Mechanical Theorem Proving in Tarski's geometry.

International audienceThis paper describes the mechanization of the proofs of the first height chapters of Schwabäuser, Szmielew and Tarski's book: Metamathematische Methoden in der Geometrie. The goal of this development is to provide foundations for other formalizations of geometry and implementations of decision procedures. We compare the mechanized proofs with the informal proofs. We also compare this piece of formalization with the previous work done about Hilbert's Gründlagen der Geometrie. We analyze the differences between the two axiom systems from the formalization point of view

Formalization and automation of Euclidean geometry

Напредак геометрије кроз векове се може разматрати кроз развој различитих аксиоматских система који је описују. Употреба аксиоматских система започиње са Хилбертом и Тарским али се ту не завршава. Чак и данас се развијају нови аксиоматски ситеми за рад са еуклидском геометријом...The advance of geometry over the centuries can be observed through the development of dierent axiomatic systems that describe it. The use of axiomatic systems begins with Euclid, continues with Hilbert and Tarski, but it doesn't end there. Even today, new axiomatic systems for Euclidean geometry are developed..

Constraint-Based Graphic Statics - A geometrical support for computer-aided structural equilibrium design

This thesis introduces “constraint-based graphic statics”, a geometrical support for computer-aided structural design. This support increases the freedom with which the designer interacts with the plane static equilibriums being shaped. Constraint-based graphic statics takes full advantage of geometry, both its visual expressiveness and its capacity to solve complex problems in simple terms. Accordingly, the approach builds on the two diagrams of classical graphic statics: a form diagram describing the geometry of a strut-and-tie network and a force diagram vectorially representing its inner static quilibrium. Two new devices improve the control of these diagrams: (1) nodes — considered as the only variables — are constrained within Boolean combinations of graphical regions; and (2) the user modifies these diagrams by means of successive operations whose geometric properties do not at any time jeopardise the static equilibrium of the strut-and-tie network. These two devices offer useful features, such as the ability to describe, constrain and modify any static equilibrium using purely geometric grammar, the ability to compute and handle multiple solutions to a problem at the same time, the ability to switch the hierarchy of constraint dependencies, the ability to execute dynamic conditional statements graphically, the ability to compute full interdependency and therefore the ability to remove significantly the limitations of compass-and-straightedge constructions and, finally the ability to propagate some solution domains symbolically. As a result, constraint-based graphic statics encourages the emergence of new structural design approaches that are highly interactive, precognitive and chronology-free: highly interactive because forces and geometries are simultaneously and dynamically steered by the designer; precognitive because the graphical region constraining each points marks out the set of available solutions before they are even explored by the user; and chronology-free because the deductive process undertaken by the designer can be switched whenever desired

Towards an Independent Version of Tarski's System of Geometry

It has been accepted for publication in the Electronic Proceedings in Theoretical Computer Science (EPTCS) volume dedicated to the ADG 2023 proceedings.International audience1926-1927, Tarski designed a set of axioms for Euclidean geometry which reached its nal form in a manuscript by Schwabhäuser, Szmielew and Tarski in 1983. The dierences amount to simplicationsobtained by Tarski and Gupta. Gupta presented an independent version of Tarski's system of geometry, thus establishing that his version could not be further simplied without modifying the axioms. To obtain the independence of one of his axioms, namely Pasch's axiom, he proved the independence of one of its consequences: the previously eliminated symmetry of betweenness. However, an independence model forthe non-degenerate part of Pasch's axiom was provided by Szczerba for another version of Tarski's system of geometry in which the symmetry of betweenness holds. This independence proof cannot be directly used for Gupta's version as the statements of the parallel postulate dier. In this paper, we present our progress towards obtaining an independent version of a variant of Gupta's system. Compared to Gupta's version, we split Pasch's axiom into this previously eliminated axiom and its non-degenerate part and change the statement of the parallel postulate. We veried the independence properties by mechanizing counter-models using the Coq proof-assistant

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

International audienceThis 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

Formalization of the Arithmetization of Euclidean Plane Geometry and Applications

International audienceThis paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamath-ematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser Szmielew and Tarski Metamathematische Methoden in der Geometrie. 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. Moreover , we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method