Tarski Geometry Axioms – Part II

• Autorzy: Roland Coghetto , Adam Grabowski
• Języki publikacji: EN

Abstrakty

EN

In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

Słowa kluczowe

EN

Tarski’s geometry axioms; foundations of geometry; Euclidean plane

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 51A05: General theory and projective geometries
• 51M04: Elementary problems in Euclidean geometries

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2016
• Tom: 24
• Numer: 2
• Strony: 157-166

Daty

wydano

2016-06-01

otrzymano

2016-06-30

online

2016-12-08

Twórcy

autor

Roland Coghetto

• Rue de la Brasserie 5, 7100 La Louvière,

autor

Adam Grabowski

• Institute of Informatics, University of Białystok, Ciołkowskiego 1M, 15-245 Białystok,

Identyfikatory

DOI

10.1515/forma-2016-0012