Abstract. We compare fifteen systems for the formalizations of mathematics with the computer. We present several tables that list various properties of these programs. The three main dimensions on which we compare these systems are: the size of their library, the strength of their logic and their level of automation. 1

We de ne the notion of formal proof sketch for the mathematical language Mizar. We show by examples that formal proof sketches are very close to informal mathematical proofs. We discuss some ways in which formal proof sketches might be used to improve mathematical proof assistants. 1

There are two dierent approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system).

We report on how mathematics is done in the Mizar system. Mizar oers a language for writing mathematics and provides software for proof-checking. Mizar is used to build Mizar Mathematical Library (MML). This is a long term project aiming at building a comprehensive library of mathematical knowledge. The language and the checking software evolve and the evolution is driven by the growing MML.

The correctness of proofs is increasingly being veried with computer programs called `proof checkers'. Examples of such proof checkers are Mizar, ACL2, PVS, Nuprl, HOL, Isabelle and Coq. This paper addresses what is one of the most important problems for that kind of system, which is how to deal with partial functions and the related issue of how to treat undened terms. In many systems the problem is avoided by articially making all functions total. However that does not correspond to the practice of every day mathematics. In type theory partial functions are modeled by giving functions extra arguments which are proof objects. Because of that it is not possible to apply a function outside its domain. However having proofs as rst class objects makes the logic non-standard. This has the disadvantages that it is unfamiliar to most mathematicians and that many proof tools won't be usable for it. For instance a theorem prover like Otter cannot be easily used for this kind of logic. Also expressions in type theoretical systems get clumsy because they contain proof objects. The PVS system solves the problem of partial functions dierently. PVS generates type-correctness conditions or TCCs for statements in its language. These are proof obligations that have to be satised `on the side' to show that the statements are well-formed. In this paper we relate the type theoretical approach to one resembling the PVS approach. We add domain conditions to ordinary rst order logic (which in this paper will be classical and one-sorted) and we show that the combination corresponds precisely to a rst order system that treats partial functions in the style of type theory. 1

We present an overview of the current state of formalization of mathematics, and argue what will be needed to make the vision from the QED manifesto come true.

A `mathematical vernacular' is a formal language for writing mathematical proofs which resembles the natural language from mathematical texts. Several systems (Hyperproof, Mizar, Isabelle/Isar) all basically have the same proof language. It consists of the combination of natural deduction with rst order inference steps. In this note we compare these three languages and present a simplied common version. 1 Mathematical Vernaculars The term `mathematical vernacular' (`wiskundige omgangstaal' or wot in Dutch) was introduced by de Bruijn [2] in the eighties. With this term he didn't mean the language that mathematicians actually use to communicate their work in practice (which is a somewhat stylized variant of natural language interspersed with formulas.) Instead he meant a formal language: a system that he had developed to represent mathematics. Supposedly, it was close in style to the actual way that mathematicians communicate, hence the name. `Vernacular' doesn't seem a word that readily admits a plural: each natural language (English, Dutch, etc.) only has one vernacular. However, because many people turned out to have dierent ideas about the `best' way to have a formal language resemble ordinary mathematics, one started talking about `mathematical vernaculars' in the plural. So because many had their own thoughts about what the `mathematical vernacular' should look like when formalized, the term became the name of a species of formal language. This paper doesn't want to promote an `own' variant of the mathematical vernacular concept. Instead, it just makes an observation about already existing formal languages. It turns out that in a signicant number of systems (`proof assistants') one encounters languages that look almost the same. Apparently there is a canonical...

Abstract. Mathematical documents, and their instrumentation by computers, have rich structure at the layers of presentation, metadata and semantics, as objects in a system for formal mathematical logic. Semantic Web tools [2] support the first two of these, with little, if any, contribution to the third, while Proof Assistants [17] instrument the third layer, typically with bespoke approaches to the first two. Our position is that a web of mathematical documents, definitions and proofs should be given a fully-fledged semantics in terms of the third layer. We propose a “Math-Wiki ” to harness Web 2.0 tools and techniques to the rich semantics furnished by contemporary Proof Assistants. 1 Background and state of the art We can identify four worlds of mathematical discourse available on the Web: – Traditional mathematical practice: a systematic body of knowledge, organised around documents written by experts, most often in L ATEX, to varying degrees of sophistication. The intended audience is an expert readership, and

Abstract. We present statistics on the standard libraries of four major

We construct a logic-enriched type theory LTTw that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalise many results from that book in LTTw, including Weyl’s definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics.