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A recursion combinator for nominal datatypes implemented in Isabelle/HOL

Published: 17 August 2006 Publication History

Abstract

The nominal datatype package implements an infrastructure in Isabelle/HOL for defining languages involving binders and for reasoning conveniently about alpha-equivalence classes. Pitts stated some general conditions under which functions over alpha-equivalence classes can be defined by a form of structural recursion and gave a clever proof for the existence of a primitive-recursion combinator. We give a version of this proof that works directly over nominal datatypes and does not rely upon auxiliary constructions. We further introduce proving tools and a heuristic that made the automation of our proof tractable. This automation is an essential prerequisite for the nominal datatype package to become useful.

References

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H. Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1981.
[2]
S. Berghofer and M. Wenzel. Inductive Datatypes in HOL - Lessons Learned in Formal-Logic Engineering. In Proc. of the 12th International Conference Theorem Proving in Higher Order Logics (TPHOLs), number 1690 in LNCS, pages 19-36, 1999.
[3]
M. J. Gabbay and A. M. Pitts. A New Approach to Abstract Syntax Involving Binders. In Logic in Computer Science, pages 214-224. IEEE Computer Society Press, 1999.
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M. Gordon. From LCF to HOL: a short history. In G. Plotkin, C. P. Stirling, and M. Tofte, editors, Proof, Language, and Interaction, pages 169-186. MIT Press, 2000.
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P. Homeier. A Design Structure for Higher Order Quotients. In Proc. of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 3603 of LNCS, pages 130-146, 2005.
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T. Melham. Automating Recursive Type Definitions in Higher Order Logic. In G. Birtwistle and P. A. Subrahmanyam, editors, Current Trends in Hardware Verification and Automated Theorem Proving, pages 341-386. Springer-Verlag, 1989.
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A. M. Pitts. Nominal Logic, A First Order Theory of Names and Binding. Information and Computation, 186:165-193, 2003.
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A. M. Pitts. Alpha-Structural Recursion and Induction (Extended Abstract). In Proc. of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 3603 of LNCS, pages 17-34, 2005.
[9]
A. M. Pitts. Alpha-Structural Recursion and Induction. Journal of the ACM, 200X. to appear.
[10]
C. Urban and M. Norrish. A Formal Treatment of the Barendregt Variable Convention in Rule Inductions. In Proc. of the 3rd International ACM Workshop on Mechanized Reasoning about Languages with Variable Binding and Names, pages 25-32, 2005.
[11]
C. Urban and C. Tasson. Nominal Techniques in Isabelle/HOL. In Proc. of the 20th International Conference on Automated Deduction (CADE), volume 3632 of LNCS, pages 38-53, 2005.
[12]
M. Wenzel. Using Axiomatic Type Classes in Isabelle. Manual in the Isabelle distribution.

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  1. A recursion combinator for nominal datatypes implemented in Isabelle/HOL

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    cover image Guide Proceedings
    IJCAR'06: Proceedings of the Third international joint conference on Automated Reasoning
    August 2006
    679 pages
    ISBN:3540371877
    • Editors:
    • Ulrich Furbach,
    • Natarajan Shankar

    Sponsors

    • NEC
    • cadence: cadence
    • Microsoft Research: Microsoft Research
    • IBM: IBM

    Publisher

    Springer-Verlag

    Berlin, Heidelberg

    Publication History

    Published: 17 August 2006

    Author Tags

    1. lambda-calculus
    2. nominal logic
    3. primitive recursion
    4. proof assistants

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    • (2024)Nominal Recursors as Epi-RecursorsProceedings of the ACM on Programming Languages10.1145/36328578:POPL(425-456)Online publication date: 5-Jan-2024
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    • (2022)Rensets and Renaming-Based Recursion for Syntax with BindingsAutomated Reasoning10.1007/978-3-031-10769-6_36(618-639)Online publication date: 8-Aug-2022
    • (2019)Bindings as bounded natural functorsProceedings of the ACM on Programming Languages10.1145/32903353:POPL(1-34)Online publication date: 2-Jan-2019
    • (2011)Recursion principles for syntax with bindings and substitutionProceedings of the 16th ACM SIGPLAN international conference on Functional programming10.1145/2034773.2034819(346-358)Online publication date: 19-Sep-2011
    • (2011)Recursion principles for syntax with bindings and substitutionACM SIGPLAN Notices10.1145/2034574.203481946:9(346-358)Online publication date: 19-Sep-2011
    • (2011)Reasoning about constants in nominal isabelle or how to formalize the second fixed point theoremProceedings of the First international conference on Certified Programs and Proofs10.1007/978-3-642-25379-9_9(87-102)Online publication date: 7-Dec-2011
    • (2010)Nominal system TACM SIGPLAN Notices10.1145/1707801.170632145:1(159-170)Online publication date: 17-Jan-2010
    • (2010)Nominal system TProceedings of the 37th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages10.1145/1706299.1706321(159-170)Online publication date: 17-Jan-2010
    • (2009)A Simple Nominal Type TheoryElectronic Notes in Theoretical Computer Science (ENTCS)10.1016/j.entcs.2008.12.115228(37-52)Online publication date: 1-Jan-2009
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