# Compiler-Guided Adaptive Proof Search with Cross-Model Synergy on Context-Dependent Theorem Proving > Compiler-guided proof search alternating between two complementary LLMs with pairwise comparison and stagnation detection improves Lean 4 theorem proving pass rates by up to 12.66 points over single-model baselines. - **Source:** [arXiv](https://arxiv.org/abs/2608.18084) - **Published:** 2026-09-05 - **Permalink:** https://picx.dev/p/U6gAy6 - **Whiteboard:** https://picx.dev/p/U6gAy6/image ## Summary # Compiler-Guided Proof Search with Cross-Model Synergy for Context-Dependent Theorem Proving ## Summary (Overview) - **Problem addressed**: Theorem proving in real-world Lean 4 projects is difficult because proofs depend on project-specific context (local definitions, project-specific lemmas, naming conventions). Standard independent sampling (pass@k) treats the compiler as a final filter rather than a source of guidance. - **Key contribution**: The authors propose a **compiler-guided search framework** that balances exploration and exploitation. It combines two complementary LLMs (a Lean-specialized model and a general reasoning model), uses compiler feedback for structured error reporting, and employs a pairwise comparison mechanism to preserve promising proof states during refinement. - **Core mechanism**: The system alternates between exploration (drawing new candidates from both models) and exploitation (repairing the current-best proof). A stagnation counter triggers a return to exploration when refinement stops making progress. - **Main findings**: On multiple Lean 4 projects, the framework outperforms both base models and agentic baselines, achieving e.g. 12.66% absolute pass-rate improvement over the best single model at lower cost on miniCTX-v2, and 3.8 points over the best single-model pass@32 on RLMEval-FLT3. The gains are attributed to explicit exploration–exploitation balancing, not to additional model calls alone. - **Analysis**: Ablation experiments confirm that all components contribute; pairwise comparison especially keeps the system from accumulating errors and wasting model calls on unpromising refinements. --- ## Introduction and Theoretical Foundation ### Background and Motivation Theorem proving in formal systems such as Lean has seen major advances recently. Systems like DeepSeek-Prover-V2 and AlphaProof have excelled on Olympiad-style problems, but real-world, project-level theorem proving remains challenging because: - The proof context includes project-specific definitions, lemmas, notations, and dependencies. - Theorems often rely on highly specific naming conventions and proof patterns not present in pretraining data. - The theorem and its context jointly determine the correct proof strategy; hence the problem is **context-dependent**. ### Problem with Existing Approaches - **Pass@k sampling**: Independent samples rarely benefit from each other; every failed attempt is discarded inaccessible feedback. This is wasteful for hard theorems that require iterative repair. - **Agentic approaches (ReAct-style)**: Use an LLM controller to decide what to do next. - **Tactic-level search**: Expensive and brittle for context-dependent problems. **Observation motivating the design**: failed proofs are not useless: they contain a nearly correct structure or a promising partial sequence of tactics nasally. But whether the reuse is helpful depends on both (a) the starting point and (b) the sequence of refinements, which can degrade a promising proof. This motivates the pairwaise comparison for selecting and preserving the best current proof state. ## Summary and Conclusions The paper introduces CARP (Compiler-Guided Adaptive Refinement Protocol), a method for context-dependent theorem proving in Lean 4. Key elements: - **Two complementary models**: a generalist (strong context usage) and a specialist (strong syntax and formal tactics). - **Compiler feedback**: structured errors from Lean are used for both repair and comparison. - **Pairwise comparison**: an LLM judge decides whether a new refinement replaces the current best proof, or if the old one should be kept. - **Stagnation counter**: triggers resampling when refinement no longer improves the proof. - **AutoSolve**: a fast deterministic tactic that closes goals without LLM calls. Experiments on 7 real-world Lean 4 projects (miniCTX-v2) show significant improvements over both base models and a ReAct-style agent baseline<｜begin▁of▁file｜> ## Reference [1] Aditya Kurniawan et al. Minimax-01: Scaling Mixture-of-Experts with Sub-1-Bit Compression. [2] Anand Rao et al. RL, but don't do RL! Better alignment via automated evaluation and LLM feedback. [3] Clara Loh and Richard Pither. 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