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Some lambda calculus and type theory formalized
 Journal of Automated Reasoning
, 1999
"... 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 ..."
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Cited by 54 (7 self)
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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.
A Simply Typed Context Calculus with FirstClass Environments
, 2002
"... . We introduce a simply typed calculus " which has both contexts and environments as firstclass 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 mechani ..."
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Cited by 12 (1 self)
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. We introduce a simply typed calculus " which has both contexts and environments as firstclass 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 ficalculus, 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...
External and internal syntax of the λcalculus
 In: Buchberger, Ida, Kutsia (Eds.), Proc. of the AustrianJapanese Workshop on Symbolic Computation in Software Science, SCSS 2008. No. 08–08 in RISCLinz Report Series
"... 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 a ..."
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Cited by 2 (1 self)
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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
Viewing λterms through Maps
"... In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λterm. We use binary trees over 0 and 1 as maps, but some wellformedness conditions are required. We develop a representation of lambda terms usi ..."
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In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λterm. We use binary trees over 0 and 1 as maps, but some wellformedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λterm). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λterms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively. The map representation has good properties. Substitution does not require adjustment of binding information: neither αconversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation.
RISCLinz Report Series No. 0808
, 2008
"... Science (SCSS 2008), held in the Castle of Hagenberg, Austria, in July 12– ..."
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Science (SCSS 2008), held in the Castle of Hagenberg, Austria, in July 12–
ELearning of Foundation of Computer Science
"... 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 fo ..."
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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 elearning 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.