A Machine-Checked Proof of Birkhoff's Variety Theorem in Martin-Löf Type Theory

The Agda Universal Algebra Library is a project aimed at formalizing the foundations of universal algebra, equational logic and model theory in dependent type theory using Agda. In this paper we draw from many components of the library to present a self-contained, formal, constructive proof of Birkhoff's HSP theorem in Martin-Löf dependent type theory. This achieves one of the project's initial goals: to demonstrate the expressive power of inductive and dependent types for representing and reasoning about general algebraic and relational structures by using them to a significant theorem in the field.