Pascal’s Theorem in Real Projective Plane

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Summary

In this article we check, with the Mizar system [2], Pascal’s theorem in the real projective plane (in projective geometry Pascal’s theorem is also known as the Hexagrammum Mysticum Theorem)1. Pappus’ theorem is a special case of a degenerate conic of two lines.

For proving Pascal’s theorem, we use the techniques developed in the section “Projective Proofs of Pappus’ Theorem” in the chapter “Pappus’ Theorem: Nine proofs and three variations” [11]. We also follow some ideas from Harrison’s work. With HOL Light, he has the proof of Pascal’s theorem2. For a lemma, we use PROVER93 and OTT2MIZ by Josef Urban4 [12, 6, 7]. We note, that we don’t use Skolem/Herbrand functions (see “Skolemization” in [1]).

[1] Jesse Alama. Escape to Mizar for ATPs. arXiv preprint arXiv:1204.6615, 2012.

[2] 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: 10.1007/978-3-319-20615-817.

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[11] Jürgen Richter-Gebert. Pappos’s Theorem: Nine Proofs and Three Variations, pages 3–31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi: 10.1007/978-3-642-17286-11.

[12] Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.

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Formalized Mathematics

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SCImago Journal Rank (SJR) 2017: 0.119
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researchers in the fields of formal methods and computer-checked mathematics

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