Further human + AI + proof assistant work on Knuth's "Claude Cycles" problem

5 Steps

1. 1

   Formalize the Problem Definition: Review the problem (e.g., Knuth's 'Claude Cycles'), define its scope, known properties, and initial conjectures. Translate these into a formal language suitable for both AI processing and proof assistant input.

2. 2

   Generate Conjectures with AI: Utilize AI systems (LLMs or specialized search algorithms) to explore the problem space. Craft detailed prompts for LLMs to generate insights, hypotheses, potential solution paths, or counter-examples.

3. 3

   Human Review and Refinement: Human experts analyze the AI-generated conjectures and insights. Filter out irrelevant suggestions, refine promising hypotheses, and provide feedback to guide subsequent AI exploration, iterating on prompts or search parameters.

4. 4

   Rigorously Verify with Proof Assistants: Translate refined conjectures or potential proofs into the language of a formal proof assistant (e.g., Coq, Lean, Isabelle). Use the proof assistant to rigorously verify the correctness of the hypotheses, ensuring mathematical soundness and eliminating errors.

5. 5

   Iterate and Document Findings: Continuously iterate through problem formalization, AI exploration, human refinement, and formal verification. Document all findings, successful proofs, and unresolved challenges to advance research and build upon discoveries.