- Albert Qiaochu Jiang, S. Welleck, J. Zhou, Wen-Ding Li, Jiacheng Liu, M. Jamnik, Timothée Lacroix, Guillaume Lample, Yuhuai Wu. 2023. D RAFT , S KETCH , AND P ROVE : G UIDING F ORMAL T HEOREM P ROVERS WITH I NFORMAL P ROOFS. Abstract: The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce wellstructured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems. Figure 1: Draft, Sketch, and Prove. Starting with an informal statement, our framework yields a formal proof through a three-stage process: drafting informal proofs, mapping them into formal sketches, and proving the remaining conjectures. Concretely, an informal statement is a mathematical problem described in a mixture of natural and mathematical languages (e.g., formulae in LTEX). Then, we use a large language model to autoformalize each informal proof into a formal sketch, which is a skeleton of the formal proof with open conjectures left unproven (indicated by the <proof> blocks). The formal sketch mirrors the structure of the informal proof. Finally, the open conjectures/gaps inside each formal sketch are proved by an off-the-shelf prover. †Equal contributions as leading authors. Correspondence to: qj213@cam.ac.uk. ‡Equal contributions as senior authors.
