# Studying Algebraic Structures using Prover9 and Mace4

## TL;DR

This paper demonstrates how automated theorem proving and counter-example generation tools like Prover9 and Mace4 can accelerate the exploration and verification of algebraic structures, specifically hoops, complementing traditional methods.

## Contribution

It introduces a case study applying Prover9 and Mace4 to study hoops, showcasing their effectiveness in discovering new theorems and improving proofs in algebraic research.

## Key findings

- Tools rapidly ruled out false conjectures
- Automated verification outperformed manual checking
- Discovered new theorems and improved proofs

## Abstract

In this chapter we present a case study, drawn from our research work, on the application of a fully automated theorem prover together with an automatic counter-example generator in the investigation of a class of algebraic structures. We will see that these tools, when combined with human insight and traditional algebraic methods, help us to explore the problem space quickly and effectively. The counter-example generator rapidly rules out many false conjectures, while the theorem prover is often much more efficient than a human being at verifying algebraic identities. The specific tools in our case study are Prover9 and Mace4; the algebraic structures are generalisations of Heyting algebras known as hoops. We will see how this approach helped us to discover new theorems and to find new or improved proofs of known results. We also make some suggestions for how one might deploy these tools to supplement a more conventional approach to teaching algebra.