Article Dans Une Revue Archive of Formal Proofs Année : 2025

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.

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Dates et versions

hal-05273348 , version 1 (22-09-2025)

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  • HAL Id : hal-05273348 , version 1

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Benjamin Puyobro, Benoît Ballenghien, Burkhart Wolff. A Proof of Hilbert Basis Theorem and an Extension to Formal Power Series. Archive of Formal Proofs, 2025. ⟨hal-05273348⟩
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