Tarski's geometry modelled in Mizar computerized proof assistant

Adam Grabowski

doi:10.15439/2016f290

Abstract

In the paper, we discuss the formal approach to Tarski geometry axioms modelled with the help of the Mizar computerized proof assistant system. Although our basic development was inspired by Julien Narboux's Coq pseudo-code and is dated back to 2014, there are significant steps in the formalization of geometry done in the last decade of the previous century. Taking this into account, we will propose the reuse of existing results within this new framework (including Hilbert's axiomatic approach), with the ultimate future goal to encode the textbook Metamathematische Methoden in der Geometrie by Schwabhauser, Szmielew and Tarski. We try however to go much further from the use of simple predicates in the direction of the use of structures with their inheritance, attributes as a tool of more human-friendly namespaces for axioms, and registrations of clusters to obtain more automation (with the possible use of external equational theorem provers like Otter/Prover9).

Highlights

F OR YEARS, foundations of geometry attracted a lot of interest of resarchers from various areas of mathematics
Apart from the discussion whether the non-emptiness of types in Mizar is real difficulty, and how much more can be attained if the reimplementation of the Mizar type system will be done in the foreseeable future, we have to cope with the limitations of the existing type structure
Some of them are written in a language which is not as expressive as contemporary Mizar language is; in the time of the beginnings of the Mizar Mathematical Library (MML) as a tight collection of Mizar articles covering various branches of mathematics, geometry was an area which was developed quite dynamically

Summary

INTRODUCTION

F OR YEARS, foundations of geometry attracted a lot of interest of resarchers from various areas of mathematics. An important example is the possibility of ruler-and-compass construction: impossibility of trisecting the angle and doubling the cube (as two out of four problems of antiquity), where the treatment of constructible numbers is way more efficient from the formal point of view Euclid and his Elements are often recalled as one of the first successful uses of an axiomatic method in mathematics, and such an approach can be formalized efficiently with the use of computer proof-assistants.

MIZARING AFFINE GEOMETRY

THE CHOICE OF FORMAL FRAMEWORK

THE ORIGINAL SELF-CONTAINED APPROACH

INTRODUCING METRIC STRUCTURE

Findings

VIII. CONCLUSIONS AND FUTURE WORK