A case study on formalizing algebra in a module system

We present a case study on a modular formal representation of algebra in the recently developed module system for the Twelf implementation of the Edinburgh Logical Framework LF. The module system employs signature morphisms as its main primitive concept, which makes it particularly useful to reason about structural translations between mathematical concepts. The mathematical content is encoded in the usual way using LF's higher order abstract syntax and judgments-as-types paradigm, but using the module system to treat all algebraic structures independently. Signature morphisms are used to give an explicit yet simple representation of modular dependency between the algebraic structures. Our results demonstrate the feasibility of comprehensively formalizing large-scale theorems and proofs and thus promise significant future applications. Copyright 2009 ACM.