In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λ-calculus enriched by pattern-matching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.

In this paper we present the algebraic--cube, an extension of Barendregt's -cube with first- and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraic--cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraic--cube. We also prove that local confluence is a modular property of all the systems in the algebraic--cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...

A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the left-hand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years, several paradigms have been investigated for the implementatio...

In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types -- substitution, expansion, and lifting -- are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the type-constant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) head-normal form, and that terms whose type does not contain ! are normalizable.

This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial type assignment on Curryfied Term Rewrite Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes in the tree representation of terms. Two operations on types are specified that are used to define type assignment on terms and rewrite rules, and are proven to be sound on both terms and rewrite rules. Using a more liberal approach to recursion, a general scheme for recursive definitions is presented, that generalizes primitive recursion, but has full Turing-machine computational power. It will be proved that, for all systems that satisfy this scheme, every typeable term is strongly normalizable. Introduction Most functional programming languages, like Miranda [23] or ML [19] for instance, although implemented through an extended Lambda Calculus ...

Abstract. In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments à la Tait and Girard, therefore explaining the name of the improved ordering. 1

Abstract. In this paper we study normalization properties of rewrite systems that are typeable using intersection types with and with sorts. We prove two normalization properties of typeable systems. On one hand, for all systems that satisfy a variant of the Jouannaud-Okada Recursion Scheme, every term typeable with a type that is not is head normalizable. On the other hand, non-Curryfied terms that are typeable with a type that does not contain, are normalizable.

In this paper we present the algebraic-λ-cube, an extension of Barendregt's λ-cube with first- and higher-order algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraic-λ-cube, provided that the first-order rewrite rules are non-duplicating and the higher-order rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraic-λ-cube. We also prove that local confluence is a modular property of all the systems in the algebraic-λ-cube, provided that the higher-order rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence.

. We introduce the framework of algebraic type systems, a generalisation of pure type systems with higher order rewriting `a la JouannaudOkada, and initiate a generic study of the modular properties of these systems. We give a general criterion for one system of this framework to be strongly normalising. As an application of our criterion, we recover all previous strong normalisation results for algebraic type systems. 1 Introduction Algebraic-functional languages, introduced by Jouannaud and Okada in [19], are based on a very powerful paradigm combining type theory and higher-order rewriting systems. These languages embed in typed -calculi higher-order rewriting and hence allow the definition of abstract data types as it is done in equational languages such as OBJ. Examples of such languages which have been studied in the literature include the algebraic simply typed -calculus ([19]), algebraic type assignments systems ([2]) and the algebraic calculus of constructions ([3]). In this ...

. We introduce the framework of algebraic pure type systems, a generalisation of pure type systems with higher order rewriting `a la Jouannaud-Okada, and initiate a generic study of the modular properties of these systems. We give a general criterion for a system of this framework to be strongly normalising. As an application of our criterion, we recover all previous strong normalisation results for algebraic pure type systems. 1 Introduction Algebraico-functional languages, introduced by Jouannaud and Okada in [18], are based on a very powerful paradigm combining type theory and higher-order rewriting systems. These languages embed in typed -calculi higher-order rewriting and hence allow the definition of abstract data types as it is done in equational languages such as OBJ. Examples of such languages which have been studied in the literature include the algebraic simply typed -calculus ([18]), algebraic type assignments systems ([1]) and the algebraic calculus of constructions ([2]...