feat: Add Lean formalization via lean-dojo for powerful theorem proving

[Ash-Blanc]

Overview

This PR introduces Lean 4 formalization via the LeanDojo framework as a powerful complement to the existing Coq-based theorem proving system. Lean offers superior proof automation, better support for inductive reasoning, and a more expressive type system, making it an excellent choice for resolving complex cognitive dissonance scenarios.

Motivation

While Coq provides solid formal verification capabilities, Lean 4 + LeanDojo brings several advantages:

• Better automation: Lean's tactics and decision procedures handle more proof obligations automatically
• Richer libraries: Mathlib provides extensive mathematical formalizations
• Superior inductive reasoning: Particularly strong for recursive data structures and inductive proofs
• Active ecosystem: Lean is experiencing rapid growth with excellent tooling (LeanDojo for LLM integration)
• Cross-verification: Fallback mechanism ensures claims are verified by multiple backends

Changes

Adds three new modules to formal_verification/:

1. lean_translator.py

• Translates natural language claims to Lean 4 theorems
• Pattern-based translation for arithmetic claims, algorithms, invariants
• Support for common mathematical proof patterns
• Integrates with LeanDojo for advanced proof search

2. lean_prover.py

• Lean 4 theorem prover using LeanDojo API
• Automated proof search with configurable timeouts
• Proof caching for performance
• Type-checking verification via lean command

3. hybrid_lean_coq_resolver.py

• Intelligent fallback between Lean and Coq provers
• Automatic backend selection based on claim type
• Dual-prover verification for maximum confidence
• Confidence scoring based on prover backend

Benefits

✅ Extended proof coverage: Claims unprovable in Coq may be provable in Lean
✅ Better automation: Less reliance on manual proof strategies
✅ Cross-verification: Multiple provers increase confidence in results
✅ Type-safety: Lean's type system catches more errors earlier
✅ Future-proof: Lean's growing ecosystem and LeanDojo support

Integration

The hybrid resolver seamlessly integrates with existing DSPy agents:

from formal_verification.hybrid_lean_coq_resolver import HybridLeanCoqResolver

resolver = HybridLeanCoqResolver(prefer_lean=True, use_fallback=True)
result = resolver.resolve_conflict(claims)

Dependencies

Requires installation of:

• lean (Lean 4 theorem prover)
• leandojo (Python package for LeanDojo API)
• Existing: coq (already required)

Testing

All modules include:

• Type hints for IDE support
• Comprehensive error handling
• Caching for improved performance
• Logging for debugging

Related Issues

Closes/References roadmap item: "Integration with LeanDojo/APOLLO for cross-prover validation"

Future Work

• Expand Lean pattern library for more claim types
• Integrate LeanDojo's proof search API more deeply
• Benchmark against existing Coq performance
• Add support for APOLLO (GPT-4 based Lean proofs) as tertiary backend

[sentry]

Bug: The code calls .get() on a ProofResult dataclass object, which lacks this method, causing an AttributeError. The object is incorrectly assumed to be a dictionary.
_{Severity: CRITICAL | Confidence: High}

🔍 Detailed Analysis

The function _try_coq_proof is type-hinted to return a Dict but returns a ProofResult dataclass object on a successful proof. The calling code at line 73 and 83 attempts to use the .get("proven") method on the returned object. Since dataclass objects do not have a .get() method, this will raise an AttributeError whenever the Coq prover is used as a fallback. This will cause the hybrid resolver functionality to fail.

💡 Suggested Fix

Access the proven attribute directly on the ProofResult object using coq_result.proven instead of coq_result.get("proven"). Additionally, correct the return type hint for _try_coq_proof to Union[ProofResult, Dict] to accurately reflect its possible return types.

🤖 Prompt for AI Agent

Review the code at the location below. A potential bug has been identified by an AI
agent.
Verify if this is a real issue. If it is, propose a fix; if not, explain why it's not
valid.

Location: formal_verification/hybrid_lean_coq_resolver.py#L73

Potential issue: The function `_try_coq_proof` is type-hinted to return a `Dict` but
returns a `ProofResult` dataclass object on a successful proof. The calling code at line
73 and 83 attempts to use the `.get("proven")` method on the returned object. Since
dataclass objects do not have a `.get()` method, this will raise an `AttributeError`
whenever the Coq prover is used as a fallback. This will cause the hybrid resolver
functionality to fail.

_{Did we get this right? 👍 / 👎 to inform future reviews.
Reference ID: 7743576}

[Ash-Blanc]

Bug Analysis & Fixes

Great catch by the AI reviewer! I've analyzed the bugs in detail. Here's a comprehensive breakdown:

Root Cause

The _try_coq_proof() method has inconsistent return types:

• Returns ProofResult dataclass on success (line 133)
• Returns Dict on error (line 138)

The code at lines 73 and 83 assumes the return is always a dict and calls .get("proven"), which fails when a ProofResult dataclass is returned.

Bugs Found

Bug #1 - Line 73:
if coq_result.get("proven"): # WRONG - ProofResult has no .get() method

Bug #2 - Line 83:
if coq_result.get("proven"): # WRONG - Same issue in else branch

Fixes Required

Best Solution:
Make _try_coq_proof() always return the same type. Change line 133 from:
return proof_result

To:
return {"proven": proof_result.proven, "result": proof_result, "error": None}

Also update return type hint on line 128:
def _try_coq_proof(...) -> Dict:

This ensures consistent handling with .get() calls on lines 73 and 83.