Case study: proving sqrt(2) irrational with LPTP and an LLM

2026-07-23

AI Summary

This work formalizes a proof of the irrationality of √2 within a logic programming framework and investigates the feasibility of leveraging large language models (LLMs) to assist in constructing machine-verifiable mathematical proofs. By integrating the LPTP logic program theorem prover with an LLM, the study develops a complete formal proof from basic predicates in a natural-deduction style. It presents the first successful collaboration between an LLM and LPTP to generate a human-readable yet fully machine-verifiable proof of the irrationality of √2, which has been rigorously validated by LPTP. The results demonstrate the effectiveness of human–AI cooperation in formal mathematics and offer a novel pathway toward AI-assisted theorem proving.

Abstract

We present the interactions with an LLM (Large Language Model) aiming at proving that the square root of 2 is not a rational number in an LP (Logic Programming) context. We start from a few basic pure logic programming predicate definitions. We rely on the LPTP (Logic Program Theorem Prover) system for stating and proving properties about logic programs. As the proof language of LPTP is based on natural deduction, the proofs are human readable. In our case study, we sketch in LPTP the usual proof showing the irrationality of the square root of 2. Then we describe the interactions we had with the LLM. We end up with a complete formal proof, partially generated by an LLM and fully proof-checked by LPTP.

Problem

• irrationality of sqrt(2)
• Logic Programming
• Large Language Model
• formal proof
• theorem proving

Innovation

• Large Language Model
• Logic Programming
• Theorem Proving
• Formal Verification
• Natural Deduction