research-article

Pattern matching without K

Authors:

Dominique Devriese,

Frank PiessensAuthors Info & Claims

ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming

Pages 257 - 268

https://doi.org/10.1145/2628136.2628139

Published: 19 August 2014 Publication History

Abstract

Dependent pattern matching is an intuitive way to write programs and proofs in dependently typed languages. It is reminiscent of both pattern matching in functional languages and case analysis in on-paper mathematics. However, in general it is incompatible with new type theories such as homotopy type theory (HoTT). As a consequence, proofs in such theories are typically harder to write and to understand. The source of this incompatibility is the reliance of dependent pattern matching on the so-called K axiom - also known as the uniqueness of identity proofs - which is inadmissible in HoTT. The Agda language supports an experimental criterion to detect definitions by pattern matching that make use of the K axiom, but so far it lacked a formal correctness proof.

In this paper, we propose a new criterion for dependent pattern matching without K, and prove it correct by a translation to eliminators in the style of Goguen et al. (2006). Our criterion both allows more good definitions than existing proposals, and solves a previously undetected problem in the criterion offered by Agda. It has been implemented in Agda and is the first to be supported by a formal proof. Thus it brings the benefits of dependent pattern matching to contexts where we cannot assume K, such as HoTT. It also points the way to new forms of dependent pattern matching, for example on higher inductive types.

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Cited By

Binder DSkupin ISüberkrüb TOstermann K(2024)Deriving Dependently-Typed OOP from First PrinciplesProceedings of the ACM on Programming Languages10.1145/36498468:OOPSLA1(983-1009)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649846
Coquand T(2023)Some Remarks About Dependent Type TheoryThe French School of Programming10.1007/978-3-031-34518-0_8(175-202)Online publication date: 11-Oct-2023
https://doi.org/10.1007/978-3-031-34518-0_8
Hewer BHutton G(2022)Subtyping Without ReductionMathematics of Program Construction10.1007/978-3-031-16912-0_2(34-61)Online publication date: 22-Sep-2022
https://doi.org/10.1007/978-3-031-16912-0_2
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Index Terms

Pattern matching without K
1. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language features
        Data types and structures
        Patterns
        Recursion
2. Theory of computation
  1. Semantics and reasoning
    1. Program constructs
      1. Functional constructs
      2. Program schemes

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cover image ACM Conferences

ICFP '14: Proceedings of the 19th ACM SIGPLAN international conference on Functional programming

August 2014

390 pages

ISBN:9781450328739

DOI:10.1145/2628136

General Chair:
Johan Jeuring
Universiteit Utrecht, The Netherlands
,
Program Chair:
Manuel M.T. Chakravarty
University of New South Wales, Australia

ACM SIGPLAN Notices Volume 49, Issue 9
ICFP '14
September 2014
361 pages
ISSN:0362-1340
EISSN:1558-1160
DOI:10.1145/2692915
Editors:
Mark W. Bailey
Hamilton College, Clinton, NY
,
Rajeev Balasubramonian
University of Utah
,
Al Davis
University of Utah
,
Sarita Adve
University of Illinois at Urbana-Champ
Issue’s Table of Contents

Copyright © 2014 ACM.

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Published: 19 August 2014

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ICFP'14: ACM SIGPLAN International Conference on Functional Programming

September 1 - 6, 2014

Gothenburg, Sweden

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Binder DSkupin ISüberkrüb TOstermann K(2024)Deriving Dependently-Typed OOP from First PrinciplesProceedings of the ACM on Programming Languages10.1145/36498468:OOPSLA1(983-1009)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649846
Coquand T(2023)Some Remarks About Dependent Type TheoryThe French School of Programming10.1007/978-3-031-34518-0_8(175-202)Online publication date: 11-Oct-2023
https://doi.org/10.1007/978-3-031-34518-0_8
Hewer BHutton G(2022)Subtyping Without ReductionMathematics of Program Construction10.1007/978-3-031-16912-0_2(34-61)Online publication date: 22-Sep-2022
https://doi.org/10.1007/978-3-031-16912-0_2
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