These are the notes for a course which is delivered to final year undergraduates and MSc students in the mathematics department at Imperial College London. I hope that they will also be useful to other mathematicians.
Aims and objectives
The aim of the course is to teach people how to formalise undergraduate and MSc level mathematics in a computer proof assistant. A computer proof assistant is a computer program which knows the axioms of mathematics and is capable of understanding and checking mathematical proofs. To formalise mathematics means to type it into a computer proof assistant.
Another way of looking at it is that a computer proof assistant turns the statement of a theorem of mathematics into a level of a puzzle game, and proving the theorem corresponds to solving the level. Hence another way of thinking about the aims and objectives of this course are that I am teaching you how to set and how to solve levels of a computer puzzle game.
The methods used in this course are extremely “hands-on”. You will be learning by doing. Every topic will be introduced via examples and you will be expected to learn it by solving problems.
My personal belief is the following. Software of this nature will become more normalised in mathematics departments, and libraries of formalised theorems will begin to grow quite large. Computer theorem provers will ultimately give rise to tools which will help humans to learn and to do mathematics, from undergraduate level to the frontiers of modern research. Complex tactics, perhaps backed by AI, will in the future start to help human researchers. PhD students will be able to search these libraries to get hints on how to proceed, or find references for claims which they hope are true. Other applications will appear as mathematicians begin to understand the potential of formalised libraries of mathematics and of computer programs which can understand them.
The Lean Theorem Prover
There are many computer proof assistants out there; the one we shall be using is called Lean. We will be using not just Lean, but also Lean’s mathematics library
mathlib, which contains a substantial number of proofs of mathematical theorems already, as well as many high-powered tactics, tools which make theorem proving easier. There are many other theorem provers out there; Isabelle/HOL and Coq are two other provers which have also been used to formalise a substantial amount of mathematics. The reason we are using Lean is simply that it is the prover which I myself am most familiar with; I do not see any obstruction in theory to porting this entire course over to another theorem prover.
We will be using Lean 3 this year, because right now Lean 4 is still a work in progress and in particular it does not yet have a powerful mathematics library. When Lean 4 is ready I will port this course over.
Experience has shown that trying to teach people new mathematics at the same time as teaching them Lean is asking too much from most people. I will hence be assuming that the reader/player is comfortable with the material covered in the first and second year of a traditional mathematics degree. Later on, when the reader is more familiar with Lean I will introduce more mathematically challenging material.
This course is focussed on mathematics, or to be more precise, classical mathematics. We will not be concentrating on the non-mathematical side of the story. Examples of topics we will say very little about are: type theories, functional programming, the lambda calculus, and constructivism.
How to read this document
This book comes in three parts: Part A, Part B and Part C. Part A is the mathematical meat of the book. In each of the various sections of Part A we will be talking about a mathematical idea and how to implement it and work with it in Lean. We will be learning by doing: you will be writing the code and proving the theorems yourself. To do this you need to download and install the course repository .
Part B is the non-mathematical background which you will need in order to make sense of what is going on. It covers basic material of a more “computer-science” nature. I will flag in the exercises the time when it might be helpful to read sections from this part.
Part C is a glossary of many common tactics. The problem sheets in the course repository will flag which tactics you need to learn about and when.