Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Acknowledgements
- Greek alphabet
- 1 Untyped lambda calculus
- 2 Simply typed lambda calculus
- 3 Second order typed lambda calculus
- 4 Types dependent on types
- 5 Types dependent on terms
- 6 The Calculus of Constructions
- 7 The encoding of logical notions in λC
- 8 Definitions
- 9 Extension of λC with definitions
- 10 Rules and properties of λD
- 11 Flag-style natural deduction in λD
- 12 Mathematics in λD: a first attempt
- 13 Sets and subsets
- 14 Numbers and arithmetic in λD
- 15 An elaborated example
- 16 Further perspectives
- Appendix A Logic in λD
- Appendix B Arithmetical axioms, definitions and lemmas
- Appendix C Two complete example proofs in λD
- Appendix D Derivation rules for λD
- References
- Index of names
- Index of definitions
- Index of symbols
- Index of subjects
9 - Extension of λC with definitions
Published online by Cambridge University Press: 05 November 2014
- Frontmatter
- Contents
- Foreword
- Preface
- Acknowledgements
- Greek alphabet
- 1 Untyped lambda calculus
- 2 Simply typed lambda calculus
- 3 Second order typed lambda calculus
- 4 Types dependent on types
- 5 Types dependent on terms
- 6 The Calculus of Constructions
- 7 The encoding of logical notions in λC
- 8 Definitions
- 9 Extension of λC with definitions
- 10 Rules and properties of λD
- 11 Flag-style natural deduction in λD
- 12 Mathematics in λD: a first attempt
- 13 Sets and subsets
- 14 Numbers and arithmetic in λD
- 15 An elaborated example
- 16 Further perspectives
- Appendix A Logic in λD
- Appendix B Arithmetical axioms, definitions and lemmas
- Appendix C Two complete example proofs in λD
- Appendix D Derivation rules for λD
- References
- Index of names
- Index of definitions
- Index of symbols
- Index of subjects
Summary
Extension of λC to the system λD0
In the present chapter we investigate the formal aspects of adding definitions to a type system. In this we follow the pioneering work of N.G. de Bruijn (cf. de Bruijn, 1970). As the basic system we take λC, the most powerful system in the λ-cube. System λC is suitable for the PAT-interpretation, because it encapsulates λP. But it also covers the nice second order aspects of λ2. Therefore, λC appears to be enough for the purpose of ‘coding’ mathematics and mathematical reasonings and is an excellent candidate for the natural extension we want, being almost inevitable for practical applications: the addition of definitions.
We start with an extension leading from λC to a system called λD0. This system contains a formal version of definitions in the usual sense, the so-called descriptive definitions, so it can be used for a great amount of applications in the realm of logic and mathematics. But λD0 does not yet allow a satisfactory representation of axioms and axiomatic notions; these will be considered in the following chapter, in which a small, further extension of λD0 leads to our final system λD. (We have noticed before that we do not consider inductive and recursive definitions, since we can do without them; see Section 8.2.)
In order to give a proper description of λD0, we first extend our set of expressions, as given in Definition 6.3.1 for λC.
- Type
- Chapter
- Information
- Type Theory and Formal ProofAn Introduction, pp. 189 - 210Publisher: Cambridge University PressPrint publication year: 2014