This article The purpose of this paper is to introduce b, a simply typed -calculus that supports type-based recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and Coupet-Grimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of type-based termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper

The paradigm of type-based termination is explored for functional programming with recursive data types. The article introduces , a lambda-calculus with recursion, inductive types, subtyping and bounded quanti cation. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.

We show that the call-by-name monad translation of simply typed lambda calculus extended with sum and product types extends to special and general inductive and coinductive types so that its crucial property of preserving typings and - and commuting reductions is maintained. Speci c similar-purpose translations such as CPS translations follow from the general monad translations by specialization for appropriate concrete monads.

We de ne a sound and complete proof system for ane -retractions in simple types (built over many atoms), and we state a necessary condition for arbitrary -retractions in simple types. We also show a simple necessary condition for polymorphic -retractability and we disprove an earlier conjecture about a stronger necessary condition.

This paper is a comparative study of a number of (intensional-semantically distinct) least and greatest fixed point operators that natural-deduction proof systems for intuitionistic logics can be extended with in a proof-theoretically defendable way. Eight pairs of such operators are analysed. The exposition is centered around a cube-shaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof- and reduction-preserving encoding of one pair in terms of another. The three dimensions of the cube reflect three orthogonal binary options: conventional-style vs. Mendler-style, basic (``[co]iterative'') vs. enhanced (``primitive-[co]recursive''), simple vs. course-of-value [co]induction. Some of the axiomatizations and encodings are well-known; others, however, are novel; the classification into a cube is also new. The differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types.