  Human mathematicians are often good at recognizing modular and reusable
theorems that make complex mathematical results within reach. In this paper, we
propose a novel method called theoREm-from-prooF extrACTOR (REFACTOR) for
training neural networks to mimic this ability in formal mathematical theorem
proving. We show on a set of unseen proofs, REFACTOR is able to extract 19.6%
of the theorems that humans would use to write the proofs. When applying the
model to the existing Metamath library, REFACTOR extracted 16 new theorems.
With newly extracted theorems, we show that the existing proofs in the MetaMath
database can be refactored. The new theorems are used very frequently after
refactoring, with an average usage of 733.5 times, and help shorten the proof
lengths. Lastly, we demonstrate that the prover trained on the new-theorem
refactored dataset proves more test theorems and outperforms state-of-the-art
baselines by frequently leveraging a diverse set of newly extracted theorems.
Code can be found at https://github.com/jinpz/refactor.
