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Implementing Euclid’s straightedge and compass constructions in type theory

Abstract

Constructions are central to the methodology of geometry presented in the Elements. This theory therefore poses a unique challenge to those concerned with the practice of constructive mathematics: can the Elements be faithfully captured in a modern constructive framework? In this paper, we outline our implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant. A result of our intuitionistic treatment of Euclidean geometry is a proof of the second proposition from Book I of the Elements in its full generality; a result that differs from other formally constructive accounts of Euclidean geometry. Our formalization of the straightedge and compass utilizes a predicate for orientation, which enables a concise and intuitive expression of Euclid’s constructions.

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Correspondence to Ariel Kellison.

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This material is based upon work supported by the National Science Foundation under Grant No.1650069.

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Kellison, A., Bickford, M. & Constable, R. Implementing Euclid’s straightedge and compass constructions in type theory. Ann Math Artif Intell 85, 175–192 (2019). https://doi.org/10.1007/s10472-018-9603-0

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Keywords

  • Constructive type theory
  • Constructive geometry
  • Foundations of geometry

Mathematics Subject Classification (2010)

  • 68T15
  • 03B35
  • 03F55