(Auto)formalization is supposed to be easy: Trellis process semantics for spelling out rigorous proofs

While Large Language Models (LLMs) have demonstrated exceptional capabilities in mathematical reasoning, they frequently produce subtle errors that evade human detection. Formal mathematical languages like Lean 4 offer mechanical proof checking, strongly motivating the need for autoformalization: the automatic translation of natural language mathematics into verifiable code. Recent trends indicate that general-purpose LLMs, heavily optimized for standard programming, now outperform smaller models explicitly fine-tuned for Lean. Leveraging this shift, we introduce an agentic autoformalization framework powered by general coding LLMs. At the core of our system is an orchestrator that manages a multi-agent pipeline tailored for research-level mathematics. Because cutting-edge research frequently relies on concepts outside the scope of existing libraries like Mathlib, our system dynamically extends necessary type definitions and validates them via a novel Auxiliary Lemma technique before formalizing the primary theorems. We applied our approach to PutnamBench, producing machine-checked Lean proofs for a random sample of 32 problems. Furthermore, we evaluate our system on five papers from the ACM Symposium on Theory of Computing (STOC) spanning combinatorics, communication complexity, mechanism design, and learning theory, successfully formalizing their main theorems and validating the generated formalizations with human experts; for all five we also formalize the proofs alongside the statements, and notably two of them are proved with no axioms beyond Lean's kernel. All of our formalizations are available at https://beyondthelibrary.github.io/formal_arxiv .

Proof autoformalization aims to translate a mathematical informal proof written in natural language into a formal proof in a formal language such as Lean~4. Several works have developed LLM-based models for proof autoformalization. However, existing evaluations have typically focused on translating well-formed informal proofs from curated datasets. We argue that a robust proof autoformalizer must remain faithful even for informal proofs that diverge from these idealized ones, and we present the first study on the robustness of proof autoformalization models. We formulate two categories of perturbations and evaluate robustness under each: a global perturbation paraphrases the informal proof in a different style, under which the formalization should remain consistent; a local perturbation alters a value, symbol, or proof step, possibly in a counterfactual way, and a robust formalization should faithfully reflect the perturbation rather than reverting to the original one or inferring a different one on its own. We build a benchmark with both perturbations on miniF2F and MATH-500, and automatically measure how stable a proof autoformalization's correctness is under global perturbations and how faithfully its output reflects local perturbations. We evaluate seven recent models, all of which are sensitive to global perturbations and mostly fail to remain faithful under local perturbations. Code and data are available via https://github.com/ucr-rai/robust-proof-autoformalization.

Accurate auto-formalization of theorem statements is essential for advancing automated discovery and verification of research-level mathematics, yet remains a major bottleneck for LLMs due to hallucinations, semantic mismatches, and their inability to synthesize new definitions. To tackle these issues, we present Aria (Agent for Retrieval and Iterative Autoformalization), a system for conjecture-level formalization in Lean that emulates human expert reasoning via a two-phase Graph-of-Thought process: recursively decomposing statements into a dependency graph and then constructing formalizations from grounded concepts. To ensure semantic correctness, we introduce AriaScorer, a checker that retrieves definitions from Mathlib for term-level grounding, enabling rigorous and reliable verification. We evaluate Aria on diverse benchmarks. On ProofNet, it achieves 91.6% compilation success rate and 68.5% final accuracy, surpassing previous methods. On FATE-X, a suite of challenging algebra problems from research literature, it outperforms the best baseline with 44.0% vs. 24.0% final accuracy. On a dataset of homological conjectures, Aria reaches 42.9% final accuracy while all other models score 0%.