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Containers, monads and induction recursion

Published online by Cambridge University Press:  20 November 2014

NEIL GHANI  and
PETER HANCOCK
NEIL GHANI
Affiliation:
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XH, Scotland Email: ng@cis.strath.ac.uk, hancock@spamcop.net
PETER HANCOCK
Affiliation:
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XH, Scotland Email: ng@cis.strath.ac.uk, hancock@spamcop.net
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Abstract

Induction recursion offers the possibility of a clean, simple and yet powerful meta-language for the type system of a dependently typed programming language. At its crux, induction recursion allows us to define a universe, that is a set U of codes and a decoding function T : U → D which assigns to every code u : U, a value T, u of some type D, e.g. the large type Set of small types or sets. The name induction recursion refers to the build-up of codes in U using inductive clauses, simultaneously with the definition of the function T, by structural recursion on codes.

Our contribution is to (i) bring out explicitly algebraic structure which is less visible in the original type-theoretic presentation – in particular showing how containers and monads play a pivotal role within induction recursion; and (ii) use these structures to present a clean and high level definition of induction recursion suitable for use in functional programming.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 26 , Special Issue 1: Special Issue: Dependently Typed Programming , January 2016 , pp. 89 - 113
Copyright
Copyright © Cambridge University Press 2014 

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