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A Case Study in Systematic Theory Exploration: Natural Numbers 1
"... In this paper, we present a case study of computer supported exploration of the theory of natural numbers, using a theory exploration model based on knowledge schemes, proposed by Bruno Buchberger. We illustrate with examples from the exploration: (i) the invention of new concepts (functions, predic ..."
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In this paper, we present a case study of computer supported exploration of the theory of natural numbers, using a theory exploration model based on knowledge schemes, proposed by Bruno Buchberger. We illustrate with examples from the exploration: (i) the invention of new concepts (functions, predicates) in the theory, using definition knowledge schemes, (ii) the invention of new propositions, using proposition schemes, (iii) the invention of problems, using algorithm knowledge schemes, (iv) the introduction of new reasoning rules, by lifting knowledge to the inference level (or using inference schemes), after their correctness was proved. In particular, we apply the schemebased model to notions from the natural numbers theory, such as: function symbols (addition, multiplication, exponentiation, predecessor, subtraction, quotient, remainder, greatest common divisor), predicate symbols (weak lessequal, strict lessequal, divides, proper divides, isprime), propositions about these notions (such as: quotientremainder decomposition theorem, prime decomposition theorem), illustrating how the model can lead to the systematic
Knowledge Archives in Theorema: A LogicInternal Approach
"... Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching la ..."
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Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higherorder predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logicinternal in the sense that they have a natural translation to higherorder logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.