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Effect Polymorphism in Higher-Order Logic (Proof Pearl)

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Interactive Theorem Proving (ITP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10499))

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Abstract

The notion of a monad cannot be expressed within higher-order logic (HOL) due to type system restrictions. We show that if a monad is used with values of only one type, this notion can be formalised in HOL. Based on this idea, we develop a library of effect specifications and implementations of monads and monad transformers. Hence, we can abstract over the concrete monad in HOL definitions and thus use the same definition for different (combinations of) effects. We illustrate the usefulness of effect polymorphism with a monadic interpreter.

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Notes

  1. 1.

    Type variables that appear in the signature of locale parameters are fixed for the whole locale. In particular, the value type cannot be instantiated inside the locale or its extension . The interpreter , however, returns s. For this reason, is defined in an extension of that merely specialises . For readability, we usually omit this detail in this paper.

  2. 2.

    Such environments can be nicely handled by applying a reader monad transformer on top (Sect. 4).

  3. 3.

    Isabelle’s feature, which resolves overloading during type checking, cannot be used either as it does not support recursive resolutions. For example, resolving takes two steps: first to and then to . The second step fails due to the intricate interleaving of type checking and resolution. Even if this is just an implementation issue, resolving overloading during type checking prevents definitions that are generic in the monad, which general overloading supports.

  4. 4.

    Following the “as abstract as possible” spirit of this paper, we actually proved the identities in the locale of commutative monads and showed that is commutative if its inner monad is.

  5. 5.

    http://www.infsec.ethz.ch/research/projects/FCSPI.html.

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Acknowledgements

We thank Dmitriy Traytel and the anonymous reviewers for suggesting many improvements to the presentation. This work is supported by the Swiss National Science Foundation grant 153217 “Formalising Computational Soundness for Protocol Implementations”.

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Authors and Affiliations

  1. Institute of Information Security, Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Andreas Lochbihler

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  1. Andreas Lochbihler
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Correspondence to Andreas Lochbihler .

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Editors and Affiliations

  1. University of Brasília, Brasília D.F., Brazil

    Mauricio Ayala-Rincón

  2. NASA, Hampton, VA, USA

    César A. Muñoz

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Lochbihler, A. (2017). Effect Polymorphism in Higher-Order Logic (Proof Pearl). In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_25

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  • DOI: https://doi.org/10.1007/978-3-319-66107-0_25

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