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Representing Nested Inductive Types using W-types

Summary: Representing Nested Inductive Types
using W-types
Michael Abbott1, Thorsten Altenkirch2, and Neil Ghani1
1 Department of Mathematics and Computer Science, University of Leicester
michael@araneidae.co.uk, ng13@mcs.le.ac.uk
2 School of Computer Science and Information Technology, Nottingham University
Abstract. We show that strictly positive inductive types, constructed from
polynomial functors, constant exponentiation and arbitrarily nested inductive
types exist in any Martin-Lof category (extensive locally cartesian closed
category with W-types) by exploiting our work on container types. This
generalises a result by Dybjer (1997) who showed that non-nested strictly positive
inductive types can be represented using W-types. We also provide a detailed
analysis of the categorical infrastructure needed to establish the result.
1 Introduction
Inductive types play a central role in programming and constructive reasoning. From an
intuitionistic point of view we can understand strictly positive inductive types (SPITs)
as well-founded trees, which may be infinitely branching. The language of SPITs is
built from polynomial types and exponentials, enriched by a constructor for inductive
types. In this language we can conveniently construct familiar types such as the natural


Source: Altenkirch, Thorsten - School of Computer Science, University of Nottingham


Collections: Computer Technologies and Information Sciences