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.

A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry

“…This is crucial to obtain a coordinate-free version of the proof of this theorem, because this theorem is the main ingredient for building a field and defining a coordinate system. The coordinatization of geometry allows the use of the algebraic approaches for automated deduction in the context of an axiom system for synthetic geometry as shown in [Bee13,BBN16]. The overall proof consists of approximately 10k lines of proof compared to the proof in [Hil60] which is three pages long and the version in [SST83] which is nine pages long.…”

“…Note, however that for arithmetization of geometry we will need to use this axiom to obtain the standard axioms of an ordered field expressed using functions instead of relations[BBN16].…”

Proof-checking Euclid

“…They verified that Hilbert's axioms follow from Tarski's [10]. With Pierre Boutry, they verified that Tarski's axioms follow from Hilbert's [9], and completed [6] the verification of the theorems in Szmielew's part of [40], which the second author began (with other co-authors) in [41]. Work is currently being done toward checking Euclid's propositions from Hilbert/Tarski axioms within Coq.…”

“…However, the same cannot be said of the proofs. Many of these have problems like those of I.9 and I.7; that is, we could fix these problems only by proving some other propositions first, and the propositions of the first half of Book I had to be proved in a different order, namely 1,3,15,5,4,10,12,7,6,8,9,11, and in some cases the proofs are much more difficult than Euclid thought. After proving those early propositions, we could follow Euclid's order better, and things went well until Prop.…”

Improving QED-Tutrix by Automating the Generation of Proofs

“…Practical applications include the recent integration of a deduction engine in Ge-oGebra [14], which is based on the internal representation of geometrical elements in complex numbers inside GeoGebra. Other examples include the systems based on the area method [15,30], the full-angle method [21,22], and many others. These systems seldom provide readable proofs, and when they do, they are far from what a high-school student would write.…”