ProofCompass: Hybrid Theorem Prover

• ProofCompass is a hybrid framework for formal theorem proving that integrates a guiding LLM with a specialized neural prover to streamline proof discovery.
• It employs natural language proof sketches and precise lemma extraction to decompose complex theorems into actionable subgoals.
• Empirical results on benchmarks like miniF2F demonstrate improved pass rates and fewer prover calls, highlighting enhanced resource efficiency.

ProofCompass is a hybrid computational framework for formal theorem proving that strategically combines a large general-purpose LLM (“Guiding LLM”) with a specialized neural prover (specifically, DeepSeek-Prover-v1.5-RL or DSP-v1.5), orchestrating both natural language and tactic-level reasoning to optimize proof success and computational resource usage. It achieves superior proof rates on standard benchmarks such as miniF2F with dramatically fewer prover calls compared to previous methods, establishing a reference paradigm where broad-semantics LLM planning is systemically coupled to domain-specialized provers without the need for additional model training (Wischermann et al., 18 Jul 2025).

1. System Architecture

ProofCompass consists of two main components operating in a tightly integrated workflow:

• Guiding LLM: A large general-purpose LLM responsible for generating natural language proof sketches, analyzing failed attempts, summarizing strategies, extracting and ranking intermediate lemmas, and generating informal subgoal proofs. In the principal evaluation, Gemini 2.0 Flash Thinking Experimental and Gemini 2.0 Flash are used for initial and downstream LLM tasks, respectively.
• Specialized Prover (DSP-v1.5): A 7B-parameter model trained on Lean 4, supporting Chain-of-Thought (CoT) input that interleaves informal and formal reasoning steps, and invoked for all candidate proof generation and verification tasks.

The overall control flow is designed such that the LLM injects high-level strategy and problem decomposition prior to, and in response to, chains of failed attempts by the specialized prover, operating within a fixed attempt budget. Information flows from the LLM to DSP-v1.5 during initial guidance and at each lemma/proof subgoal step. All proof candidates are type-checked by Lean 4 to ensure formal correctness. This architecture eliminates the need for retraining either model and permits plug-and-play LLM substitution (Wischermann et al., 18 Jul 2025).

2. Algorithmic Workflow

The ProofCompass protocol proceeds in distinct stages, each substantiated by explicit equations and pseudocode:

1. Input: Accepts the formal theorem statement ($T_{\mathrm{formal}}$) and, optionally, an informal description ($T_{\mathrm{informal}}$).
2. Natural Language Guidance: The LLM first produces a full natural language proof ($p_{NL}$, Eq. 1) and then a summary ($p_{\mathrm{summary}}$, Eq. 2), which focuses the downstream search space.
3. Initial Prover Calls: The specialized prover, in CoT mode, attempts up to $N$ proofs, each guided by the LLM-generated summary; if successful, the process terminates.
4. Lemma Extraction and Selection: If all $N$ initial attempts fail, every “have” lemma in the failed outputs forms the candidate set ($L_{\mathrm{extract}}$, Eq. 4), syntax-filtered to $L_{\mathrm{valid}}$. The LLM ranks and selects up to $k$ lemmas ($L_{\mathrm{select}}$, Eq. 5) for prioritized decomposition.
5. Subgoal Proof Synthesis: For each selected lemma $T_{\mathrm{informal}}$0, the LLM generates informal proof sketches. Already completed (“proven”) lemmas are identified and isolated ($T_{\mathrm{informal}}$1, Eq. 6).
6. Hierarchical Proof Plan: DSP-v1.5 attempts proofs for sublemmas and, when all are proven, attempts the final theorem, reusing successful elements.
7. Completion or Exhaustion: The iterative process continues up to the overall budget $T_{\mathrm{informal}}$2, or until a successful, type-checked proof is assembled.

The core workflow is formalized in detailed pseudocode, with all transitions and data flows specified (Wischermann et al., 18 Jul 2025).

3. Mathematical Formalization and Metrics

ProofCompass introduces several mathematical constructs to model and evaluate performance:

• Success Probability: For a fixed budget $T_{\mathrm{informal}}$3, empirical success rate is given by

$T_{\mathrm{informal}}$4

This metric, standard in Pass@k evaluation, enables rigorous benchmarking against prior art.

• Expected Attempts: With per-attempt success rate $T_{\mathrm{informal}}$5, the expected attempts required scales as $T_{\mathrm{informal}}$6, modulated adaptively by LLM-guided decomposition.
• Resource Usage: Total computational overhead is modeled as

$T_{\mathrm{informal}}$7

with measured values $T_{\mathrm{informal}}$8 s (one-time) and $T_{\mathrm{informal}}$9–4 s per prover call.

• Heuristic Lemma Scoring: Lemmas are ranked by an attention-like score:

$p_{NL}$0

This criterion prefers lemmas that facilitate proof progress with minimally nested dependencies, as encoded in the LLM prompting (Wischermann et al., 18 Jul 2025).

4. Empirical Results and Benchmarking

All experimental evaluation is performed on the standard miniF2F-test benchmark comprising 488 Lean 4 problems, using Pass@k metrics and running on Nvidia RTX 4090 hardware with Lean 4.15 for formal verification.

• Performance Comparisons: With attempt budgets of 32 and 128, ProofCompass yields:
  • Pass@32: 52.5% (+2.5 points over DSP-v1.5@32)
  • Pass@128: 55.3% (+3.7 points over DSP-v1.5@128, +0.4 over DSP-v1.5@3200 with 25× fewer attempts)
• Baselines: Outperforms other leading systems measured under similar budgets, including BC-Prover, LEGO-Prover, and Lyra.
• Ablation Studies: Isolate the individual contributions of lemma guidance (51.6%/53.3%) and informal proof guidance (52.0%/52.9%). The synergy of both components explains the observed improvement in resource-constrained regimes.

Empirical timing diagrams illustrate that after about 46 total attempts, ProofCompass amortizes its additional LLM cost and becomes overall faster than the baseline solely prover-driven approach (Wischermann et al., 18 Jul 2025).

5. Efficiency Mechanisms and Comparative Analysis

ProofCompass achieves its improvements through several coordinated mechanisms:

• NL Sketch Injection: High-quality LLM-generated proof sketches significantly narrow the search space, decreasing DSP-v1.5's average attempt time from 4.05 s to 3.21 s.
• Lemma-Driven Decomposition: Selection and prioritization of intermediate goals allow partitioning of difficult theorems into tractable subgoals, each with higher individual provability.
• Syntactic Alignment: Lemmas are extracted directly from DSP-v1.5 CoT outputs, ensuring subgoals remain within the prover’s training distribution. Direct LLM lemma generation without this compatibility showed degraded performance due to format mismatch.
• Resource Efficiency: The architecture is designed to maximize proof rate per computational unit, scaling practically up to moderate attempt budgets; a timing advantage is realized beyond a modest number of calls, particularly for problems not obviously easy for the base prover.

This approach is positioned relative to systems such as BC-Prover, ReProver, LLMStep, and Lyra, where comparable or lower Pass@k rates are achieved despite higher computational expenditure (Wischermann et al., 18 Jul 2025).

6. Limitations and Outlook

Identified limitations include:

• Domain Specificity: The system is tailored to Chain-of-Thought-style Lean provers; generalization to other proof search paradigms (e.g., BFS-Prover) is non-trivial and demands novel natural language prompting techniques.
• Budget Scaling: Evaluations thus far are limited to budgets up to 128; it remains to be seen whether the architecture maintains its advantage when extended to the very large budgets routinely used in some prover studies.
• Fixed Parameters: No adaptive or learned tuning of lemma selection scoring weights ($p_{NL}$1) is employed; such meta-optimization could further enhance effectiveness for challenging theorems.
• Static LLM Overhead: For trivially provable theorems, the LLM invocation cost may not be justified. This suggests the utility of adaptive scheduling that invokes LLM guidance only after initial prover failures.

Overall, ProofCompass exemplifies a strategy for maximizing the synergy between global proof planning using LLMs and efficient tactic-level search by domain-trained provers, providing a template for future research in collaborative, multi-model formal reasoning (Wischermann et al., 18 Jul 2025).

7. Context in Verifiable Computation and Proof System Landscape

ProofCompass operates in the broader context of verifiable computation, where different proof systems (including homomorphic authenticators, verifiable delay functions, zk-SNARKs, zk-STARKs, and others) are compared by properties such as universality, zero-knowledge (ZK), transparency, post-quantum (PQ) safety, setup requirements, prover/verifier time and proof size (Barbara, 2019). Synergistic LLM–specialized prover architectures such as ProofCompass constitute a methodological advance at the proof search and automation layer, orthogonal yet complementary to the developments in cryptographic proof system design, and notably relevant to universal proof compilers and all-purpose verifiable computation workflows.

A plausible implication is that as LLM guidance frameworks improve, their integration with general-purpose universal proof compilers and interaction with advanced VC systems could further enhance the automation and transparency of complex formal mathematical reasoning (Barbara, 2019, Wischermann et al., 18 Jul 2025).