Seed Prover 1 5

Overview

Seed-Prover 1.5 bridges natural language mathematical reasoning and formal theorem proving by decomposing the task into three specialized agents: natural language proving, sketch-to-Lean translation, and formal verification. This hierarchical approach enables parallel sub-goal solving and efficient test-time scaling.

Core Technique

The system employs three coordinated agents with distinct roles:

Agent Orchestration: Rather than monolithic end-to-end proving, the framework decomposes work into independent stages that enable parallelization and leverage each agent's strengths.

# Three-agent orchestration pattern
def prove_theorem(problem):
    # Stage 1: Natural language prover
    nl_proof = nl_prover_agent.prove(problem)

    # Stage 2: Sketch translator (Rubric RL trained)
    lean_sketch = sketch_translator.translate(nl_proof, problem)

    # Stage 3: Agentic verifier
    verified_lemmas = verifier_agent.verify_and_build(lean_sketch)

    return verified_lemmas if verified_lemmas else None

Tool-Augmented Verification: The verifier agent uses three key tools: Lean compilation for correctness checking, Mathlib search for relevant lemmas, and Python execution for computational validation. Tool usage patterns are learned through PPO-based training.

# Verifier's tool integration
class AgenticProver:
    def __init__(self):
        self.tools = {
            'lean_verify': compile_lean_proof,
            'search_mathlib': find_relevant_lemmas,
            'compute': execute_python
        }

    def prove_incremental(self, sketch):
        while not complete:
            action = select_tool()  # Learned via PPO
            result = self.tools[action](current_state)
            update_internal_state(result)

Hierarchical Reward Structure: Large-scale RL training uses outcome rewards from Lean compilation. Rubric RL for the sketch model decomposes complex proofs into lemma-style sub-goals with separate reward signals.

When to Use This Technique

Use Seed-Prover 1.5 when:

• Proving Lean theorems from mathematical problem statements
• You want interpretable intermediate (sketch) representations
• Problems decompose naturally into independent lemmas
• Combining natural language and formal reasoning is beneficial

When NOT to Use This Technique

Avoid this approach if:

• Working with non-Lean formal systems (requires agent retooling)
• Proof states are highly interdependent (decomposition ineffective)
• Real-time single-pass proving is required
• Mathematical domain is completely novel (limited Mathlib coverage)

Implementation Notes

The framework requires:

• Three separately trained agents with coordinated interfaces
• PPO training infrastructure for tool-use learning
• Rubric RL for intermediate sketch-to-Lean translation
• Integration with Lean compiler and Mathlib repository
• Handles both undergraduate (PutnamBench) and graduate (FATE-H) difficulty levels

Key Performance

• 88% on PutnamBench (undergraduate-level)
• 80% on FATE-H (graduate-level)
• Efficient through hierarchical decomposition

References

• Multi-agent orchestration for theorem proving
• Tool-augmented LLM agents with PPO training
• Rubric-based reward learning for structured outputs