Algorithm Synthesis by Lazy Thinking: Examples and Implementation in Theorema

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.