Formalization of the arithmetization of Euclidean plane geometry and applications

Abstract

This 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 metamathematical 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.

Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

“…Also, it may seem difficult to trust computer algebra systems. We studied this issue in the context of 2D some years ago [13], but to be exhaustive, we also had to prove the correctness of the link from synthetic geometry to algebra (this is partially done in the Boutry et al paper [3] in the context of 2D). To generate a Coq proof of the whole process presented in this section, we still need to adapt the above-mentionned approaches to 3D.…”

“…Table 6 presents the lemma for the Pappus to Dandelin-Gallucci implication and Table 7 presents the reciprocal statement Dandelin-Gallucci to Pappus. 3 Both statements and proofs can be found in the git repository already mentioned. Note that only the basic axioms of incidence geometry are used by our prover, and, for instance, the Pappus configuration which define points , and is not discovered by our method.…”

Hybrid dynamical type theories for navigation

“…For the second task, the main obstacle is the reliance of the proof of safety on basic Euclidean geometry which requires some effort to formalize. Fortunately, it may be possible to leverage GeoCoq [9], a dependently typed formalization Tarski's axiomatic system [35] in the Coq proof assistant [11,4], previously used to formalize the first book of Euclid's Elements [5,20].…”

Towards Intuitive Reasoning in Axiomatic Geometry

“…The axioms such as congpseudo reflexivity are then straightforward versions of the axioms we already got to know in Figure 3. A current overview of the status of GEOCOQ can be found in [3,4].…”