The Agda Universal Algebra Library and Birkhoff's Theorem in Dependent Type Theory
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The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof assistant. This paper describes the UALib and demonstrates that Agda is accessible to working mathematicians (such as ourselves) as a tool for formally verifying nontrivial results in general algebra and related fields. The library includes a substantial collection of definitions, theorems, and proofs from universal algebra and equational logic and as such provides many examples that exhibit the power of inductive and dependent types for representing and reasoning about general algebraic and relational structures. The first major milestone of the UALib project is a complete proof of Birkhoff's HSP theorem. To the best of our knowledge, this is the first time Birkhoff's theorem has been formulated and proved in dependent type theory and verified with a proof assistant.
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