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...

from the last declaration in \Delta (which is p:'). (oe-E) In fact the ([\Theta]) is not exactly the ([\Theta]) that is found by induction. Possibly some of the free variables in ([\Theta]) are renamed. What happens is the following: 1. Consider the proof-context \Delta 1 ] \Delta 2 and especially the renaming of the declared variables in \Delta 2 that has been caused by the operation ]. 2. Rename the free proof-variables in ([\Theta]) accordingly, obtaining say, ([\Theta 0 ]). 3. Apply ([\Sigma]) to ([\Theta 0 ]). (There will in practice be no confusion if we just write ([\Theta]) instead.) Of course the intended meaning is that the judgement below the double lines is derivable if the judgement above the lines is. This will be proved later in Theorem 3.2.8. It should be clear at this point however that there is a one-to-one correspondence between the occurrences of ' as a (non-discharged) premise in the deduction and declarations p:' in \Delta. Notation. If for \Sigma a deducti...

One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. Acknowledgement. The following persons provided help in various ways. Erik Barendsen, Jon Barwise, Johan van Benthem, Andreas Blass, Olivier Danvy, Wil Dekkers, Marko van Eekelen, Sol Feferman, Andrzej Filinski, Twan Laan, Jan Kuper, Pierre Lescanne, Hans Mooij, Robert Maron, Rinus Plasmeijer, Randy Pollack, Kristoffer Rose, Richard Shore, Rick Statman and Simon Thompson. Partial support came from the European HCM project Typed lambda calculus (CHRXCT-92-0046), the Esprit Working Group Types (21900) and the Dutch NWO project WINST (612-316-607). 1. Introduction This paper is written to honor Church's gr...

Abstract. Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are

Formal reasoning is notoriously long and arduous; in order to use it to reason effectively in the construction of programs it is, therefore, paramount that we design our notations to be both clear and economical. Taking examples from AI, from imperative programming, from the use of the Bird-Meertens formalism and from category theory we demonstrate how the right choice of what to denote and how it is denoted can make significant improvements to formal calculations. Brief mention is also made of the connection between economical notation and properties of type. 1 2 Foreword Earlier this year I was an invited speaker at the 5th British Computer Society Theoretical Computer Science Colloquium held at Royal Holloway and Bedford New College, London. Before you is the text of my lecture, almost but not quite as given at the conference. (Perhaps the best way to describe the present paper is as the lecture that I should have given.) The publication of the text of the lecture will, ...

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Contemporary proof veri cators often use a command language to construct proofs. These commands are often called tactics. This new generation of theorem provers is a substantial improvement over earlier ones such asAUTOMATH. Based on experience with these new provers we feel the need to study these languages further, especially, because we think that these may be improved in their adequateness to express proofs closer to the established mathematical vernacular. We also feel that a systematic treatment of these vernaculars may lead to an improvement towards the automatic inference of trivial proof steps. In any case a systematic treatment will lead to a better understanding of the command languages. This exercise is carried out in the setting of Pure Type Systems (PTSs) in which a whole range of logics can be embedded. We rstidentify a subclass of PTSs, called the PTSs for logic. For this class we de ne a formal mathematical vernacular and we prove elementary sound- and completeness. Via an elaborate example we try to assess how easy proofs in mathematics can be written down in our vernacular along the lines of the original proofs. 1

We give an overview of issues surrounding computerverified theorem proving in the standard pure-mathematical context.

Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an in-depth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends and integrates state-of-the-art technologies from Natural Language Processing (Discourse Representation Theory) and Automated Reasoning (Proof Planning) in a novel and promising way, having the potential to initiate progress in both of these disciplines.

. Our long-range goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and deduction perspective and give an in-depth analysis for a sample textbook proof. A three phase model for proof understanding is developed: parsing, structuring and refining. It shows that the combined application of techniques from both NLP and AR is quite successful. Moreover, it allows to uncover interesting insights that might initiate progress in both AI disciplines. Keywords: automated reasoning, natural language processing, discourse analysis 1 Introduction In [12], John McCarthy notes that "Checking mathematical proofs is potentially one of the most interesting and useful applications of automatic computers". In the first half of the 1960s, one of his students, namely Paul Abrahams, implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, "was primarily directed towar...