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.
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Notes
- 1.
Prover9 input uses only ASCII characters.
- 2.
See goal script sl-pr1.gl.
- 3.
See output file sl-pr1.txt.
- 4.
para stands for “paramodulation”, an inference rule that performs a form of equational reasoning generalising the usual notion of using an equation to rewrite a term within formula.
- 5.
See goal script sl-trans.gl.
- 6.
See output file sl-trans.txt.
- 7.
In a Heyting algebra one normally uses \(x \rightarrow y\) for \(y \ominus x\), and \(x \wedge y\) for \(x \oplus y\).
- 8.
Strictly speaking this is a bounded hoop: an unbounded hoop omits the constant 1 and axiom (8). We are only concerned with bounded hoops in this book, so for brevity, we drop the word “bounded”.
- 9.
See output file hp-semilattice.txt.
- 10.
See output file hp-ge-sl.txt.
- 11.
See output file hp-plus-mono.txt, hp-sub-mono-left.txt and hp-sub-mono-right.txt.
- 12.
See output file hp-res.txt.
- 13.
See output file pc-egs.txt.
- 14.
We call these hoops \(\mathbf {L}_n\) in honour of Łukasiewicz and Tarski (1930) whose multi-valued logics have a natural semantics with values in these hoops.
- 15.
See output file hp-linear-egs.txt.
- 16.
See output file hp-egs.txt.
- 17.
See output file hp-sum-lemma.txt.
- 18.
Prover9 is a theorem-prover for finitely axiomatisable first-order theories: it is not designed to work with something like the principle of induction that can only be expressed either as an infinite axiom schema or as a second-order property. The use of interactive proof assistants that can handle induction is of potential interest in mathematics education, but is not the focus of the present chapter. There has been research on fully automated proof in higher-order logic, but this is in its early days.
- 19.
See output files conjectureNNSNNSNN.txt and conjecturePNNNNPNN.txt.
- 20.
See output file theoremNNSNNSNN-eq-expanded.txt.
- 21.
See output file theoremNNSNNSNN-eq-basic-lemmas.txt.
- 22.
See file theoremPNNNNPNN-eq.txt.
- 23.
See file theoremPNNNNPNN-eq-lemmas.txt.
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Arthan, R., Oliva, P. (2019). Studying Algebraic Structures Using Prover9 and Mace4. In: Hanna, G., Reid, D., de Villiers, M. (eds) Proof Technology in Mathematics Research and Teaching . Mathematics Education in the Digital Era, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-28483-1_5
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