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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]