Seed Prover Automated Theorem Proving

Seed-Prover: Automated Theorem Proving with Formal Verification

Seed-Prover represents a major advancement in automated theorem proving by combining reinforcement learning with formal verification feedback from the Lean proof assistant. The system enables language models to iteratively refine mathematical proofs through multiple inference strategies, achieving state-of-the-art performance on formal mathematics.

Core Concept

The key insight is that formal verification provides clear, unambiguous supervision signals: a proof is either correct (accepts in Lean) or indicates specific errors that guide refinement. Rather than relying solely on language model outputs, Seed-Prover uses this structured feedback to:

• Iteratively refine proofs based on Lean error messages
• Leverage previously proved lemmas to build progressively more complex proofs
• Apply self-summarization to extract essential proof structures
• Employ three test-time inference strategies for both deep (single proof exploration) and broad (multiple proof attempts) reasoning

Architecture Overview

The system consists of the following components:

• Proof Generation Module: Language model generates proof attempts in Lean formal syntax
• Formal Verification Loop: Lean type-checker provides binary feedback (accept/reject) with specific error messages
• Lemma Library: Repository of previously proved theorems enabling compositional proof building
• Refinement Agent: Takes verification feedback and reformulates proof attempts
• Test-Time Strategies:
  • Deep reasoning: Extends single proof branch with detailed exploration
  • Broad reasoning: Generates multiple proof hypotheses in parallel
  • Hybrid: Combines deep and broad strategies adaptively

Implementation Steps

Step 1: Set up formal verification infrastructure

The system requires integration with the Lean proof assistant to provide structured feedback on proof attempts:

import subprocess
import json
from typing import Tuple

class LeanVerifier:
    """Wrapper for Lean formal verification"""
    def __init__(self, lean_path: str = "lean"):
        self.lean_path = lean_path

    def verify_proof(self, theorem: str, proof: str) -> Tuple[bool, str]:
        """
        Verify a proof in Lean and return status and feedback.

        Args:
            theorem: The theorem statement in Lean syntax
            proof: The proof attempt in Lean syntax

        Returns:
            (is_valid, feedback) where is_valid is bool and feedback is error msg or success
        """
        lean_code = f"{theorem}\n{proof}"

        try:
            result = subprocess.run(
                [self.lean_path, "--stdin"],
                input=lean_code.encode(),
                capture_output=True,
                timeout=5
            )

            if result.returncode == 0:
                return True, "Proof accepted"
            else:
                error_msg = result.stderr.decode()
                return False, error_msg

        except subprocess.TimeoutExpired:
            return False, "Proof verification timeout"

The LeanVerifier class handles communication with Lean, capturing whether the proof type-checks and returning specific error messages that guide refinement.

Step 2: Implement the lemma library and proof state tracker

Maintain a database of previously proved lemmas to enable compositional proof construction:

class LemmaLibrary:
    """Stores and retrieves previously proved lemmas"""
    def __init__(self):
        self.lemmas: dict[str, str] = {}  # name -> proof code
        self.theorem_statements: dict[str, str] = {}  # name -> statement

    def add_lemma(self, name: str, theorem: str, proof: str):
        """Store a successfully verified lemma"""
        self.lemmas[name] = proof
        self.theorem_statements[name] = theorem

    def get_applicable_lemmas(self, goal: str, top_k: int = 5) -> list[Tuple[str, str]]:
        """
        Retrieve lemmas relevant to current goal using semantic matching.
        In practice, use embeddings to find related lemmas.
        """
        # Simplified: could use embedding similarity in production
        relevant = []
        for name, statement in self.theorem_statements.items():
            if self._relevance_score(statement, goal) > 0.7:
                relevant.append((name, self.lemmas[name]))

        return relevant[:top_k]

    def _relevance_score(self, lemma: str, goal: str) -> float:
        """Simple relevance scoring; use embeddings in production"""
        # Placeholder for semantic similarity computation
        common_terms = set(lemma.split()) & set(goal.split())
        return len(common_terms) / max(len(lemma.split()), 1)

This component enables the system to build on previous successes, reducing redundant proof search.

Step 3: Implement refinement through verification feedback

Use Lean's error messages to guide proof reformulation:

class ProofRefinement:
    """Refines proofs based on verification feedback"""
    def __init__(self, llm, verifier: LeanVerifier, lemma_library: LemmaLibrary):
        self.llm = llm  # Language model for generation
        self.verifier = verifier
        self.lemma_library = lemma_library

    def refine_proof(self, theorem: str, proof_attempt: str,
                    feedback: str, context_lemmas: list[str]) -> str:
        """
        Given a failed proof and Lean's error message, generate improved version.

        Args:
            theorem: The theorem statement
            proof_attempt: Previous attempt that failed
            feedback: Error message from Lean
            context_lemmas: Available lemmas to use

        Returns:
            Refined proof attempt
        """
        prompt = f"""You are a formal proof assistant. Fix the following Lean proof.

Theorem: {theorem}

Previous proof attempt:
{proof_attempt}

Error feedback:
{feedback}

Available lemmas:
{chr(10).join(context_lemmas)}

Generate a corrected proof:"""

        refined = self.llm.generate(prompt, max_tokens=1500)
        return refined

    def iterative_refinement(self, theorem: str, max_iterations: int = 5) -> Tuple[bool, str]:
        """
        Iteratively refine proof until it verifies or max iterations reached.
        """
        # Get relevant lemmas
        lemmas = self.lemma_library.get_applicable_lemmas(theorem)
        lemma_strs = [f"-- {name}: {stmt}" for name, stmt in lemmas]

        # Generate initial proof
        proof = self.llm.generate(f"Prove: {theorem}", max_tokens=1500)

        for iteration in range(max_iterations):
            is_valid, feedback = self.verifier.verify_proof(theorem, proof)

            if is_valid:
                return True, proof

            # Refine based on feedback
            proof = self.refine_proof(theorem, proof, feedback, lemma_strs)

        return False, proof

This refinement loop embodies the core of Seed-Prover: using formal verification signals to guide iterative improvement.

Step 4: Implement test-time inference strategies

Implement both deep and broad reasoning modes:

class TestTimeStrategies:
    """Multiple reasoning strategies at inference time"""
    def __init__(self, prover: ProofRefinement):
        self.prover = prover

    def deep_reasoning(self, theorem: str, max_depth: int = 10) -> Tuple[bool, str]:
        """
        Deep reasoning: explore a single proof path exhaustively.
        Uses extended chain-of-thought with detailed intermediate steps.
        """
        prompt = f"""Prove this theorem with detailed step-by-step reasoning:

{theorem}

Provide extensive intermediate steps and justifications. Show your complete reasoning process."""

        proof = self.prover.llm.generate(prompt, max_tokens=3000)  # Extended context
        is_valid, _ = self.prover.verifier.verify_proof(theorem, proof)
        return is_valid, proof

    def broad_reasoning(self, theorem: str, num_attempts: int = 3) -> Tuple[bool, str]:
        """
        Broad reasoning: generate multiple proof hypotheses in parallel.
        Returns the first valid proof found.
        """
        for attempt in range(num_attempts):
            prompt = f"""Generate a proof for: {theorem}

Attempt {attempt + 1} - Try a different approach from previous attempts."""

            proof = self.prover.llm.generate(prompt, max_tokens=1500)
            is_valid, _ = self.prover.verifier.verify_proof(theorem, proof)

            if is_valid:
                return True, proof

        return False, None

    def hybrid_reasoning(self, theorem: str) -> Tuple[bool, str]:
        """
        Hybrid: apply broad reasoning first to find viable approaches,
        then deep reasoning to refine the best candidate.
        """
        # Phase 1: Broad search for viable approaches
        is_valid, candidate = self.broad_reasoning(theorem, num_attempts=2)

        if not is_valid:
            # Phase 2: If broad search fails, apply deep reasoning
            return self.deep_reasoning(theorem)

        # Phase 3: Refine the candidate with deep reasoning
        theorem_with_hint = f"{theorem}\n-- Hint: Start with the approach above"
        return self.deep_reasoning(theorem_with_hint)

These strategies allow the system to adapt its reasoning depth based on proof complexity.

Step 5: Integrate self-summarization for proof compression

Enable the system to extract and summarize key proof structures:

class ProofSummarization:
    """Summarizes proofs to extract essential structures"""
    def __init__(self, llm):
        self.llm = llm

    def summarize_proof(self, theorem: str, proof: str) -> str:
        """
        Extract high-level proof structure, removing low-level tactics.
        """
        prompt = f"""Summarize the key logical steps of this proof, ignoring low-level Lean tactics:

Theorem: {theorem}

Proof:
{proof}

Provide a high-level summary of the proof's logical structure:"""

        summary = self.llm.generate(prompt, max_tokens=500)
        return summary

    def extract_subgoals(self, proof: str) -> list[str]:
        """
        Parse proof to extract intermediate subgoals.
        These become candidates for lemmatization.
        """
        # Simple heuristic: find "have" and "show" statements
        lines = proof.split('\n')
        subgoals = []

        for line in lines:
            if 'have ' in line or 'show ' in line:
                subgoals.append(line.strip())

        return subgoals

This enables iterative proof building where intermediate results are captured as lemmas.

Practical Guidance

When to use Seed-Prover:

• Formal mathematics problems with well-specified Lean definitions (best case)
• Competition problems (IMO style) with rich mathematical structure
• Multi-step reasoning where lemma reuse provides significant value
• When proof search benefit from multiple diverse attempts

When NOT to use Seed-Prover:

• Informal, natural language mathematical reasoning
• Domains without formal verification infrastructure
• Real-time applications where iterative refinement is too slow
• Problems solvable by simple symbolic computation

Key hyperparameters and tuning:

• max_iterations: Control refinement depth (5-10 typically sufficient)
• num_attempts (broad reasoning): Balance exploration cost vs success rate (2-5 attempts)
• max_depth (deep reasoning): Tradeoff between detailed reasoning and computation
• Lemma library size: More lemmas help but slow retrieval; maintain top-k relevance
• Timeout for Lean verification: 5 seconds standard; increase for complex proofs

Expected performance characteristics:

• IMO problems: ~78% success rate with best configuration
• Requires ~1.5-3 verification calls per proof on average
• Each iteration typically produces incremental progress (not jumping to solution)
• Performance improves significantly with relevant lemma library (26-35% boost observed)

Reference

Seed-Prover: Deep and Broad Reasoning for Automated Theorem Proving. arXiv:2507.23726