Constructing (Co)inductive Types via Large Sizes

TL;DR

This paper introduces a new approach to constructing inductive and coinductive types using a large size type in intensional type theory, enhancing the expressiveness and consistency of sized types.

Contribution

It extends type theory with a large size type and parametric quantifiers, enabling the construction of both inductive and coinductive types, including finitely-branching types, with a consistency proof.

Findings

Constructs inductive and coinductive types in extended type theory.

Provides a consistency proof via an impredicative realisability model.

Improves on previous work by handling finitely-branching types.

Abstract

To ensure decidability and consistency of its type theory, a proof assistant should only accept terminating recursive functions and productive corecursive functions. Most proof assistants enforce this through syntactic conditions, which can be restrictive and non-modular. Sized types are a type-based alternative where (co)inductive types are annotated with additional size information. Well-founded induction on sizes can then be used to prove termination and productivity. An implementation of sized types exists in Agda, but it is currently inconsistent due to the addition of a largest size. We investigate an alternative approach, where intensional type theory is extended with a large type of sizes and parametric quantifiers over sizes. We show that inductive and coinductive types can be constructed in this theory, which improves on earlier work where this was only possible for the finitely-branching inductive types. The consistency of the theory is justified by an impredicative realisability model, which interprets the type of sizes as an uncountable ordinal.