Complete independence of an axiom system for central translations

A recently proposed axiom system for Andr\'e's central translation structures is improved upon. First, one of its axioms turns out to be dependent (derivable from the other axioms). Without this axiom, the axiom system is indeed independent. Second, whereas most of the original independence models were infinite, finite independence models are available. Moreover, the independence proof for one of the axioms employed proof-theoretic techniques rather than independence models; for this axiom, too, a finite independence model exists. For every axiom, then, there is a finite independence model. Finally, the axiom system (without its single dependent axiom) is not only independent, but completely independent.Comment: 10 pages. Submitted to Note di Matematic

Tarski Geometry Axioms

This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. Specifically, we obtain Gupta’s amazing proof which implies Hilbert’s axiom I1 that two points determine a line. The primary Mizar coding was heavily influenced by [9] on axioms of incidence geometry. The original development was much improved using Mizar adjectives instead of predicates only, and to use this machinery in full extent, we have to construct some models of Tarski geometry. These are listed in the second section, together with appropriate registrations of clusters. Also models of Tarski’s geometry related to real planes were constructed.Richter William - Departament of Mathematics Nortwestern University Evanston, USAGrabowski Adam - Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok PolandAlama Jesse - Technical University of Vienna AustriaGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607–610, 1990.Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869, pages 139–156, 2007.Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.Alfred Tarski and Steven Givant. Tarski’s system of geometry. Bulletin of Symbolic Logic, 5(2):175–214, 1999.Wojciech A. Trybulec. Axioms of incidence. Formalized Mathematics, 1(1):205–213, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. Received June 16, 201

Eliciting implicit assumptions of proofs in the MIZAR Mathematical Library by property omission

When formalizing proofs with interactive theorem provers, it often happens that extra background knowledge (declarative or procedural) about mathematical concepts is employed without the formalizer explicitly invoking it, to help the formalizer focus on the relevant details of the proof. In the contexts of producing and studying a formalized mathematical argument, such mechanisms are clearly valuable. But we may not always wish to suppress background knowledge. For certain purposes, it is important to know, as far as possible, precisely what background knowledge was implicitly employed in a formal proof. In this note we describe an experiment conducted on the MIZAR Mathematical Library of formal mathematical proofs to elicit one such class of implicitly employed background knowledge: properties of functions and relations (e.g., commutativity, asymmetry, etc.).Comment: 11 pages, 3 tables. Preliminary version presented at the 3rd Workshop on Modules and Libraries for Proof Assistants (MLPA-11), affiliated with the 2nd Conference on Interactive Theorem Proving (ITP-2011), Nijmegen, the Netherland

Dependencies in Formal Mathematics: Applications and Extraction for Coq and Mizar

Two methods for extracting detailed formal dependencies from the Coq and Mizar system are presented and compared. The methods are used for dependency extraction from two large mathematical repositories: the Coq Repository at Nijmegen and the Mizar Mathematical Library. Several applications of the detailed dependency analysis are described and proposed. Motivated by the different applications, we discuss the various kinds of dependencies that we are interested in,and the suitability of various dependency extraction methods