Abstract. We illustrate with concrete examples the systematic exploration of Tuple Theory in a bottom-up and top-down way. In the bottom-up exploration we start from two axioms, add new notions and in this way we build the theory, check all the new notions introduced and prove some of them by the new prover which we created in the TH∃OREM ∀ system. In order to synthesize some algorithms on tuples (like e. g. insertionsort) we use an approach based on proving. Namely, we start from the specification of the problem (input and output condition) and we construct an inductive proof of the fact that for each input there exists a solution which satisfies the output condition. The problem will be reduced to smaller problems, the method will be applied like in a ”cascade” and finally the problem is so simple that the corresponding algorithm (function) already exists in the knowledge. The algorithm can be then extracted immediately from the proof. We present an experiment on synthesis of the insertion-sort algorithm on tuples, based on the proof existence of the solution. This experiment is paralleled with the construction (exploration) of the appropriate theory of tuples. The main purpose of this research is to concretely construct examples of theories and to reveal the typical activities which occur in theory exploration, in the context of a specific application – in this case algorithm synthesis by proving. 1

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