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

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 [SST83]. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a " back-translation " from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method.
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Contributor : Pierre Boutry <>
Submitted on : Thursday, March 3, 2016 - 6:57:37 PM
Last modification on : Tuesday, April 2, 2019 - 1:38:39 AM
Long-term archiving on: : Saturday, June 4, 2016 - 11:30:39 AM

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  • HAL Id : hal-01282550, version 1

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Pierre Boutry, Gabriel Braun, Julien Narboux. From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry. SCSS 2016, the 7th International Symposium on Symbolic Computation in Software Science, Mar 2016, Tokyo, Japan. pp.14-28. ⟨hal-01282550⟩

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