Article

Deep Induction: Induction Rules for (Truly) Nested Types

Authors:

Patricia Johann,

Andrew PolonskyAuthors Info & Claims

Foundations of Software Science and Computation Structures: 23rd International Conference, FOSSACS 2020, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020, Dublin, Ireland, April 25–30, 2020, Proceedings

Pages 339 - 358

https://doi.org/10.1007/978-3-030-45231-5_18

Published: 25 April 2020 Publication History

Abstract

This paper introduces deep induction, and shows that it is the notion of induction most appropriate to nested types and other data types defined over, or mutually recursively with, (other) such types. Standard induction rules induct over only the top-level structure of data, leaving any data internal to the top-level structure untouched. By contrast, deep induction rules induct over all of the structured data present. We give a grammar generating a robust class of nested types (and thus ADTs), and develop a fundamental theory of deep induction for them using their recently defined semantics as fixed points of accessible functors on locally presentable categories. We then use our theory to derive deep induction rules for some common ADTs and nested types, and show how these rules specialize to give the standard structural induction rules for these types. We also show how deep induction specializes to solve the long-standing problem of deriving principled and practically useful structural induction rules for bushes and other truly nested types. Overall, deep induction opens the way to making induction principles appropriate to richly structured data types available in programming languages and proof assistants. Agda implementations of our development and examples, including two extended case studies, are available.

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Cited By

Grégoire BLéchenet JTassi EKrebbers RTraytel DPientka BZdancewic S(2023)Practical and Sound Equality Tests, Automatically: Deriving eqType Instances for Jasmin’s Data Types with Coq-ElpiProceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3573105.3575683(167-181)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3573105.3575683
Johann PGhiorzi EPopescu AZdancewic S(2022)(Deep) induction rules for GADTsProceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3497775.3503680(324-337)Online publication date: 17-Jan-2022
https://dl.acm.org/doi/10.1145/3497775.3503680

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cover image Guide Proceedings

Foundations of Software Science and Computation Structures: 23rd International Conference, FOSSACS 2020, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020, Dublin, Ireland, April 25–30, 2020, Proceedings

Apr 2020

656 pages

ISBN:978-3-030-45230-8

DOI:10.1007/978-3-030-45231-5

Editors:
Jean Goubault-Larrecq
ENS/CNRS, Université Paris-Saclay, Cachan, France
,
Barbara König
University of Duisburg-Essen, Duisburg, Germany

© The Author(s) 2020.

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 25 April 2020

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Cited By

Grégoire BLéchenet JTassi EKrebbers RTraytel DPientka BZdancewic S(2023)Practical and Sound Equality Tests, Automatically: Deriving eqType Instances for Jasmin’s Data Types with Coq-ElpiProceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3573105.3575683(167-181)Online publication date: 11-Jan-2023
https://dl.acm.org/doi/10.1145/3573105.3575683
Johann PGhiorzi EPopescu AZdancewic S(2022)(Deep) induction rules for GADTsProceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3497775.3503680(324-337)Online publication date: 17-Jan-2022
https://dl.acm.org/doi/10.1145/3497775.3503680

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