The refereed post-proceedings of the International Workshop of the Types Working Group are presented in this volume. The 17 papers address all current issues in formal reasoning and computer programming based on type theory, including languages and computerized tools for reasoning; applications in several domains, such as analysis of programming languages; certified software; formalization of mathematics; and mathematics education.

This volume is published in Honor of Philip Wadler on the occasion of his 60th birthday, and the collection of papers form a Festschrift for him. The contributions are made by some of the many who know Phil and have been influenced by him. The research papers included here represent some of the areas in which Phil has been active, and the editors thank their colleagues for agreeing to contribute to this Festschrift. We attempt to summarize Phil Wadler's scientific achievements. In addition, we describe the personal style and enthusiasm that Phil has brought to the subject.

An introduction to dependent types, demonstrating the most beautiful aspects, one step at a time. A program's type describes its behavior. Dependent types are a first-class part of a language, and are much more powerful than other kinds of types; using just one language for types and programs allows program descriptions to be as powerful as the programs they describe. The Little Typer explains dependent types, beginning with a very small language that looks very much like Scheme and extending it to cover both programming with dependent types and using dependent types for mathematical reasoning. Readers should be familiar with the basics of a Lisp-like programming language, as presented in the first four chapters of The Little Schemer. The first five chapters of The Little Typer provide the needed tools to understand dependent types; the remaining chapters use these tools to build a bridge between mathematics and programming. Readers will learn that tools they know from programming-pairs, lists, functions, and recursion-can also capture patterns of reasoning. The Little Typer does not attempt to teach either practical programming skills or a fully rigorous approach to types. Instead, it demonstrates the most beautiful aspects as simply as possible, one step at a time.