Scheme-Based Systematic Exploration of Natural Numbers

In this paper, we report 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, relations) in the theory, using knowledge scheme;, (ii) the invention of new propositions, using proposition schemes; (iii) the invention of problems, using knowledge schemes; (iv) the introduction of new reasoning rules, by lifting knowledge to the inference level, after their correctness was proved