction and recursion principles for HIITs

Implemented a formalization and translation tool in Agda and Haskell

Abstract

Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code.