From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry

15 pages•Published: March 27, 2016

Abstract

This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant.
As a basis for this work, Tarski’s system of geometry was chosen for its well-known metamathematical properties.
This work completes our formalization of the two-dimensional results contained in part one of Metamathematische Methoden in der Geometrie.
We define the arithmetic operations geometrically and prove that they verify the properties of an ordered field.
Then, we introduce cartesian coordinates, and provide an algebraic characterization of the main geometric predicates.
In order to prove the characterization of the segment congruence relation, we provide a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems.
The arithmetization of geometry justifies the use the algebraic automated deduction methods in geometry.
We give an example of the use this formalization by deriving from Tarski's system of geometry a formal proof of theorems of nine points using Gröbner basis.

Keyphrases: arithmetization, Coq, formalization, geometry, Tarski's system of geometry

In: James H. Davenport and Fadoua Ghourabi (editors). SCSS 2016. 7th International Symposium on Symbolic Computation in Software Science, vol 39, pages 14--28

Links:	https://easychair.org/publications/paper/ZvfW
	https://doi.org/10.29007/k47p