Commutative Algebra in the Mizar System

Abstract

We report on the development of algebra in the Mizar system. This includes the construction of formal multivariate power series and polynomials as well as the definition of ideals up to a proof of the Hilbert basis theorem. We present how the algebraic structures are handled and how we inherited the past developments from the Mizar Mathematical Library (MML). The MML evolves and past contributions are revised and generalized. Our work on formal power series caused a number of such revisions. It seems that revising past developments with an intent to generalize them is a necessity when building a database of formalized mathematics. This poses a question: how much generalization is best?

On algebraic hierarchies in mathematical repository of Mizar

“…The hierarchy of structures available in the Mizar Mathematical Library was described in [35] and was enriched during the formalization of the proof of Fundamental Theorem of Algebra [29]. However in 2007 a big refinement (a revision [19]) took place, and parts of the net of structures together with corresponding attributes were a subject for refactoring.…”

“…Note that doubleLoopStr inherits from both addLoopStr and multLoopStr_0, that is it joins the signatures of additive and multiplicative groups. Particular properties such as commutativity or the existence of inverse elements are described by attribute definitions (see [35] Observe that because the Axiom of Choice is hardcoded in the Mizar checker, the collection of attributes clustered in the above definition of type should be shown to exist for at least one object; otherwise (with the illustrative example of infinite empty set) this should be contradictory. This is called the paradigm of non-emptiness of types in Mizar.…”

Computational logic: its origins and applications

“…De Bruijn's approach was based on dependenttype theories of his own devising. At around the same time, Andrzej Trybulec had the idea of formalizing mathematics using a form of typed set theory; a substantial amount of mathematics was translated into his Mizar formalism, with a particularly elegant treatment of algebraic structures [49].…”

Four Decades of Mizar

“…In the MML there are also systematically developed branches of mathematics [33], e.g., set theory [21], topology, functional analysis, abstract algebra [20], category theory, and lattice theory. Some of the results proven for lattices were obtained with the help of equational provers-like EQP/Otter (now Prover9).…”