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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