Theory Exploration in Theorema: Case Study on Lists

We present the systematic build-up in the Theorema system of the theory of lists. This was carried out in parallel with the process of synthesis of some sorting algorithms in the same system. We use appropriate induction principles for lists and we construct a collections of properties of lists which are necessary for the automatic synthesis of sorting algorithms. In contrast with another version of the list theory in the Theorema system, which is based on higher order logic and uses sequence variables, our approach uses first order predicate logic (which is semi-decidable). This approach opens the way for the effective automation of proofs, of the exploration of theories and of the synthesis of the algorithms applied on lists. This case study in theory exploration can be also used in teaching, especially because it is completely supported by the Theorema system.