The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed -calculus with dependent types. Syntax is treated in a style similar to, but more general than, Martin-Lof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higher-order judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logic-independent tools such as proof editors and proof checkers can be constructed.

We present a higher-order calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice proof-theoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically.

this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs

Abstract not found

Various formulations of constructive type theories have been proposed to serve as the basis for machine-assisted proof and as a theoretical basis for studying programming languages. Many of these calculi include a cumulative hierarchy of "universes," each a type of types closed under a collection of type-forming operations. Universes are of interest for a variety of reasons, some philosophical (predicative vs. impredicative type theories), some theoretical (limitations on the closure properties of type theories), and some practical (to achieve some of the advantages of a type of all types without sacrificing consistency.) The Generalized Calculus of Constructions (CC ! ) is a formal theory of types that includes such a hierarchy of universes. Although essential to the formalization of constructive mathematics, universes are tedious to use in practice, for one is required to make specific choices of universe levels and to ensure that all choices are consistent. In this pa...

Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lL-calculus; the representation mechanism is judgements-as-types, developed for relevant logics. The lL-calculus type theory is a first-order dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required proof-theoretic meta-theory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I -calculus and ML with references. We describe the Curry-Howard-de Bruijn correspondence of the lL-calculus with a s...

In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation.

Presentation of Rewriting and Typing 13 2 Abstract Rewriting Systems 15 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.2 Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.3 Morphisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 2.4 Properties of Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : 18 2.5 Strategies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.6 Criteria : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 2.7 Conclusions and Related Work : : : : : : : : : : : : : : : : : : : : : : : : : 24 3 Topology 27 3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.2 Topology : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 3.3 Equivalence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 3.4 Topological Characte...

The purpose of a logical framework such as LF is to provide a language for defining logical systems suitable for use in a logic-independent proof development environment. All inferential activity in an object logic (in particular, proof search) is to be conducted in the logical framework via the representation of that logic in the framework. An important tool for controlling search in an object logic, the need for which is motivated by the difficulty of reasoning about large and complex systems, is the use of structured theory presentations. In this paper a rudimentary language of structured theory presentations is presented, and the use of this structure in proof search for an arbitrary object logic is explored. The behaviour of structured theory presentations under representation in a logical framework is studied, focusing on the problem of "lifting" presentations from the object logic to the metalogic of the framework. The topic of imposing structure on logic presentations...

In the area of foundations of mathematics and computer science, three related topics dominate. These are -calculus, type theory and logic.