Babel-formal: Translation of Proofs between Lean and Rocq

In this work, we investigate using proof terms (the low-level representation of formal proofs) as a pivot language for translating proof scripts between proof assistants and across tactic sets. Unlike direct proof translation, this approach does not require an aligned training corpus; it only needs aligned context at inference time so that both systems elaborate comparable terms. We compare two strategies: (1) direct script-to-script translation with an off-the-shelf LLM (GPT-5), and (2) proof term translation, where an LLM turns a proof term into a proof script in the target language. We build a small benchmark of aligned sources (14 files, 117 lemmas) across Lean and Rocq, and train models to map Lean and Rocq proof terms back to their native proof scripts. Our experiments show that proof term translation works for cross-assistant translation and for translation between tactic sets. It is complementary to using a SoTA off-the-shelf LLM for direct proof script translation (combining both performs best), scales easily in terms of training data, and handles tactic-set translation better (\eg vanilla Rocq $\to$ SSReflect).