ProofBridge: Auto-Formalization of Natural Language Proofs in Lean via Joint Embeddings

2025-10-17

AI Summary

Addressing the end-to-end formalization of natural-language mathematical theorems and proofs into Lean 4, existing stepwise approaches suffer from semantic misalignment between theorems and proofs. This work proposes ProofBridge, a unified framework that (1) introduces the first joint theorem-proof embedding model to achieve cross-modal semantic alignment; (2) integrates retrieval-augmented fine-tuning with an iterative repair mechanism guided by Lean’s type checker and bidirectional equivalence verification, jointly optimizing semantic fidelity and type correctness. Evaluated on miniF2F-Test-PF, ProofBridge achieves a 3.28× improvement in Recall@1, a 31.14% gain in semantic correctness, and a 1.64 percentage-point increase in type correctness (pass@32), significantly outperforming strong baselines.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Large language models for search
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Translating human-written mathematical theorems and proofs from natural language (NL) into formal languages (FLs) like Lean 4 has long been a significant challenge for AI. Most state-of-the-art methods address this separately, first translating theorems and then generating proofs, creating a fundamental disconnect vis-a-vis true proof auto-formalization. This two-step process and its limitations were evident even in AlphaProof's silver-medal performance at the 2024 IMO, where problem statements needed manual translation before automated proof synthesis. We present ProofBridge, a unified framework for automatically translating entire NL theorems and proofs into Lean 4. At its core is a joint embedding model that aligns NL and FL (NL-FL) theorem-proof pairs in a shared semantic space, enabling cross-modal retrieval of semantically relevant FL examples to guide translation. Our training ensures that NL-FL theorems (and their proofs) are mapped close together in this space if and only if the NL-FL pairs are semantically equivalent. ProofBridge integrates retrieval-augmented fine-tuning with iterative proof repair, leveraging Lean's type checker and semantic equivalence feedback to ensure both syntactic correctness and semantic fidelity. Experiments show substantial improvements in proof auto-formalization over strong baselines (including GPT-5, Gemini-2.5, Kimina-Prover, DeepSeek-Prover), with our retrieval-augmented approach yielding significant gains in semantic correctness (SC, via proving bi-directional equivalence) and type correctness (TC, via type-checking theorem+proof) across pass@k metrics on miniF2F-Test-PF, a dataset we curated. In particular, ProofBridge improves cross-modal retrieval quality by up to 3.28x Recall@1 over all-MiniLM-L6-v2, and achieves +31.14% SC and +1.64% TC (pass@32) compared to the baseline Kimina-Prover-RL-1.7B.

Problem

• Automatically translating natural language proofs into Lean 4 formal language
• Addressing the disconnect between theorem translation and proof generation
• Ensuring semantic equivalence and syntactic correctness in formalization

Innovation

• Joint embedding model aligns NL-FL theorem-proof pairs
• Retrieval-augmented fine-tuning with iterative proof repair
• Cross-modal retrieval guides translation using semantic equivalence