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Equational Reasoning with Applicative Functors

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Interactive Theorem Proving (ITP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9807))

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Abstract

In reasoning about effectful computations, it often suffices to focus on the effect-free parts. We present a package for automatically lifting equations to effects modelled by applicative functors. It exploits properties of the concrete functor thanks to a modular classification based on combinators. We formalise the meta theory and demonstrate the usability of our Isabelle/HOL package with two case studies. This is a first step towards practical reasoning with effectful computations.

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Notes

  1. 1.

    As lifting wraps the types of free variables in F, it does not look into abstractions, but treats them like constants. For example, is lifted to rather than . Thus, lifting effectively operates on first-order terms.

  2. 2.

    Hinze briefly considers functors with the combinators S and C and notes that the case with only the combinator C might be interesting, too, but omits the details.

  3. 3.

    The following shows the equivalence (bold face denotes doubling and underlining dropping of a repetition): .

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Acknowledgements

Peter Gammie triggered our interest in reasoning about applicative functors and helped with the Stern-Brocot tree. We thank Dmitriy Traytel, Joachim Breitner, and the anonymous reviewers for suggesting many textual improvements. The first author was supported by SNSF 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, Zürich, Switzerland

    Andreas Lochbihler & Joshua Schneider

Authors
  1. Andreas Lochbihler
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  2. Joshua Schneider
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Corresponding authors

Correspondence to Andreas Lochbihler or Joshua Schneider .

Editor information

Editors and Affiliations

  1. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Stephan Merz

A Definitions of Applicative Functors

A Definitions of Applicative Functors

This appendix lists Isabelle/HOL definitions for the idioms mentioned in this paper. The definitional packages and their syntaxes are documented in the Isabelle/Isar reference manual. The proofs of the applicative laws and combinators are available online [23].

Environment (Reader)

figure h

Stream

figure i

Infinite binary tree

figure j

Non-standard numbers as used in non-standard analysis in Isabelle/HOL [17]. The type \(\alpha \) star is the quotient of the environment idiom (\(\alpha \), nat) env over equality in some free ultrafilter \(\mathcal {U}\) on nat.

figure k

Option

figure l

Zip list

figure m

Probability

figure n

Subprobability

figure o

Set

figure p

Non-empty set

figure q

Monoid, commutative monoid, idempotent monoid

figure r

The type variable \(\alpha \) must have sort monoid-add. If \(\alpha \) has sort comm-monoid-add, then monoid-ap has C. If \(\alpha \) has sort idemp-monoid-add, then monoid-ap has W.

Either

figure s

Distinct non-empty list. The function remdups removes duplicates from a list by retaining only the last occurrence of each element.

figure t

State

figure u

List

figure v

Parser. The function apfst applies a function to the first component of a tuple.

figure w

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Lochbihler, A., Schneider, J. (2016). Equational Reasoning with Applicative Functors. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_16

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  • DOI: https://doi.org/10.1007/978-3-319-43144-4_16

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