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DRIFT: Decompose, Retrieve, Illustrate, then Formalize Theorems

Automating the formalization of mathematical statements for theorem proving remains a major challenge for Large Language Models (LLMs). LLMs struggle to identify and utilize the prerequisite mathematical knowledge and its corresponding formal representation in languages like Lean. Current retrieval-augmented autoformalization methods query external libraries using the informal statement directly, but overlook a fundamental limitation: informal statements lack direct mappings to mathematical theorems and lemmata, nor do those theorems translate trivially into the formal primitives of languages like Lean. To address this, we introduce DRIFT, a novel framework that enables LLMs to decompose informal mathematical statements into smaller, more tractable "sub-components". This facilitates targeted retrieval of premises from mathematical libraries such as Mathlib. Additionally, DRIFT retrieves illustrative theorems to help models use premises more effectively in formalization tasks. We evaluate DRIFT across diverse benchmarks (ProofNet, ConNF, and MiniF2F-test) and find that it consistently improves premise retrieval, nearly doubling the F1 score compared to the DPR baseline on ProofNet. Notably, DRIFT demonstrates strong performance on the out-of-distribution ConNF benchmark, with BEq+@10 improvements of 42.25% and 37.14% using GPT-4.1 and DeepSeek-V3.1, respectively. Our analysis shows that retrieval effectiveness in mathematical autoformalization depends heavily on model-specific knowledge boundaries, highlighting the need for adaptive retrieval strategies aligned with each model's capabilities.

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