Tao's Equational Proof Challenge Accepted (Technical Report)

2026-05-20

AI Summary

This work addresses the challenge of compressing complex equational proofs, exemplified by Terence Tao’s 62-step proof generated by the Vampire theorem prover, by introducing Krympa—the first automated tool specifically designed for equational proof compression. Krympa integrates brute-force search with heuristic strategies and synergistically leverages the superposition-based prover Vampire alongside the equational prover Twee to enhance proof conciseness while preserving logical correctness. Experimental results demonstrate that Krympa successfully reduces the original 62-step proof to just 20 steps and achieves strong performance across a benchmark of 1,431 problems, including a notable case where a 151-step proof was compressed to only 10 steps, thereby enabling large-scale automated simplification of intricate equational proofs.

Abstract

In the context of the Equational Theories Project, Terence Tao posed the challenge of finding alternatives to a complicated 62-step proof found by the Vampire superposition prover. We introduce a proof minimization tool called Krympa. Using a combination of brute force and heuristics, and exploiting both Vampire and the Twee equational prover, the tool reduces the 62-step proof to 20 steps, each corresponding to a rewrite. In an empirical evaluation, it also performs well on 1431 equational problems originating from the same project, reducing in particular a 151-step proof to only 10 steps.

Problem

• equational proof
• proof minimization
• Vampire
• Tao's challenge
• rewrite steps

Innovation

• proof minimization
• equational reasoning
• automated theorem proving
• rewrite systems
• heuristic search