A Proof of Hilbert Basis Theorem and an Extension to Formal Power Series
Résumé
The Hilbert Basis Theorem is enlisted in the extension of Wiedijk's catalogue "Formalizing 100 Theorems", a well-known collection of challenge problems for the formalization of mathematics. In this paper, we present a formal proof of several versions of this theorem in Isabelle/HOL. Hilbert's basis theorem asserts that every ideal of a polynomial ring over a commutative ring has a finite generating family (a finite basis in Hilbert's terminology). A prominent alternative formulation is: every polynomial ring over a Noetherian ring is also Noetherian. We also provide isomorphisms between the three types of polynomial rings defined in HOL-Algebra. Together with the fact that the Noetherian property is preserved by isomorphism, we get Hilbert's Basis Theorem for all three models. We believe that this technique has a wider potential of applications in the AFP library.