Grr Obner Bases in Type Theory

We describe how the theory of Grr obner bases, an important part of computational algebra, can be developed within Martin-LL of's type theory. In particular, we aim for an integrated development of the algorithms for computing Grr obner bases: we want to prove, constructively in type theory, the existence of Grr obner bases and from such proofs extract the algorithms. Our main contribution is a reformulation of the standard theory of Grr obner bases which uses generalised inductive deenitions. We isolate the main non{constructive part, a minimal bad sequence argument, and use the open induction principle Rao88,Coq92] to interpret it by induction. This leads to short constructive proofs of Dickson's lemma and Hilbert's basis theorem, which are used to give an integrated development of Buchberger's algorithm. An important point of this work is that the elegance and brevity of the original proofs are maintained while the new proofs also have a direct constructive content. In the appendix we present a computer formalisation of Dickson's lemma and an abstract existence proof of Grr obner bases.