What are the Right Symmetries for Formal Theorem Proving?

LLMs have recently achieved strong results on formal proving benchmarks. However, existing evaluations remain heavily concentrated on competition-style problems and often fail to capture how models behave on longer, more dependency-rich mathematical developments. We introduce TheoremBench, a Lean4 benchmark designed to evaluate theorem provers beyond contest settings. The benchmark is built from nearly one hundred classical theorems and is released in two complementary forms: a plain main version containing one target theorem per instance, and a premised version that expands each theorem into a structured family of related proving tasks consisting of the main theorem together with automatically extracted supporting subtheorems. This design enables evaluation of not only whether the final theorem was proved from scratch, but also of partial progress through the internal proof structure of a theorem. Our experiments show that explicit premises substantially improve performance for Lean4-capable prover models. To provide a comprehensive evaluation, we introduce theorem-level coverage and token-efficiency metrics that expose qualitative differences in proof behavior. The results show that current provers remain strongly biased toward easy subtheorems and often solve theorems through long and inefficient tactic traces rather than compact proof plans. TheoremBench therefore provides a more fine-grained view of formal reasoning ability and highlights the importance of structural benchmark design for evaluating Lean4 theorem provers.

Recent advances in large language models (LLMs) have shown promise in formal theorem proving, yet evaluating semantic correctness remains challenging. Existing evaluations rely on indirect proxies such as lexical overlap with human-annotated proof, or expensive manual inspection. Inspired by the shift from lexical comparison to test-based evaluation in code generation, we propose T , a framework that evaluates the semantic correctness of formal theorems: a generated theorem is considered correct only if all dependent successor theorems compile successfully, analogous to integration testing. We construct a benchmark from 5 real-world Lean 4 repositories, comprising 2,206 problems paired with 41 successor theorems on average, automatically extracted without human effort. Experiments demonstrate that while state-of-the-art models achieve high compilation success, they perform significantly worse under our semantic metric. The best model, Claude-Sonnet-4.5, achieves only 38.9% Testing Accuracy on the full set, given both natural language proof and successor theorems as context, revealing a critical gap in current theorem generation capabilities.

Recent neural theorem provers use reinforcement learning with verifiable rewards (RLVR), where proof assistants provide binary correctness signals. While verifiable rewards are cheap and scalable without reward hacking issues, they suffer from sparse credit assignment: models receive no learning signal from difficult problems where partial progress goes unrewarded. This motivates learned reward models that can evaluate proof quality beyond binary verification. However, comparing reward models is challenging since it typically requires expensive RL training ablations. To address this, we introduce \textbf{FormalRewardBench}, the first benchmark for evaluating reward models in formal theorem proving with Lean 4. Our benchmark consists of 250 preference pairs where correct proofs are paired with incorrect variants generated through five expert curated error injection strategies: forced mistakes, minimal single-point variations, verbose incorrect proofs, natural language justification, and Python code injection. We evaluate frontier LLMs (e.g., Claude Opus 4.5), judge LLMs (e.g., CompassJudger-1-14B), general-purpose LLMs (e.g., Qwen2.5-72B-Instruct), and specialized theorem proving models (e.g., DeepSeek-Prover-V2-7B). Our results reveal that frontier LLMs achieve the highest performance (59.8%) while specialized theorem provers perform the worst (24.4%), suggesting that theorem proving ability does not transfer to proof evaluation. We provide further insights on various error injection mechanisms, highlighting the challenging nature of most injection mechanisms. We release \textbf{FormalRewardBench} publicly to encourage more research on developing reward models in formal mathematics.