This book develops a type theory, studies its properties, and explains its uses in computer science. The book focuses in particular on how the study of type theory may offer a powerful and uniform language for programming, program specification and development, and logical reasoning. The type theory developed here reflects a conceptual distinction between logical propositions and computational data types. Starting from an introduction of the basic concepts, the author explains the meaning and use of the type-theoretic language with proof-theoretic justifications, and discusses various issues in the study of type theory. The practical use of the language is illustrated by developing an approach to specification and data refinement in type theory, which supports modular development of specification, programs, and proofs. Students and researchers in computer science and logic will welcome this exciting new book.

This book contains a selection of papers presented at the ?rst annual workshop of the TYPES Working Group (Computer-Assisted Reasoning Based on Type Theory, EU IST project 29001), which was held 8th 12th of December, 2000 at the University of Durham, Durham, UK. It was attended by about 80 researchers. The workshop follows a series of meetings organised in 1993, 1994, 1995, 1996, 1998, and 1999 under the auspices of the Esprit BRA6435 and the - prit Working Group 21900 for the previous TYPES projects. Those proceedings were also published in the LNCS series, edited by Henk Barendregt and Tobias Nipkow (Vol. 806, 1993), by Peter Dybjer, Bengt Nordstr]om, and Jan Smith (Vol. 996, 1994), by Stefano Berardi and Mario Coppo (Vol. 1158, 1995), by Christine Paulin-Mohring and Eduardo Gimenez (Vol. 1512, 1996), by Thorsten Altenkirch, Wolfgang Naraschewski, and Bernhard Reus (Vol. 1657, 1998), and by Thierry Coquand, Peter Dybjer, Bengt Nordstr]om, and Jan Smith (Vol. 1956, 1999). The Esprit BRA6453 was itself a continuation of the former Esprit - tion 3245, Logical Frameworks: Design, Implementation, and Experiments. The articles from the annual workshops under that Action were edited by Gerard Huet and Gordon Plotkin in the books Logical Frameworks and Logical En- ronments, both published by Cambridge University Press. Acknowledgements We are very grateful to members of Durham s Computer Assisted Reasoning Group, especially Robert Kiessling, for helping to organise the workshop. Robert s contribution was key to the success of the meeting."