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 Contents 8 Foreword 14 Preface 16 Acknowledgements 28 Greek alphabet 29 Untyped lambda calculus 30 Simply typed lambda calculus 62 Second order typed lambda calculus 98 Types dependent on types 114 Types dependent on terms 132 The Calculus of Constructions 152 The encoding of logical notions in 166 Definitions 194 Extension of λC with definitions 218 Rules and properties of 240 Flag-style natural deduction in 254 Mathematics in 286 Sets and subsets 308 Numbers and arithmetic in λD 334 An elaborated example 378 Further perspectives 408 Appendix ALogic in λD 420 Appendix B Arithmetical axioms, definitions and lemmas 426 Appendix C Two complete example proofs in 432 Appendix DDerivation rules for λD 438 References 440 Index of names 448 Index of definitions 450 Index of symbols 452 Index of subjects 454 Online,ISBN:,9781139567725,Hardback,ISBN:,9781107036505 Online ISBN: 9781139567725,Hardback ISBN: 9781107036505 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.
