siddhartha-gadgil/leanaide/invariants

Skill: Invariant Construction

Pre-condition: Do not apply this skill unless the problem involves a system that undergoes changes (e.g., iterative processes, group actions, games, or transformations) and asks for a property that remains constant or monotone.

Goal: Identify or construct a mathematical object (number, polynomial, group, etc.) that is preserved under the given operations.

Instructions:

1. Identify the State Space: Define the set of all possible configurations.
2. Analyze Transformations: List the allowed moves or operations that change the state.
3. Candidate Selection:

• Numerical: Parity, sum, product, coloring (modular arithmetic).
• Structural: Symmetries, fixed points, Euler characteristic.
• Monovariants: If an invariant is not found, look for a quantity that strictly increases or decreases.

1. Verification: Prove that for any state $S$ and any transformation $T$, $I(S) = I(T(S))$.