Statistics on Digital Libraries of Mathematics

We present statistics on the standard libraries of four major proof assistants for mathematics: HOL Light, Isabelle/HOL, Coq and Mizar. The advent of digital computers has introduced a new way of doing mathematics called 'formalized mathematics'. In this style of doing mathematics one encodes the mathematics in the computer in sufficient detail that the computer can fully check the correctness according to a small number of logical rules. This style of doing mathematics is much more precise and trustable than the traditional way of first understanding the mathematics in one's head and then just writing it on a blackboard or on paper. Also it is a very pleasurable experience to write down one's mathematics in a way that all the details are there, knowing that there is nothing left implicit. However, these positive aspects of formalized mathematics have to be paid for. Generally it takes much longer to turn mathematics into formalized form than it takes to just understand it, or even than to write it down in a traditional way. (A rough estimate might be that it takes about ten times as long to formalize something than it takes to write it down in meticulous traditional 'textbook style'.) One might wonder where this time is going, i.e., how much it is spent on the various aspects of formalization. For instance there are the aspects of formalizing the definitions, choosing good notation for the defined notions, then stating the appropriate formal statements to be proved, and finally writing the formal proofs themselves. Another question that might be posed is whether there are significant differ- ences in the time needed for these activities between the different systems for formalization of mathematics. In this paper we will study these questions. We will not do this by focusing on the activity of formalization, but rather by studying the results of this activity, the libraries of formalized mathematics that have been created by the various research communities that work on this subject. These libraries have grown into quite large human 'artifacts', which - we claim - deserve study in their own right. In this