We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science --- LICS'96 (E. Clarke editor), pp. 264--275, New Brunswick, NJ, July 27--30 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination. 1 Introduction A logical framework is a formal system desig...

We propose a refinement of the type theory underlying the LF logical framework by a form of subtypes and intersection types. This refinement preserves desirable features of LF, such as decidability of type-checking, and at the same time considerably simplifies the representations of many deductive systems. A subtheory can be applied directly to hereditary Harrop formulas which form the basis of Prolog and Isabelle. 1 Introduction Over the past two years we have carried out extensive experiments in the application of the LF Logical Framework [HHP93] to represent and implement deductive systems and their metatheory. Such systems arise naturally in the study of logic and the theory of programming languages. For example, we have formalized the operational semantics and type system of Mini-ML and implemented a proof of type preservation [MP91] and the correctness of a compiler to a variant of the Categorical Abstract Machine [HP92]. LF is based on a predicative type theory with dependent t...

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...

This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higher-order primitive recursion and higher-order intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The \Sigma-types of the calculus enable us to achieve this. There are many similarities with the subset interpretation of Martin-Lof type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical bo...

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions to the Calculus of Constructions --- we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fi-reduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However j-contractions beh...

The smallest transitive relation ! on well-typed normal terms such that if t is a strict subterm of u then t ! u and if T is the normal form of the type of t and the term t is not a sort then T ! t is well-founded in the type systems of the cube. Thus every term admits a j-long normal form. Introduction In this paper we prove that the smallest transitive relation ! on well-typed normal terms such that ffl if t is a strict subterm of u then t ! u, ffl if T is the normal form of the type of t and the term t is not a sort then T ! t is well-founded in the type systems of the cube [1]. This result is proved using the notion of marked terms introduced by de Vrijer [6]. A motivation for this theorem is to define the j-long form of a normal term in these type systems. In simply typed -calculus, to define the j-long form of a normal term we first define the j-long form of a variable x of type P 1 ! ::: ! P n ! P (P atomic) as the term [y 1 : P 1 ]:::[y n : P n ](x y 0 1 ::: y 0 n ) w...

The Calculus of Constructions (CC) ([Coquand 1985]) is a typed lambda calculus for higher order intuitionistic logic: proofs of the higher order logic are interpreted as lambda terms and formulas as types. It is also the union of Girard's system F! ([Girard 1972]), a higher order typed lambda calculus, and a first order dependent typed lambda calculus in the style of de Bruijn's Automath ([de Bruijn 1980]) or Martin-Lof's intuitionistic theory of types ([Martin-Lof 1984]). Using the impredicative coding of data types in F! , the Calculus of Constructions thus becomes a higher order language for the typing of functional programs. We shall introduce and try to explain CC by exploiting especially the first point of view, by introducing a typed lambda calculus that faithfully represent higher order predicate logic (so for this system the Curry-Howard `formulas-as-types isomorphism' is really an isomorphism.) Then we discuss some propositions that are provable in CC but not in the higher or...

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 propose a new framework for representing logics, called LF + and based on the Edinburgh Logical Framework. The new framework allows us to give, apparently for the first time, general definitions which capture how well a logic has been represented. These definitions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlying logic. This distinction does not seem to be possible with other frameworks. Using our definitions, we show that, for example, natural deduction first-order logic can be well-represented in LF + , whereas linear and relevant logics cannot. We also show that our syntactic definitions of representation have a simple formulation as indexed isomorphisms, which both confirms that our approach is a natural one and provides a link between type-theoretic and categorical approaches to frameworks. 1 Introduction Much effort has been devoted to building systems for supporting the construction of f...

. This paper presents a system for explicit substitutions in Pure Type Systems (PTS). The system allows to solve type checking, type inhabitation, higher-order unification, and type inference for PTS using purely first-order machinery. A novel feature of our system is that it combines substitutions and variable declarations. This allows as a sideeffect to type check let-bindings. Our treatment of meta-variables is also explicit, such that instantiations of meta-variables is internalized in the calculus. This produces a confluent -calculus with distinguished holes and explicit substitutions that is insensitive to ff-conversion, and allows directly embedding the system into rewriting logic. 1 Introduction Explicit substitutions provide a convenient framework for encoding higher-order typed -calculus using first-order machinery. In particular, this allows to integrate higher-order unification with first-order provers, rewriting logic, and to delay evaluation and resolve scoping...