Formal systems for proving incidence results

U ovoj tezi razvijen je formalni sistem za dokazivanje teorema incidencije u projektivnoj geometiji. Osnova sistema je Čeva/Menelaj metod za dokazivanje teorema incidencije. Formalizacija o kojoj je ovdje riječ izvedena je korišćenjem Δ-kompleksa, pa su tako u disertaciji spojene oblasti logike, geometrije i algebarske topologije. Aksiomatski sekventi proizilaze iz 2-ciklova Δ-kompleksa. Definisana je Euklidska i projektivna interpretacija sekvenata i dokazana je saglasnost i odlučivost sistema. Dati su primjeri iščitavanja teorema incidencije iz dokazivih sekvenata sistema. U tezi je data i procedura za provjeru da li je skup od n šestorki tačaka aksiomatski sekvent.In this thesis, a formal sequent system for proving incidence theorems in projective geometry is introduced. This system is based on the Ceva/Menelaus method for proving theorems. This formalization is performed using Δ-complexes, so the areas of logic, geometry and algebraic topology are combined in the dissertation. The axiomatic sequents of the system stem from 2-cycles of Δ-complexes. The Euclidean and projective interpretations of the sequents are defined and the decidability and soundness of the system are proved. Patterns for extracting formulation and proof of the incidence result from derivable sequents of system are exemplified. The procedure for deciding if set of n sextuples represent an axiomatic sequent is presented within the thesis

Mechanical Theorem Proving in Projective Geometry

We present an algorithm that is able to confirm projective incidence statements by carrying out calculations in the ring of all formal determinants (brackets) of a configuration. We will describe an implementation of this prover and present a series of examples treated by the prover, including Pappos' and Desargues' Theorems, the Sixteen Point Theorem, Saam's Theorem, the Bundle Condition, the uniqueness of a harmonic Point and Pascal's Theorem

Diagrammatic Reasoning in Projective Geometry

The heart of our thesis is that matrices of incidence can be used for mechanical theorem proving in projective geometry. To every geometrical statement of projective geometry is associated a matrix of incidence whose normal form is computed after successive identifications of rows and columns. Our main result is that the geometrical statement implies such or such property (incidence between a point and a line) if and only if the normal form of the associated matrix contains a 1 at such or such entry. 1 Introduction Diagrammatic representation appears in many human activities: mathematics, physics, economics, politics, etc. It constitutes a natural framework for the formalization and the mechanization of reasoning. As an example, Euclid, pioneer of the formal methods at the beginnings of the study of the algorithms, took advantage, proving the theorems of his geometry, of the visual properties of the geometrical figures. It is generally believed that figures are informal objects and th..

Homography in ℝℙ

The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12].Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18].In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17].Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]).Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.Rue de la Brasserie 5, 7100 La Louvière, BelgiumSusanne Apel. The geometry of brackets and the area principle. Phd thesis, Technische Universität München, Fakultät für Mathematik, 2014.Susanne Apel and Jürgen Richter-Gebert. Cancellation patterns in automatic geometric theorem proving. In Automated Deduction in Geometry, pages 1–33. Springer, 2010.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Laurent Fuchs and Laurent Thery. A formalization of Grassmann-Cayley algebra in Coq and its application to theorem proving in projective geometry. In Automated Deduction in Geometry, pages 51–67. Springer, 2010.Kanchun, Hiroshi Yamazaki, and Yatsuka Nakamura. Cross products and tripple vector products in 3-dimensional Euclidean space. Formalized Mathematics, 11(4):381–383, 2003.Wojciech Leończuk and Krzysztof Prażmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.Wojciech Leończuk and Krzysztof Prażmowski. Projective spaces – part I. Formalized Mathematics, 1(4):767–776, 1990.Xiquan Liang, Piqing Zhao, and Ou Bai. Vector functions and their differentiation formulas in 3-dimensional Euclidean spaces. Formalized Mathematics, 18(1):1–10, 2010. doi:10.2478/v10037-010-0001-2.Nicolas Magaud, Julien Narboux, and Pascal Schreck. Formalizing projective plane geometry in Coq. In Automated Deduction in Geometry, pages 141–162. Springer, 2008.Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean Axiom. Victoria University of Wellington, New Zealand, 2012. Master’s thesis.Karol Pąk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199–211, 2007. doi:10.2478/v10037-007-0024-5.Karol Pąk. Linear transformations of Euclidean topological spaces. Formalized Mathematics, 19(2):103–108, 2011. doi:10.2478/v10037-011-0016-3.Jürgen Richter-Gebert. Mechanical theorem proving in projective geometry. Annals of Mathematics and Artificial Intelligence, 13(1-2):139–172, 1995.Jürgen Richter-Gebert. Perspectives on projective geometry: a guided tour through real and complex geometry. Springer Science & Business Media, 2011.Wojciech Skaba. The collinearity structure. Formalized Mathematics, 1(4):657–659, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990