Lean Theorem Proving Guide

Lean Theorem Proving Agent Guide

Overview

LeanAgent is an LLM-based agent for automated theorem proving in Lean 4, a modern proof assistant. It combines LLM reasoning with formal verification — proposing proof steps that are verified by Lean's type checker. Can prove novel theorems, not just benchmarks, by exploring proof strategies, backtracking on failures, and learning from successful proofs.

Architecture

Theorem Statement (Lean 4)
         ↓
   Goal Analysis Agent (understand proof obligations)
         ↓
   Tactic Suggestion Agent (propose proof steps)
         ↓
   Lean 4 Verification (check tactic correctness)
         ↓
   Backtracking (if tactic fails, try alternatives)
         ↓
   Proof or timeout

Usage

from lean_agent import LeanAgent

agent = LeanAgent(
    llm_provider="anthropic",
    lean_path="/path/to/lean4",
)

# Prove a theorem
result = agent.prove(
    theorem="""
    theorem add_comm (m n : Nat) : m + n = n + m := by
      sorry
    """,
    max_attempts=50,
    timeout=120,
)

if result.proved:
    print("Proof found!")
    print(result.proof)
else:
    print(f"Failed. Best attempt:\n{result.best_attempt}")
    print(f"Remaining goals: {result.remaining_goals}")

Proof Search Strategies

# Configure search strategy
agent = LeanAgent(
    search_config={
        "strategy": "best_first",   # best_first, bfs, dfs
        "max_depth": 20,            # Max proof steps
        "beam_width": 5,            # Tactics to try per step
        "temperature": 0.7,         # LLM sampling temp
        "backtrack_on_fail": True,
    },
)

# Interactive proof mode
session = agent.interactive_prove(
    theorem="theorem my_thm : ∀ n : Nat, n + 0 = n := by"
)

while not session.done:
    print(f"Current goals:\n{session.goals}")
    tactics = session.suggest_tactics(k=5)
    for i, t in enumerate(tactics):
        print(f"  {i}: {t.tactic} (confidence: {t.score:.2f})")
    # Agent automatically picks best tactic
    session.step()

Lean 4 Tactic Library

-- Common tactics LeanAgent uses:
-- intro, apply, exact, rfl, simp, omega
-- induction, cases, constructor, ext
-- rw, calc, have, let, show

-- Example theorem + proof
theorem list_append_nil (l : List α) : l ++ [] = l := by
  induction l with
  | nil => simp
  | cons h t ih => simp [ih]

Batch Proving

# Prove multiple theorems
theorems = [
    "theorem t1 : 1 + 1 = 2 := by sorry",
    "theorem t2 (n : Nat) : n + 0 = n := by sorry",
    "theorem t3 (n m : Nat) : n + m = m + n := by sorry",
]

results = agent.prove_batch(
    theorems=theorems,
    parallel=True,
    timeout_per=60,
)

for thm, result in zip(theorems, results):
    status = "PROVED" if result.proved else "FAILED"
    print(f"[{status}] {thm[:50]}...")

Use Cases

1. Automated proving: Prove mathematical theorems formally
2. Proof assistance: Suggest tactics during manual proving
3. Verification: Formally verify mathematical claims
4. Education: Learn Lean 4 tactics with AI guidance
5. Research: Explore new proof techniques

References

• LeanAgent GitHub
• Lean 4
• Mathlib4
• LeanDojo