Per Martin-Lof's work on the development of constructive type theory has had a tremendous impact on the fields of logic and the foundations of mathematics. It also has broader philosophical significance and important applications in areas such as computing science and linguistics. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Lof over the last twenty-five years. As well as celebrating the anniversary of the birth of the subject it covers many of the diverse fields which are now influenced by type theory. It is an invaluable record of current activity and includes contributions from N. G. de Bruijn and William Tait, both important figures in the early development of the subject. Also published for the first time is one of Per Martin-Lof's earliest papers.

• 1: Yet another constructivization of classical logic


• 2: Extension of Martin-Löf's type theory with record types


• 3: Type-theoretical checking and philosophy of mathematics


• 4: The Hahn-Banach theorem in type theory


• 5: A realizability interpretation of Martin-Löf's type theory


• 6: The groupoid interpretation of type theory


• 7: An intuitionistic theory of types


• 8: Analytic program derivation in type theory


• 9: About storage operators


• 10: On universes in type theory


• 11: How to believe a machine-checked proof


• 12: Building up a toolbox for Martin-Löf's type theory: subset theory


• 13: An introduction to well-ordering proofs in Martin-Löf's type theory


• 14: Variable-free formalization of the Curry-Howard theory


• 15: The forget-restore principle: a paradigmatic example