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Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective

  • Nicolas Magaud
  • Agathe Chollet
  • Laurent Fuchs
Article
  • 96 Downloads

Abstract

This work presents a formalization of the discrete model of the continuum introduced by Harthong (1989), the Harthong-Reeb line. This model was at the origin of important developments in the Discrete Geometry field (Reveillès and Richard, Ann. Math. Artif. Intell. Math. Inform. 16(14), 89–152 (1996)). The formalization is based on the work presented in Chollet et al. (2012, 2009) where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges (1999). Laugwitz-Schmieden numbers (Laugwitz 1983) are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated (Chollet et al., Theor. Comput. Sci. 466, 2–19 (2012)). In this paper, we transpose all these definitions and properties into a formal description using the Coq proof assistant. We also show that Laugwitz-Schmieden numbers can be used to actually compute continuous functions. We hope that this work could improve techniques for both implementing numeric computations and reasoning about them in geometric systems.

Keywords

Coq Formal proofs Discrete geometry Non-standard arithmetic Arithmetisation scheme Exact computations 

Mathematics Subject Classification (2010)

03F55 03H15 03F60 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Icube UMR 7357 CNRS - Université de StrasbourgIllkirch CedexFrance
  2. 2.Laboratoire MIA - Université de La RochelleLa Rochelle CedexFrance
  3. 3.Laboratoire XLIM-SIC UMR 6172 CNRS - Université de PoitiersFuturoscope Chasseneuil CedexFrance

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