Exploring Formalisation: A Primer in Human-Readable Mathematics in Lean 3 with Examples from Simplicial Topology

152 Pages·2022·1.843 MB·English

by  Clara Löh

Surveys and Tutorials in the Applied
Mathematical Sciences
Volume 11
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Clara Löh
Exploring Formalisation
A Primer in Human-Readable Mathematics
in Lean 3 with Examples from Simplicial
Topology
Clara Löh
Fakultät für Mathematik
Universität Regensburg
Regensburg, Germany

ISSN 2199-4765 ISSN 2199-4773 (electronic)
Surveys and Tutorials in the Applied Mathematical Sciences
ISBN 978-3-031-14648-0 ISBN 978-3-031-14649-7 (eBook)
https://doi.org/10.1007/978-3-031-14649-7

Mathematics Subject Classification (2020): 68V20, 68V15, 55U10, 18A99, 55N35, 55P65

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Contents
Introduction 1
1 The Lean Proof Assistant 7
1.1 Proof Assistants 8
1.2 Foundations 9
1.3 Types 10
1.4 Proofs 12
1.E Exercises 18
2 Basic Examples 21
2.1 Injectivity and Surjectivity of Maps 22
2.1.1 Pen-and-Paper 22
2.1.2 Lean 24
2.2 Induction 31
2.2.1 Pen-and-Paper 32
2.2.2 Lean 32
2.2.3 LeanwithSums 34
2.3 Commutators 37
2.3.1 Pen-and-Paper 37
2.3.2 Lean 38
2.4 The Real Zero 40
2.4.1 Pen-and-Paper 40
2.4.2 Lean 41
2.4.3 LeanwithLimits 44
2.E Exercises 46
v
vi Contents
3 Design Choices 49
3.1 Recurring Design Options 50
3.1.1 TypesandSets 50
3.1.2 StructuresandProperties 50
3.1.3 RestrictiveTypesandCuttingCorners 51
3.1.4 ConstructingExamplesandEvaluation 51
3.2 Simplicial Complexes 52
3.2.1 Pen-and-Paper 52
3.2.2 Lean 54
3.3 Simplicial Maps 58
3.3.1 Pen-and-Paper 58
3.3.2 Lean 60
3.4 Finite Simplicial Complexes 67
3.4.1 Pen-and-Paper 67
3.4.2 Lean 67
3.5 Generating Simplicial Complexes 72
3.5.1 Pen-and-Paper 72
3.5.2 Lean 75
3.6 Combining Simplicial Complexes 78
3.6.1 Pen-and-Paper 79
3.6.2 Lean 80
3.7 The Euler Characteristic 83
3.7.1 Pen-and-Paper 83
3.7.2 Lean 85
3.8 Towards a Library 89
3.8.1 InteractionwithOtherLibraries 89
3.8.2 Completeness 89
3.8.3 SubstructuresandQuotients 90
3.8.4 Generality 91
3.E Exercises 92
4 Abstraction and Prototyping 97
4.1 Direct Formalisation: Functorial Semi-Norms 98
4.1.1 Pen-and-Paper 98
4.1.2 Lean 100
4.2 Indirect Formalisation: Amenable Multiplicity 112
4.2.1 Pen-and-Paper 112
4.2.2 Abstraction 115
4.2.3 Lean 120
4.E Exercises 136
Bibliography 137
Index 143
Introduction
Proof assistants allow us to formalise mathematical de(cid:12)nitions, examples,
theorems, and proofs, and to verify these proofs. The key feature of such a
formalisation is the associated machine-checkable guarantee for correctness
of proofs. The formalisation of mathematics in a proof assistant also has the
following bene(cid:12)ts:
• Implementing mathematics requires a very thorough understanding of
the subject and all corner cases. Therefore, a successful formalisation
leads to a better knowledge of the details.
• Aformalisationinaproofassistantcanleadtoexecutablecodeandthus
provides the opportunity to run experiments on concrete examples.
• Proof assistants encourage abstraction. It is often easier to formalise
concepts declaratively through universal properties instead of through
concrete, potentially cumbersome, constructions. This can lead to a
clearer view on core issues and ideas.
• Interacting with a proof assistant is fun. And addictive.
Allofthesebene(cid:12)tsarenotonlyvaluableforresearch,butalsoinateach-
ing setting: A better understanding of details can increase the awareness for
potentialpitfalls,short-cutsthatmightnotbeobvioustonovicesinthe(cid:12)eld,
andotherstrugglesofstudents.Ifstudentslearnbasicprooftechniquesviaa
proofassistant,theycanobtainimmediatefeedback[52]andcanexperiment
withproofsandexamples.Moreover,studentscanseeinmorepracticalterms
how abstractions simplify mathematical life.
These notes are an extended version of the material for the seminar
ProofLab: Simplicial Topology at Universit(cid:127)at Regensburg in WS 2021/22.
1
2 Introduction
The goal of this seminar was to learn the basics of formalising mathematics
in a proof assistant. In addition to basic examples from (cid:12)rst-year mathemat-
ics, as our sample theory we chose simplicial topology, which is a higher-
dimensional version of graph theory. For the implementation we used the
Lean3proofassistantandthe mathliblibrary,whichcoversasigni(cid:12)cantpart
of undergraduate mathematics.
The focus of this seminar was to (cid:12)nd straightforward formalisations that
lead to human-readable machine-veri(cid:12)able proofs. While some of the chal-
lengesarespeci(cid:12)ctosimplicialcomplexesandLean,mostofthemaregeneric
and will also occur with other (cid:12)elds and other proof assistants.
These notes are not a comprehensive treatment of the full programming
language Lean and the underlying dependent type theory. Instead, we study
thetransitionbetweenpen-and-papermathematicsandthemachinecounter-
partaswellasthedesignoptionsinthisprocess.Thisisillustratedwithvar-
ious examples, proceeding in three stages: Examples from (cid:12)rst year courses,
examples from simplicial topology, and recent sample results from algebraic
and geometric topology.
Overview of these notes
• Chapter1Westartwithaquickintroductiontoproofassistantsandto
Lean in particular. This requires a basic understanding of the founda-
tions of mathematics, i.e., how objects, statements, and proofs can be
described in a formal language. In particular, this includes the ability
to assemble and disassemble logical statements.
• Chapter 2 We then consider a (cid:12)rst selection of simple examples: In-
jectivity and surjectivity of maps, induction and sums over (cid:12)nite sets,
commutatorsingrouptheory,andafundamentalpropertyofrealnum-
bers. We always follow the same structure: We (cid:12)rst develop a pen-and-
paper version and then formalise it in Lean. The examples are chosen
in such a way that basic proof patterns occur. The later examples also
illustrate the interaction with the mathlib library.
• Chapter 3 As a larger example theory, we consider basic concepts and
examples from simplicial topology. We formalise simplicial complexes,
simplicial maps, and the Euler characteristic. During this formalisa-
tion, we discuss typical design questions and experiment with concrete
examples.
• Chapter 4 Finally, we consider situations from recent research in al-
gebraic and geometric topology. These serve as examples of how proof
assistants encourage abstraction and thus can lead to interesting per-
spectives.
PrerequisitesItriedtokeepthingsassimpleaspossible.Onthemathemati-
calside,basicexperiencewithproofsandlogicalformulaeaswellaswithbasic
Introduction 3
mathematical objects (e.g., sets, maps, groups, reals) is required. Chapter 4
uses basic terminology on categories and functors; Section 4.2 is probably
a bit mysterious without background knowledge in basic algebraic topology
(fundamental groups, universal coverings, cohomology, CW-complexes), but
the formalisation strategy will be understandable without knowing the se-
mantics of these notions.
Ontheprogrammingside,acquaintancewithbasicprogrammingconcepts
ishelpful(e.g.,declarations,types,basiccontrolstructures,recursion).Prior
exposure to the functional programming paradigm is a bonus.
ExercisesEachchapterendswithaselectionofexercisesaroundtheformali-
sation of mathematical terminology, theorems, and proofs. Skeleton (cid:12)les and
solutions to selected exercises of Chapter 2 and Chapter 3 are provided in
the git repository:
Source code All source code discussed in these notes is available in a git
repository:
https://gitlab.com/polywuisch/exploring-formalisation
Errata Comments and corrections for these notes can be submitted by email
to [email protected]; errata will be collected at
https://loeh.app.uni-regensburg.de/exploring-formalisation
Further reading These notes are merely a primer in the formalisation of
mathematics. To get a deeper understanding of proof assistants and formal-
isation it is highly recommended to dive into the literature, tutorials, and
larger formalisation projects.
• J. Avigad, L. de Moura, S. Kong. Theorem Proving in Lean, Re-
lease3.23.0,https://leanprover.github.io/theorem proving in lean/,2021.
• J. Avigad, K. Buzzard, R.Y. Lewis, P. Massot. Mathematics in Lean.
https://leanprover-community.github.io/mathematics in lean/
• A. Baanen, A. Bentkamp, J. Blanchette, J. H(cid:127)olzl, J. Limperg. The
Hitchhiker’sGuidetoLogicalVeri(cid:12)cation,2021StandardEdition,2021.
https://cs.brown.edu/courses/cs1951x/static (cid:12)les/main.pdf
• Y. Bertot, P. Cast(cid:19)eran. Interactive Theorem Proving and Program De-
velopment. Coq’Art: The Calculus of Inductive Constructions, Texts in
Theoretical Computer Science, An EATCS Series, Springer, 2004.
• K. Buzzard, J. Commelin, P. Massot. Lean perfectoid spaces.
https://leanprover-community.github.io/lean-perfectoid-spaces/
• K. Buzzard, M. Pedramfar. The natural number game.
https://www.ma.imperial.ac.uk/∼buzzard/xena/natural number game/