<abstract xmlns="http://www.w3.org/1999/xhtml">

In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . .
Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”).
In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code.
In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional).
We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).
</abstract>

In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane. It is well known that in the projective plane, the notions of points and lines are dual [11], [9], [15], [8]. Some results (analytical, synthetic, combinatorial) of projective geometry are already present in some libraries Lean/Hott [5], Isabelle/Hol [3], Coq [13], [14], [4], Agda [6], . . . .
Proofs of dual statements by proof assistants have already been proposed, using an axiomatic method (for example see in [13] - the section on duality: “[...] For every theorem we prove, we can easily derive its dual using our function swap [...]2”).
In our formalisation, we use an analytical rather than a synthetic approach using the definitions of Leończuk and Prażmowski of the projective plane [12]. Our motivation is to show that it is possible by developing dual definitions to find proofs of dual theorems in a few lines of code.
In the first part, rather technical, we introduce definitions that allow us to construct the duality between the points of the real projective plane and the lines associated to this projective plane. In the second part, we give a natural definition of line concurrency and prove that this definition is dual to the definition of alignment. Finally, we apply these results to find, in a few lines, the dual properties and theorems of those defined in the article [12] (transitive, Vebleian, at_least_3rank, Fanoian, Desarguesian, 2-dimensional).
We hope that this methodology will allow us to continued more quickly the proof started in [7] that the Beltrami-Klein plane is a model satisfying the axioms of the hyperbolic plane (in the sense of Tarski geometry [10]).

Duality Notions in Real Projective Plane

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"Duality Notions in Real Projective Plane"}

In this article, we check with the Mizar system [1], [2], the converse of Desargues’ theorem and the converse of Pappus’ theorem of the real projective plane....

Duality Notions in Real Projective Plane

Published Online: Jul 09, 2022

Page range: 161 - 173

Accepted: Sep 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0016

Keywords
Principle of Duality, duality, real projective plane, converse theorem

© 2022 Roland Coghetto, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Duality Notions in Real Projective Plane

Published Online: Jul 09, 2022

Page range: 161 - 173

Accepted: Sep 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0016

KeywordsPrinciple of Duality, duality, real projective plane, converse theorem

© 2022 Roland Coghetto, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
Principle of Duality, duality, real projective plane, converse theorem