Skip to main content
Springer Nature Link
Log in

Shall We Juggle, Coinductively?

  • Conference paper
Certified Programs and Proofs (CPP 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7679))

Included in the following conference series:

Abstract

Buhler et al. presented a mathematical theory of toss juggling by regarding a toss pattern as an arithmetic function, where the function must satisfy a condition for the pattern to be valid. In this paper, the theory is formalized in terms of coinduction, reflecting the fact that the validity of toss juggling is related to a property of infinite phenomena. A tactic is implemented for proving the validity of toss patterns in Coq. Additionally, the completeness and soundness of a well-known algorithm for checking the validity is demonstrated. The result exposes a practical aspect of coinductive proofs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

') var buybox = document.querySelector("[data-id=id_"+ timestamp +"]").parentNode var buyingOptions = buybox.querySelectorAll(".buying-option") ;[].slice.call(buyingOptions).forEach(initCollapsibles) // springerPlus roll out 10% starts here var springerPlusGroup = setLocalStorageSpringerPlus(); var rollOutSpringerPlus = springerPlusGroup === "B" function setLocalStorageSpringerPlus() { var selectUserKey = "springerPlusRollOut"; var springerPlusGroup = "X"; if (!window.localStorage) return springerPlusGroup; try { var selectUserValue = window.localStorage.getItem(selectUserKey) springerPlusGroup = selectUserValue || randomDistributionSpringerPlus(selectUserKey) } catch (err) { console.log(err) } return springerPlusGroup; } function randomDistributionSpringerPlus(selectUserKey) { var randomGroup = Math.random() < 0.7 ? "A" : "B" window.localStorage.setItem(selectUserKey, randomGroup) return randomGroup } if (rollOutSpringerPlus) { revealSpringerPlus(); } function revealSpringerPlus() { if(buybox) { document.querySelectorAll(".c-springer-plus").forEach(function(node) { node.style.display = "block" }) } } //springerPlus ends here var buyboxMaxSingleColumnWidth = 480 function initCollapsibles(subscription, index) { var toggle = subscription.querySelector(".buying-option-price") subscription.classList.remove("expanded") var form = subscription.querySelector(".buying-option-form") var priceInfo = subscription.querySelector(".price-info") var buyingOption = toggle.parentElement if (toggle && form && priceInfo) { toggle.setAttribute("role", "button") toggle.setAttribute("tabindex", "0") toggle.addEventListener("click", function (event) { var expandedBuyingOptions = buybox.querySelectorAll(".buying-option.expanded") var buyboxWidth = buybox.offsetWidth ;[].slice.call(expandedBuyingOptions).forEach(function(option) { if (buyboxWidth <= buyboxMaxSingleColumnWidth && option != buyingOption) { hideBuyingOption(option) } }) var expanded = toggle.getAttribute("aria-expanded") === "true" || false toggle.setAttribute("aria-expanded", !expanded) form.hidden = expanded if (!expanded) { buyingOption.classList.add("expanded") } else { buyingOption.classList.remove("expanded") } priceInfo.hidden = expanded }, false) } } function hideBuyingOption(buyingOption) { var toggle = buyingOption.querySelector(".buying-option-price") var form = buyingOption.querySelector(".buying-option-form") var priceInfo = buyingOption.querySelector(".price-info") toggle.setAttribute("aria-expanded", false) form.hidden = true buyingOption.classList.remove("expanded") priceInfo.hidden = true } function initKeyControls() { document.addEventListener("keydown", function (event) { if (document.activeElement.classList.contains("buying-option-price") && (event.code === "Space" || event.code === "Enter")) { if (document.activeElement) { event.preventDefault() document.activeElement.click() } } }, false) } function initialStateOpen() { var buyboxWidth = buybox.offsetWidth ;[].slice.call(buybox.querySelectorAll(".buying-option")).forEach(function (option, index) { var toggle = option.querySelector(".buying-option-price") var form = option.querySelector(".buying-option-form") var priceInfo = option.querySelector(".price-info") if (buyboxWidth > buyboxMaxSingleColumnWidth) { toggle.click() } else { if (index === 0) { toggle.click() } else { toggle.setAttribute("aria-expanded", "false") form.hidden = "hidden" priceInfo.hidden = "hidden" } } }) } initialStateOpen() if (window.buyboxInitialised) return window.buyboxInitialised = true initKeyControls() })()

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Buhler, J., Eisenbud, D., Graham, R., Wright, C.: Juggling Drops and Descents. American Mathematical Monthly 101(6), 507–519 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chlipala, A.: Certified Programming with Dependent Types. MIT Press (2011), http://adam.chlipala.net/cpdt/

  3. Giménez, E., Castéran, P.: A Tutorial on [Co-]Inductive Types in Coq (1998)

    Google Scholar 

  4. Giménez, E.: Codifying Guarded Definitions with Recursive Schemes. In: Smith, J., Dybjer, P., Nordström, B. (eds.) TYPES 1994. LNCS, vol. 996, pp. 39–59. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  5. Howard, W.A.: The Formulae-As-Types Notion of Construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 479–490. Academic Press (1980)

    Google Scholar 

  6. Leroy, X., Grall, H.: Coinductive big-step operational semantics. Information and Computation 207(2), 284–304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nakata, K., Uustalu, T.: Trace-Based Coinductive Operational Semantics for While. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 375–390. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. OEIS Foundation Inc. Sequence A010060. The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org/A010060

  9. Sangiorgi, D.: On the Origins of Bisimulation and Coinduction. ACM Transactions on Programming Languages and Systems 31(4), 15:1–15:41 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Electro-Communications, Japan

    Keisuke Nakano

Authors
  1. Keisuke Nakano
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, WA, USA

    Chris Hawblitzel

  2. INRIA Saclay and LIX, Ecole Polytechnique, Palaiseaux Cedex, France

    Dale Miller

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nakano, K. (2012). Shall We Juggle, Coinductively?. In: Hawblitzel, C., Miller, D. (eds) Certified Programs and Proofs. CPP 2012. Lecture Notes in Computer Science, vol 7679. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35308-6_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35308-6_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35307-9

  • Online ISBN: 978-3-642-35308-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation