# Birkhoff's Completeness Theorem for Multi-Sorted Algebras Formalized in Agda

@article{Abel2021BirkhoffsCT, title={Birkhoff's Completeness Theorem for Multi-Sorted Algebras Formalized in Agda}, author={Andreas Abel}, journal={ArXiv}, year={2021}, volume={abs/2111.07936} }

This document provides a formal proof of Birkhoff’s completeness theorem for multi-sorted algebras which states that any equational entailment valid in all models is also provable in the equational theory. More precisely, if a certain equation is valid in all models that validate a fixed set of equations, then this equation is derivable from that set using the proof rules for a congruence. The proof has been formalized in Agda version 2.6.2 with the Agda Standard Library version 1.7 and this…

## One Citation

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

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This paper presents a self-contained, formal, constructive proof of Birkhoff’s HSP theorem in Martin-Löf dependent type theory, to demonstrate the expressive power of inductive and dependent types for representing and reasoning about general algebraic and relational structures by using them to formalize a significant theorem in the field.

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