Implementation of Bourbaki´s Elements of Mathematics in Coq: ; Part Two ; Ordered Sets, Cardinals, Integers

French title: Implémentation de la théorie des ensembles de Bourbaki dans Coq; partie 2; Ensembles Ordonnés, cardinaux, nombres entiers

Author: José Grimm

Location: Sophia Antipolis – Méditerranée

Inria Research Report Number: 7150

Revision: 2

Team: Apics

Date: December 2009

Revised Date: April 2010

Keywords: Gaia, Coq, Bourbaki, orders, cardinals, ordinals, integers.

French keywords: Gaia, Coq, Bourbaki, ordre, cardinaux, ordinaux, entiers.

Note: Work done in collaboration with Alban Quadrat.

Email: Jose.Grimm@sophia.inria.fr

Abstract

We believe that it is possible to put the whole work of Bourbaki into a computer. One of the objectives of the Gaia project concerns homological algebra (theory as well as algorithms); in a first step we want to implement all nine chapters of the book Algebra. But this requires a theory of sets (with axiom of choice, etc.) more powerful than what is provided by Ensembles; we have chosen the work of Carlos Simpson as basis. This reports lists and comments all definitions and theorems of the Chapter “Ordered Sets, Cardinals, Integers”. The code (including some exercises) is available on the Web, under http://www-sop.inria.fr/apics/gaia.

French Abstract

Nous pensons qu´il est possible de mettre dans un ordinateur l´ensemble de l´œuvre de Bourbaki. L´un des objectifs du projet Gaia concerne l´algèbre homologique (théeorie et algorithmes); dans une première étape nous voulons implémenter les neuf chapitres du livre Algèbre. Au préalable, il faut implémenter la théorie des ensembles. Nous utilisons l´Assistant de Preuve Coq; les choix fondamentaux et axiomes sont ceux proposés par Carlos Simpson. Ce rapport liste et commente toutes les définitions et théorèmes du Chapitre “Ensembles ordonnés, cardinaux, nombres entiers”. Une petite partie des exercises a été résolue. Le code est disponible sur le site Web http://www-sop.inria.fr/apics/gaia.


Short Table of Contents


1. Introduction
2. Order relations. Ordered sets
3. Well-ordered sets
4. Equipotent Sets. Cardinals
5. Natural integers. Finite sets
6. Properties of integers
7. Infinite sets
8. The size of one
9. Exercises
10. Theorems, Notations, Definitions
Bibliography
Table of Contents
Index


1. Introduction

2. Order relations. Ordered sets

3. Well-ordered sets

4. Equipotent Sets. Cardinals

5. Natural integers. Finite sets

6. Properties of integers

7. Infinite sets

8. The size of one

9. Exercises

10. Theorems, Notations, Definitions


Bibliography

[1] Yves Bertod, Pierre Castéran. Interactive Theorem Proving and Program Development. Springer, 2004.

[2] N. Bourbaki. Elements of Mathematics, Theory of Sets. Springer, 1968.

[3] N. Bourbaki. Éléments de mathématiques, Théorie des ensembles. Diffusion CCLS, 1970.

[4] Coq Development Team. The Coq reference manual. http://coq.inria.fr.

[5] José Grimm. Implementation of Bourbaki´s Elements of Mathematics in Coq: Part One, Theory of Sets. Research Report, number RR-6999, INRIA, 2009, http://hal.inria.fr/inria-00408143/en/

[6] Douglas Hofstadter. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, 1979.

[7] Jean-Louis Krivine. Théorie axiomatique des ensembles. Presses Universitaires de France, 1972.

Table of Contents


1. Introduction
     1.1. Objectives
     1.2. Content of this document
     1.3. Terminology
     1.4. Tactics
     1.5. Removed theorems
          1.5.1. Definition of a function by induction
          1.5.2. Intervals
2. Order relations. Ordered sets
     2.1. Definition of an order relation
     2.2. Preorder relations
     2.3. Notation and terminology
     2.4. Ordered subsets. Product of ordered sets
     2.5. Increasing mappings
     2.6. Maximal and minimal elements
     2.7. Greatest element and least element
     2.8. Upper and lower bounds
     2.9. Least upper bound and greatest lower bound
     2.10. Directed sets
     2.11. Lattices
     2.12. Totally ordered sets
     2.13. Intervals
3. Well-ordered sets
     3.1. Segments of a well-ordered set
     3.2. The principle of transfinite induction
     3.3. Zermelo´s theorem
     3.4. Inductive sets
     3.5. Isomorphisms of well-ordered sets
     3.6. Lexicographic products
     3.7. Ordinals
4. Equipotent Sets. Cardinals
     4.1. The cardinal of a set
     4.2. Order relation between cardinals
     4.3. Operations on cardinals
     4.4. Properties of the cardinals 0 and 1
     4.5. Exponentiation of cardinals
     4.6. Order relation and operations on cardinals
5. Natural integers. Finite sets
     5.1. Definition of integers
     5.2. Inequalities between integers
     5.3. The set of natural integers
     5.4. The principle of induction
     5.5. Finite subsets of ordered sets
     5.6. Properties of finite character
     5.7. Finite cardinals and the type nat
          5.7.1. Pseudo-ordinals
          5.7.2. Pseudo-ordinals and the type nat
          5.7.3. Bijection between nat and the integers
6. Properties of integers
     6.1. Operations on integers and finite sets
     6.2. Strict inequalities between integers
     6.3. Intervals in sets of integers
     6.4. Finite sequences
          6.4.1. Lists as functions
          6.4.2. Contracting lists
     6.5. Characteristic functions on sets
     6.6. Euclidean Division
     6.7. Expansion to base b
     6.8. Combinatorial analysis
          6.8.1. Iterated functions
          6.8.2. Factorial
          6.8.3. Number of injections
          6.8.4. Number of coverings
          6.8.5. The binomial coefficient
          6.8.6. Number of increasing functions
          6.8.7. Number of monomials
7. Infinite sets
     7.1. The set of natural integers
     7.2. Definition of mappings by induction
     7.3. Properties of infinite cardinals
     7.4. Countable sets
     7.5. Stationary sequences
8. The size of one
                    8.1. First comments.
                    8.2. The case of Coq.
                    8.3. Premiminary computations.
                    8.4. Size of a triple.
                    8.5. Size of a graph.
                    8.6. Size of a bijection.
                    8.7. Conclusion.
9. Exercises
     9.1. Section 1
                    9.1.1. 1.
                    9.1.2. 2.
                    9.1.3. 3.
                    9.1.4. 4.
                    9.1.5. 5.
                    9.1.6. 6.
                    9.1.7. 7.
                    9.1.8. 8.
                    9.1.9. 9.
                    9.1.10. 10.
                    9.1.11. 11.
                    9.1.12. 12.
                    9.1.13. 13.
                    9.1.14. 14.
                    9.1.15. 15.
                    9.1.16. ¶ 16.
                    9.1.17. ¶ 17.
                    9.1.18. ¶ 18.
                    9.1.19. 19.
                    9.1.20. ¶ 20.
                    9.1.21. 21.
                    9.1.22. ¶ 22.
                    9.1.23. ¶ 23.
                    9.1.24. 24.
     9.2. Section 2
                    9.2.1. 1.
                    9.2.2. 2.
                    9.2.3. 3.
                    9.2.4. ¶ 4.
                    9.2.5. 5.
                    9.2.6. ¶ 6.
                    9.2.7. ¶ 7.
                    9.2.8. ¶ 8.
                    9.2.9. 9.
                    9.2.10. 10.
                    9.2.11. ¶ 11.
                    9.2.12. ¶ 12.
                    9.2.13. 13.
                    9.2.14. ¶ 14.
                    9.2.15. 15.
                    9.2.16. ¶ 16.
                    9.2.17. ¶ 17.
                    9.2.18. ¶ 18.
                    9.2.19. 19.
                    9.2.20. ¶ 20.
     9.3. Section 3
                    9.3.1. ¶ 1.
                    9.3.2. 2.
                    9.3.3. ¶ 3.
                    9.3.4. 4.
                    9.3.5. 5.
                    9.3.6. 6.
     9.4. Section 4
                    9.4.1. 1.
                    9.4.2. 2.
                    9.4.3. 3.
                    9.4.4. 4.
                    9.4.5. ¶ 5.
                    9.4.6. ¶ 6.
                    9.4.7. 7.
                    9.4.8. ¶ 8.
                    9.4.9. ¶ 9.
                    9.4.10. ¶ 10.
                    9.4.11. ¶ 11.
     9.5. Section 5
                    9.5.1. 1.
                    9.5.2. 2.
                    9.5.3. 3.
                    9.5.4. 4.
                    9.5.5. 5.
                    9.5.6. 6.
                    9.5.7. 7.
                    9.5.8. 8.
                    9.5.9. 9.
                    9.5.10. 10.
                    9.5.11. ¶ 11.
                    9.5.12. ¶ 12.
                    9.5.13. ¶ 13.
                    9.5.14. ¶ 14.
                    9.5.15. ¶ 15.
                    9.5.16. 16.
                    9.5.17. ¶ 17.
                    9.5.18. 18.
     9.6. Section 6.
                    9.6.1. 1.
                    9.6.2. 2.
                    9.6.3. 3.
                    9.6.4. 4.
                    9.6.5. 5.
                    9.6.6. 6.
                    9.6.7. 7.
                    9.6.8. 8.
                    9.6.9. 9.
                    9.6.10. ¶ 10.
                    9.6.11. ¶ 11.
                    9.6.12. ¶ 12.
                    9.6.13. ¶ 13.
                    9.6.14. ¶ 14.
                    9.6.15. ¶ 15.
                    9.6.16. 16.
                    9.6.17. 17.
                    9.6.18. ¶ 18.
                    9.6.19. ¶ 19.
                    9.6.20. ¶ 20.
                    9.6.21. ¶ 21.
                    9.6.22. ¶ 22.
                    9.6.23. ¶ 23.
                    9.6.24. ¶ 24.
                    9.6.25. ¶ 25.
                    9.6.26. ¶ 26.
                    9.6.27. ¶ 27.
                    9.6.28. ¶ 28.
                    9.6.29. 29.
                    9.6.30. 30.
                    9.6.31. ¶ 31.
                    9.6.32. ¶ 32.
                    9.6.33. ¶ 33.
10. Theorems, Notations, Definitions
                    10.1. Theorems of Chapter 1
                    10.2. Theorems of Chapter 2
                    10.3. Theorems of Chapter 3
                    10.4. Theorems of Chapter 4
                    10.5. Theorems of Chapter 5
                    10.6. Theorems of Chapter 6
                    10.7. Symbols
                    10.8. Letters
                    10.9. Words
Bibliography
Index