From LLM-Generated Conjectures to Lean Formalizations: Automated Polynomial Inequality Proving via Sum-of-Squares Certificates

2026-05-14

AI Summary

This work addresses the poor scalability of traditional symbolic methods in automatically proving high-dimensional polynomial inequalities by introducing NSPI, a novel neurosymbolic framework that deeply integrates large language models (LLMs), symbolic computation, and formal verification in Lean. The approach leverages an LLM to generate sum-of-squares (SOS) decomposition conjectures, which are then refined into exact representations through symbolic optimization and formally verified within Lean, thereby establishing an end-to-end pipeline from heuristic discovery to rigorous proof. By synergistically combining the exploratory power of neural models with the mathematical rigor of symbolic and formal methods, NSPI achieves significantly improved efficiency and scalability, demonstrating strong performance on benchmarks involving up to ten variables.

Abstract

Automated proving of polynomial inequalities is a fundamental challenge in automated mathematical reasoning, where rich algebraic structure and a rapidly growing certificate search space hinder scalability. Purely symbolic approaches provide strong guarantees but often scale poorly as the number of variables or the degree increases, due to expensive algebraic manipulations and rapidly growing intermediate expressions. In parallel, LLM-guided methods have made notable progress, particularly on competition-style inequalities with a small number of variables. To address the remaining scalability challenges, we propose NSPI, a neuro-symbolic framework that combines the complementary strengths of LLMs and symbolic computation for polynomial-inequality proving. Concretely, an LLM proposes a conjecture in the form of an approximate polynomial Sum-Of-Squares (SOS) decomposition; we refine it via symbolic computation to obtain an exact polynomial SOS representation, which directly proves the target inequality, and we further certify the proof in Lean, yielding an end-to-end pipeline from heuristic discovery to machine-checked proof. Experiments on challenging benchmarks involving polynomials with up to 10 variables demonstrate the effectiveness and scalability of the proposed method.

Problem

• polynomial inequality
• automated proving
• scalability
• Sum-of-Squares
• formal verification

Innovation

• neuro-symbolic
• sum-of-squares
• polynomial inequality proving
• large language models
• formal verification