Higher-Order Intuitionistic Formalization and Proofs in Hilbert's Elementary Geometry
ADG '00 Revised Papers from the Third International Workshop on Automated Deduction in Geometry
Interactive Theorem Proving and Program Development
Interactive Theorem Proving and Program Development
Geometric constraints solving: some tracks
Proceedings of the 2006 ACM symposium on Solid and physical modeling
Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)
On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic
Journal of Automated Reasoning
Mechanical theorem proving in Tarski's geometry
ADG'06 Proceedings of the 6th international conference on Automated deduction in geometry
Proof certificates for algebra and their application to automatic geometry theorem proving
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Formalizing projective plane geometry in Coq
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Using three-valued logic to specify and verify algorithms of computational geometry
ICFEM'05 Proceedings of the 7th international conference on Formal Methods and Software Engineering
Journal of Automated Reasoning
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Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the non-degeneracy conditions lead to long technical proofs. In addition, when considering higher-dimensions, the amount of incidence relations (e.g. point-line, point-plane, line-plane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a two-dimensional setting. Regarding at least three-dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and non-incidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.