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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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Cited by 23 (0 self)
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Analysis and Transformation of Proof Procedures
, 1994
"... Automated theorem proving has made great progress during the last few decades. Proofs of more and more difficult theorems are being found faster and faster. However, the exponential increase in the size of the search space remains for many theorem proving problems. Logic program analysis and transfo ..."
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Cited by 8 (2 self)
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Automated theorem proving has made great progress during the last few decades. Proofs of more and more difficult theorems are being found faster and faster. However, the exponential increase in the size of the search space remains for many theorem proving problems. Logic program analysis and transformation techniques have also made progress during the last few years and automated theorem proving can benefit from these techniques if they can be made applicable to general theorem proving problems. In this thesis we investigate the applicability of logic program analysis and transformation techniques to automated theorem proving. Our aim is to speed up theorem provers by avoiding useless search. This is done by detecting and deleting parts of the theorem prover and theory under consideration that are not needed for proving a given formula. The analysis and transformation techniques developed for logic programs can be applied in automated theorem proving via a programming technique called ...
Formalizing integration theory with an application to probabilistic algorithms
 Proceedings of TPHOLs 2004. Number 3223 in LNCS, Pack City
, 2004
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An Approach of Formalizing Mathematics by Reformulations  A Proposal for QED 
, 1995
"... ly logical metalevel characterizations of all admitted logical systems are necessary. These characterization would make use of metaformulae of the kind formula j ("' j ") which stands for the fact that ' j is a formula in the formal system S j . Correspondingly predicates \Pi j ("\Delt ..."
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ly logical metalevel characterizations of all admitted logical systems are necessary. These characterization would make use of metaformulae of the kind formula j ("' j ") which stands for the fact that ' j is a formula in the formal system S j . Correspondingly predicates \Pi j ("\Delta ` j \Gamma"; ß j ), standing for ß j is a proof for \Delta ` j \Gamma in the formal system S j , can be employed. Of course, the metalanguage has to be rich enough that the notions of formula and proof can be defined in it. For instance, if we have some derivability relations like \Delta 1 ` j \Gamma 1 ; : : : ; \Delta n ` j \Gamma n \Delta ` j \Gamma RULE j k and ; \Delta ` j \Gamma AXIOM j k we have the following formulae in the metalanguage in order to axiomatize the notion of proof \Pi j ("h\Delta 1 i ` j h\Gamma 1 i"; ß 1 ) : : : \Pi j ("h\Delta n i ` j h\Gamma n i"; ß n ) ! \Pi j ("h\Deltai ` j h\Gammai"; RULE j k (ß 1 ; : : : ; ß n )) \Pi...
A Language for describing the Mathematical Knowledge
"... Introduction Mathematics is generally regarded as the exact subject for excellence. But the language commonly used by mathematicians (in paper, monographs, and even textbooks) can be remarkably vague. In particular, the language of mathematical texts invariably contains a substantial admixture of n ..."
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Introduction Mathematics is generally regarded as the exact subject for excellence. But the language commonly used by mathematicians (in paper, monographs, and even textbooks) can be remarkably vague. In particular, the language of mathematical texts invariably contains a substantial admixture of natural language, and that has all the usual potential for ambiguity and imprecision. Moreover, there is the question of the correctness of mathematical reasoning: there are plenty of welldocumented cases where published results turned out to be faulty. The formalization of mathematics addresses both these questions, precision and correctness, although formal systems can be syntactically too constricting and too verbose for human reasoning. By formalization we mean expressing mathematics, both statements and proofs, in a (usually small and simple) formal language with strict rules of grammar and unambiguous semantics. In other words, in formalizing mathematics, we m