A Dynamic Poincare Principle

摘要

Proofs contain important mathematical knowledge and for mathematical knowledge management it is important to represent them adequately. They can be given at different levels of abstraction and writing a proof is typically a compromise between two extremes. On the one hand it should be in full detail so that it can be checked without using any intelligence, on the other hand it should be concise and informative. Making everything fully explicit is not adequate for most mathematical fields since easy parts do not need any communication. In particular in traditional proofs, computations are typically not made explicit, but a reader is expected to check them for him- or herself. Barendregt formulated a principle, the Poincare Principle, which allows to separate reasoning and computation. However, should any computation be hidden? Or only easy computations? What is easy? How can we be sure that computations are correct? In this contribution, relevant notions are discussed and a principle is introduced which allows for checkable proofs which give a choice to see on request two different types of argument. The first type of argument states why any computation of this kind is correct. The second type states a (typically lengthy) detailed low-level proof of a trace of the computation.