{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,3]],"date-time":"2024-09-03T22:30:42Z","timestamp":1725402642569},"reference-count":0,"publisher":"EasyChair","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant.<\/jats:p><jats:p>As a basis for this work, Tarski\u2019s system of geometry was chosen for its well-known metamathematical properties.<\/jats:p><jats:p>This work completes our formalization of the two-dimensional results contained in part one of Metamathematische Methoden in der Geometrie.<\/jats:p><jats:p>We define the arithmetic operations geometrically and prove that they verify the properties of an ordered field.<\/jats:p><jats:p>Then, we introduce cartesian coordinates, and provide an algebraic characterization of the main geometric predicates.<\/jats:p><jats:p>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.<\/jats:p><jats:p>The arithmetization of geometry justifies the use the algebraic automated deduction methods in geometry.<\/jats:p><jats:p>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 Gr\u00f6bner basis.<\/jats:p>","DOI":"10.29007\/k47p","type":"proceedings-article","created":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T18:04:17Z","timestamp":1516730657000},"page":"14--2","source":"Crossref","is-referenced-by-count":1,"title":["From Tarski to Descartes: Formalization of the Arithmetization of Euclidean Geometry"],"prefix":"10.29007","volume":"39","author":[{"given":"Pierre","family":"Boutry","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gabriel","family":"Braun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Julien","family":"Narboux","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"11545","event":{"name":"SCSS 2016. 7th International Symposium on  Symbolic Computation in Software Science"},"container-title":["EPiC Series in Computing"],"original-title":[],"deposited":{"date-parts":[[2018,1,23]],"date-time":"2018-01-23T18:04:20Z","timestamp":1516730660000},"score":1,"resource":{"primary":{"URL":"https:\/\/easychair.org\/publications\/paper\/ZvfW"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[null]]},"references-count":0,"URL":"https:\/\/doi.org\/10.29007\/k47p","relation":{},"ISSN":["2398-7340"],"issn-type":[{"type":"print","value":"2398-7340"}],"subject":[]}}