Formalising Mathematics – in Praxis; A Mathematician’s First Experiences with Isabelle/HOL and the Why and How of Getting Started

AbstractThis is an account of a mathematician’s first experiences with the proof assistant (interactive theorem prover) Isabelle/HOL, including a discussion on the rationale behind formalising mathematics and the choice of Isabelle/HOL in particular, some instructions for new users, some technical and conceptual observations focussing on some of the first difficulties encountered, and some thoughts on the use and potential of proof assistants for mathematics.</jats:p

Case Studies in Proof Checking

The aim of computer proof checking is not to find proofs, but to verify them. This is different from automated deduction, which is the use of computers to find proofs that humans have not devised first. Currently, checking a proof by computer is done by taking a known mathematical proof and entering it into the special language recognized by a proof verifier program, and then running the verifier to hopefully obtain no errors. Of course, if the proof checker approves the proof, there are considerations of whether or not the proof checker is correct, and this has been complicated by the fact that so many systems have sprung into being. The two main challenges in using a proof checker today are the time needed to learn the syntax and general usage of the system and the time needed to formalize a proof in the system even when the user is already proficient with it. As mathematicians are not yet using proof checkers regularly, we wanted to evaluate the validity of this reluctance by analyzing these main obstacles. Judging by Dr. Wiedijk’s Formalizing 100 Theorems list, which gives an overview of the headway various proof systems have made in mathematics, Coq and Mizar are two of the most successful systems in use today (Wiedijk, 2007). I simultaneously formalized two fairly involved theorems in these two systems while I was at approximately the same level of familiarity with each. I kept track of my experiences with learning the systems and analyzed their comparative strengths and weaknesses. The analysis and summary of experiences should also give a general idea of the current state of computer-aided proof checking

Formalization of Real Analysis: A Survey of Proof Assistants and Libraries

International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis

Formalization and automation of Euclidean geometry

Напредак геометрије кроз векове се може разматрати кроз развој различитих аксиоматских система који је описују. Употреба аксиоматских система започиње са Хилбертом и Тарским али се ту не завршава. Чак и данас се развијају нови аксиоматски ситеми за рад са еуклидском геометријом...The advance of geometry over the centuries can be observed through the development of dierent axiomatic systems that describe it. The use of axiomatic systems begins with Euclid, continues with Hilbert and Tarski, but it doesn't end there. Even today, new axiomatic systems for Euclidean geometry are developed..

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom

A change-oriented architecture for mathematical authoring assistance

The computer-assisted authoring of mathematical documents using a scientific text-editor requires new mathematical knowledge management and transformation techniques to organize the overall workflow of anassistance system like the &#937;MEGAsystem.The challenge is that, throughout the system, various kinds of given and derived knowledge units occur in different formats and with different dependencies. If changes occur in these pieces of knowledge, they need to be effectively propagated. We present a Change-Oriented Architecture for mathematical authoring assistance. Thereby, documents are used as interfaces and the components of the architecture interact by actively changing the interface documents and by reacting on changes. In order to optimize this style of interaction, we present two essential methods in this thesis. First, we develop an efficient method for the computation of weighted semantic changes between two versions of a document. Second, we present an invertible grammar formalism for the automated bidirectional transformation between interface documents. The presented architecture provides an adequate basis for the computer-assisted authoring of mathematical documents with semantic annotations and a controlled mathematical language

Gradual computerisation and verification of mathematics : MathLang's path into Mizar

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)