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

We present a basis for a category-theoretic account of Mendler-style inductive types. The account is based on suitably defined concepts of Mendler-style algebra and algebra homomorphism; Mendler-style inductive types are identified with initial Mendler-style algebras. We use the identification to obtain a reduction of conventional inductive types to Mendler-style inductive types and a reduction in the presence of certain restricted existential types of Mendler-style inductive types to conventional inductive types.

This paper is part of an ongoing research programme at the Computer Science Department of the University of Udine on proof editors, started in 1992, based on HOAS encodings in dependent typed #-calculus for program logics [15, 21, 16]. In this paper, we investigate the applicability of this approach to the modal -calculus. Due to its expressive power, we adopt the Calculus of Inductive Constructions (CIC), implemented in the system Coq. Beside its importance in the theory and verification of processes, the modal -calculus is interesting also for its syntactic and proof theoretic peculiarities. These idiosyncrasies are mainly due to a) the negative arity of "" (i.e., the bound variable x ranges over the same syntactic class of x#); b) a context-sensitive grammar due the condition on x#; c) rules with complex side conditions (sequent-style "proof " rules). These anomalies escape the "standard" representation paradigm of CIC; hence, we need to accommodate special techniques for enforcing these peculiarities. Moreover, since generated editors allow the user to reason "under assumptions", the designer of a proof editor for a given logic is urged to look for a Natural Deduction formulation of the system. Hence, we introduce a new proof system N # K in Natural Deduction style for K. This system should be more natural to use than traditional Hilbert-style systems; moreover, it takes best advantage of the possibility of manipulating assumptions o#ered by CIC in order to implement the problematic substitution of formul for variables. In fact, substitutions are delayed as much as possible, and are kept in the derivation context by means of assumptions. This mechanism fits perfectly the stack discipline of assumptions of Natural Deduction, and it is neatly formalized in CIC. Bes...

Abstract We advocate the Mendler style of coding terminating recursion schemes as combinators by showing on the example of two simple and much used schemes (course-of-value iteration and simultaneous iteration) that choosing the Mendler style can sometimes lead to handier constructions than following the construction style of cata and para like combinators. 1 Introduction This paper is intended as an advert for something we call the Mendler style. This is a not too widely known style of coding terminating recursion schemes by combinators that di ers from the construction style of the famous cata and para combinators (for iteration and primitive-recursion, respectively) [Mal90,Mee92], here called the conventional style. The paper ar...

Abstract We present a basis for a categorical account of Mendler-style inductive types by introducing a notion of initial Mendler-style algebras and use it for giving a reduction of (conventional) inductive types to Mendler-style inductive types and two reductions of Mendler-style inductive types to (conventional) inductive types. 1 Introduction For a category-theoretically minded computing scientist, the phrase "inductive types" normally serves as a name for a construction that is usually conveniently modelled by means of the notion of initial algebras [Mal90,Fok92]. In typed lambda calculi (type theories), at the same time, one encounters both inductive types in this (call it conventional) sense and Mendler-style inductive types [Men...

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.

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