Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.

Abstract not found

. We introduce a simply typed -calculus " which has both contexts and environments as first-class values. In ", holes in contexts are represented by ordinary variables of appropriate types and hole filling is represented by the functional application together with a new abstraction mechanism which takes care of packing and unpacking of the term which is used to fill in the holes of the context. " is a conservative extension of the simply typed fi-calculus, enjoys subject reduction property, is confluent and strongly normalizing. The traditional method of defining substitution does not work for our calculus. So, we also introduce a new method of defining substitution. Although we introduce the new definition of substitution out of necessity, the new definition turns out to be conceptually simpler than the traditional definition of substitution. 1 Introduction Informally speaking, a context (in -calculus) is a -term with some holes in it. For example, writing [ ] for a hole, y: [ ] is a...

There is growing interest in the study of the syntactic structure of expressions equipped with a variable binding mechanism. The importance of this study can be justified for various reasons, e.g. educational, scientific and engineering reasons. This study is educationally important since in logic and computer science, we cannot avoid teaching the

Abstract. We report our teaching experience of undergraduate courses on logic and computation. The topics covered are the syntax and the semantics of basic logical and computational calculi. We put our emphasis on the treatment of formal systems and formal reasoning which is notoriously difficult for most students. To overcome this difficulty, we developed an e-learning system called CAL and have been using it for the last several of years. The CAL system turned out not only useful, but also essential to achieve the goals for our courses. In this paper we give an overview of the CAL system, describe the design principle of our courses, and discuss our teaching experiences.

Science (SCSS 2008), held in the Castle of Hagenberg, Austria, in July 12–