Gradual computerisation and verification of mathematics : MathLang's path into Mizar
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There are many proof checking tools that allow capturing mathematical knowledge into formal representation. Those proof systems allow further automatic verifica- tion of the logical correctness of the captured knowledge. However, the process of encoding common mathematical documents in a chosen proof system is still labour- intensive and requires comprehensive knowledge of such system. This makes the use of proof checking tools inaccessible for ordinary mathematicians. This thesis provides a solution for the computerisation of mathematical documents via a num- ber of gradual steps using the MathLang framework. We express the full process of formalisation into the Mizar proof checker. The first levels of such gradual computerisation path have been developing well before the course of this PhD started. The whole project, called MathLang, dates back to 2000 when F. Kamareddine and J.B. Wells started expressing their ideas of novel approach for computerising mathematical texts. They mainly aimed at developing a mathematical framework which is flexible enough to connect existing, in many cases different, approaches of computerisation mathematics, which allows various degrees of formalisation (e.g., partial, full formalisation of chosen parts, or full formalisation of the entire doc- ument), which is compatible with different mathematical foundations (e.g., type theory, set theory, category theory, etc.) and proof systems (e.g., Mizar, Isar, Coq, HOL, Vampire). The first two steps in the gradual formalisation were developed by F. Kamareddine, J.B. Wells and M. Maarek with a small contribution of R. Lamar to the second step. In this thesis we develop the third level of the gradual path, which aims at capturing the rhetorical structure of mathematical documents. We have also integrated further steps of the gradual formalisation, whose final goal is the Mizar system. We present in this thesis a full path of computerisation and formalisation of math- ematical documents into the Mizar proof checker using the MathLang framework. The development of this method was driven by the experience of computerising a number of mathematical documents (covering different authoring styles).