Alchemy: Amplifying Theorem-Proving Capability through Symbolic Mutation
Authors: name: Shaonan Wu, name: Shuai Lu, name: Yeyun Gong, name: Nan Duan, name: Ping Wei

Formal proofs are challenging to write even for experienced experts. Recent
progress in Neural Theorem Proving (NTP) shows promise in expediting this
process. However, the formal corpora available on the Internet are limited
compared to the general text, posing a significant data scarcity challenge for
NTP. To address this issue, this work proposes Alchemy, a general framework for
data synthesis that constructs formal theorems through symbolic mutation.
Specifically, for each candidate theorem in Mathlib, we identify all invocable
theorems that can be used to rewrite or apply to it. Subsequently, we mutate
the candidate theorem by replacing the corresponding term in the statement with
its equivalent form or antecedent. As a result, our method increases the number
of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore,
we perform continual pretraining and supervised finetuning on this augmented
corpus for large language models. Experimental results demonstrate the
effectiveness of our approach, achieving a 5% absolute performance improvement
on Leandojo benchmark. Additionally, our synthetic data achieve a 2.5% absolute
performance gain on the out-of-distribution miniF2F benchmark. To provide
further insights, we conduct a comprehensive analysis of synthetic data
composition and the training paradigm, offering valuable guidance for
developing a strong theorem prover.