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34
Mechanizing set theory: Cardinal arithmetic and the axiom of choice
 Journal of Automated Reasoning
, 1996
"... Abstract. Fairly deep results of ZermeloFrænkel (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 resu ..."
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Cited by 16 (9 self)
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Abstract. Fairly deep results of ZermeloFrænkel (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, orderisomorphisms, 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 Wellordering 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
Verifying the Unification Algorithm in LCF
 Science of Computer Programming
, 1985
"... Manna and Waldinger's theory of substitutions and unification has been verified using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an example of interaction with LCF. Translating the theory into LCF's domaintheoretic logic is largely st ..."
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Cited by 9 (0 self)
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Manna and Waldinger's theory of substitutions and unification has been verified using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an example of interaction with LCF. Translating the theory into LCF's domaintheoretic logic is largely straightforward. Wellfounded induction on a complex ordering is translated into nested structural inductions. Correctness of unification is expressed using predicates for such properties as idempotence and mostgenerality. The verification is presented as a series of lemmas. The LCF proofs are compared with the original ones, and with other approaches. It appears di#cult to find a logic that is both simple and flexible, especially for proving termination.
Revisiting the Notion of Function
"... Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both pro ..."
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Cited by 6 (5 self)
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Functions play a central role in type theory, logic and computation. We describe how the notions of functionalisation (the way in which functions can be constructed) and instantiation (the process of applying a function to an argument) have been developed in the last century. We explain how both processes were implemented in Frege's Begriffschrift [17], Russell's Ramified Type Theory [42] and the lambdacalculus (originally introduced by Church [12, 13]) showing that the lambdacalculus misses a crucial aspect of functionalisation. We then pay attention to some special forms of function abstraction that do not exist in the lambdacalculus and we show that various logical constructs (e.g., let expressions and definitions and the use of parameters in mathematics), can be seen as forms of the missing part of functionalisation. Our study of the function concept leads...
A Compendium of Continuous Lattices in MIZAR  Formalizing recent mathematics
, 2002
"... This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar ..."
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Cited by 4 (0 self)
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This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar processor. This eort was originally motivated by the question of whether or not the Mizar system was suciently developed for the task of expressing advanced mathematics. The current state of the formalization57 Mizar articles written by 16 authors indicates that in principle the Mizar system has successfully met the challenge. To our knowledge it is the most sizable eort aimed at mechanically checking some substantial and relatively recent eld of advanced mathematics. However, it does not mean that doing mathematics in Mizar is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of [25] has: (i) prompted many improvements of the Mizar proof checking system; (ii) caused numerous revisions of the the Mizar data base; and (iii) contributed to the \to do" list of further changes to the Mizar system.
Some considerations on the usability of Interactive Provers
"... Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to ana ..."
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Cited by 3 (1 self)
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Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1
The De Bruijn Factor
, 2000
"... Abstract. We study de Bruijn’s ‘loss factor ’ between the size of an ordinary mathematical exposition and its full formal translation inside a computer. This factor is determined by a combination of the amount of detail present in the original text and the expressivity of the system used to do the f ..."
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Cited by 3 (1 self)
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Abstract. We study de Bruijn’s ‘loss factor ’ between the size of an ordinary mathematical exposition and its full formal translation inside a computer. This factor is determined by a combination of the amount of detail present in the original text and the expressivity of the system used to do the formalization. For three specific examples this factor turns out to be approximately equal to four. 1 Loss Factor In ‘A survey of the project Automath ’ de Bruijn wrote (p. 160 in section A.5 of [9] which is a reprint from [1]): A very important thing that can be concluded from all writing experiments is the constancy of the loss factor. The loss factor expresses what we loose in shortness when translating very meticulous ‘ordinary ’ mathematics into Automath. This factor may be quite big, something like 10 or 20, but it is constant: it does not increase if we go further in the book. It would not be too hard to push the constant factor down by efficient abbreviations. Here ⋆ we briefly study this loss factor, which we call the de Bruijn factor. When writing a ‘formal proof ’ (a proof that is entered in a computer in full detail in such a way that