🤖 AI Summary
This study investigates the reasoning capabilities and limitations of artificial intelligence in tackling formalized mathematical problems, exemplified by the “grasshopper problem” (IMO 2009, Problem 6). Using the Aristotle API to generate Lean 4 proofs, the authors construct a formal framework comprising four verified auxiliary lemmas, employing techniques such as maximality arguments, adjacent-swap strategies, and partial-sum properties. While local reasoning succeeds in establishing these lemmas, the main theorem remains incomplete due to an unresolved global counting contradiction, marked with a ‘sorry’ placeholder—highlighting a critical bottleneck in AI’s ability to conduct combinatorial global reasoning. This work presents the first verifiable, AI-generated formal proof artifact that clearly delineates proven from unproven components and releases reproducible Lean code, offering a precise diagnostic of current limitations in AI-assisted theorem proving.
📝 Abstract
AI-assisted theorem proving can now generate substantial Lean developments for olympiad-level mathematics, but the evidential status of such developments depends on which declarations are actually verified. This paper reports a Lean 4 formalization case study of an Aristotle API proof attempt for the Grasshopper problem, originally posed as IMO 2009 Problem 6. The generated artifact states a generalized Lean version of the theorem, contains four verified helper lemmas for local components of a maximality and adjacent-swap exchange strategy, and leaves the main theorem grasshopper closed directly by one unresolved sorry. The verified components establish that the final partial sum equals the total sum, that an adjacent transposition can affect only the relevant intermediate partial sum, that the changed partial sum has the expected form, and that maximality at a position admitting an adjacent successor swap forces a corresponding forbidden-set membership fact. The Aristotle output summary identifies the intended remaining mathematical step as the global counting step needed to show that these membership facts produce at least n distinct forbidden values, contradicting the cardinality assumption |M| < n; the Lean source itself does not reduce the main theorem to a separately encoded counting lemma. This case study gives an inspectable example of a central limitation in AI-assisted formalization, namely that local proof search can succeed while the global combinatorial bookkeeping required for a theorem remains unresolved. The paper contributes a reproducible Lean artifact and a precise analysis of its verified and unverified proof content.