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Monotone recursive types and recursive data representations in Cedille

Published online by Cambridge University Press:  10 December 2021

Christopher Jenkins*
Affiliation:
Computer Science, 14 MacLean Hall, The University of Iowa, Iowa City, IA, USA
Aaron Stump
Affiliation:
Computer Science, 14 MacLean Hall, The University of Iowa, Iowa City, IA, USA
*
*Corresponding author. Email: christopher-jenkins@uiowa.edu
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Abstract

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Guided by Tarksi’s fixpoint theorem in order theory, we show how to derive monotone recursive types with constant-time roll and unroll operations within Cedille, an impredicative, constructive, and logically consistent pure typed lambda calculus. This derivation takes place within the preorder on Cedille types induced by type inclusions, a notion which is expressible within the theory itself. As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding. For both encodings, we prove induction and examine the computational and extensional properties of their destructor, iterator, and primitive recursor in Cedille. For our Scott encoding in particular, we translate into Cedille a construction due to Lepigre and Raffalli (2019) that equips Scott naturals with primitive recursion, then extend this construction to derive a generic induction principle. This allows us to give efficient and provably unique (up to function extensionality) solutions for the iteration and primitive recursion schemes for Scott-encoded data.

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Paper
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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