Theorema is a project that aims at supporting the entire process of mathematical theory exploration within one coherent logic and software system. This survey paper illustrates the style of Theoremasupported mathematical theory exploration by a case study (the automated synthesis of an algorithm for the construction of Gröbner Bases) and gives an overview on some reasoners and organizational tools for theory exploration developed in the Theorema project.

Abstract. We prove that the propositional translations of the KneserLovász theorem have polynomial size extended Frege proofs and quasipolynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lovász theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.

Recently, we proposed a systematic method for top-down synthesis and verification of lemmata and algorithms called "lazy thinking method" as a part of systematic mathematical theory exploration (mathematical knowledge management). The lazy thinking method is characterized:• by using a library of theorem and algorithm schemes • and by using the information contained in failing attempts to prove the schematic theorem or the correctness theorem for the algorithm scheme for inventing lemmata or requirements for subalgorithms, respectively.In this paper, we give a couple of examples for algorithm synthesis using the lazy thinking paradigm. These examples illustrate how the synthesized algorithm depends on the algorithm scheme used. Also, we give details about the implementation of the lazy thinking algorithm synthesis method in the frame of the Theorema system. In this implementation, the synthesis of the example algorithms can be carried out completely automatically, i.e. without any user interaction.

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We investigate the proof complexity of a class of propositional formulas expressing a combinatorial principle known as the Kneser-Lovász Theorem. This is a family of propositional tautologies, indexed by an nonnegative integer parameter k that generalizes the Pigeonhole Principle (obtained for k = 1).We show, for all fixed k, 2 Ω(n) lower bounds on resolution complexity and exponential lower bounds for bounded depth Frege proofs. These results hold even for the more restricted class of formulas encoding Schrijver's strenghtening of the Kneser-Lovász Theorem. On the other hand for the cases k = 2, 3 (for which combinatorial proofs of the Kneser-Lovász Theorem are known) we give polynomial size Frege (k = 2), respectively extended Frege (k = 3) proofs. The paper concludes with a brief announcement of the results (presented in subsequent work) on the complexity of the general case of the Kneser-Lovász theorem.

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 schemes, (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.

In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily difficult, and we illustrate this in the theory of natural numbers. We consider a restriction, the decomposition in domains with a well-founded partial ordering: we introduce the notions of irreducible elements, reducible elements w.r.t. a composition operation, decomposition of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve. Natural numbers can be classified as a decomposition domain, in which we know how to solve the decomposition problem. This leads to the prime decomposition theorem.

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