Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material. "Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems culminating in the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalize mathematics. The only prerequisites are a good knowledge of undergraduate algebra and analysis. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarize themselves with the material"-- Provided by publisher Content: 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 C 8. Definitions 9. Extension of C with definitions 10. Rules and properties of D 11. Flag-style natural deduction in D 12. Mathematics in D: a first attempt 13. Sets and subsets 14. Numbers and arithmetic in D 15. An elaborated example 16. Further perspectives Appendix A. Logic in D Appendix B. Arithmetical axioms, definitions and lemmas Appendix C. Two complete example proofs in D Appendix D. Derivation rules for D References Index of names Index of technical notions Index of defined constants Index of subjects. Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This book provides a gentle step-by-step introduction in the art of formalizing mathematics on the basis of type theory. It is suitable for a broad audience, ranging from undergraduate students to researchers.
