Driven by the question,'What is the computational content of a (formal) proof?', this book
studies fundamental interactions between proof theory and computability. It provides a …

This article explores the use of non-idempotent intersection types in the framework of the λ-
calculus. Different topics are presented in a uniform framework: head normalization, weak …

§ 1. Introduction. For most of its history, mathematics was algorithmic in nature. The
geometric claims in Euclid's Elements fall into two distinct categories:“problems,” which …

We present a ty** system with non-idempotent intersection types, ty** a term syntax
covering three different calculi: the pure λ-calculus, the calculus with explicit substitutions λ …

We study the system IFP of intuitionistic fixed point logic, an extension of intuitionistic first-
order logic by strictly positive inductive and coinductive definitions. We define a realizability …

We present a ty** system for the λ-calculus, with non-idempotent intersection types. As it is
the case in (some) systems with idempotent intersections, a λ-term is typable if and only if it …

Definitions are traditionally considered to be a safe mechanism for introducing concepts on
top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in …

We present a type theory with some proof-irrelevance built into the conversion rule. We
argue that this feature is useful when type theory is used as the logical formalism underlying …

We prove the correctness of a formalised realisability interpretation of extensions of first-
order theories by inductive and coinductive definitions in an untyped λ-calculus with fixed …

When enriching the lambda-calculus with rewriting, union types may be needed to type all
strongly normalizing terms. However, with rewriting, the elimination rule (orE) of union types …