Structural Reasoning and Challenge Clinic

Attempt these four tasks before opening feedback. Keep a short contract, plan, argument or trace, and a looking-back note. These are the required P tasks; additional lesson drills are targeted repair or optional enrichment.

P09: Necessary is not sufficient

Starting from 1 with +4 or multiplication by 5, use parity and bounded search to decide whether 7 is reachable. Justify the bound.

P10: Construct and certify

For coins 2,5 and amounts 0,3,7, give a minimum witness or prove no witness exists. Explain why [] and None differ.

P11: Define sufficient state

Derive the minimum-coin recurrence for unlimited positive denominations. Explain exactly why amount alone fails for limited stock.

P12: A decreasing measure

Explain why an integer-valued nonnegative strictly decreasing variant proves termination and why positive real-valued decrease alone does not.

Record assistance and reuse relevant lab evidence; do not duplicate the same work to satisfy several checklists.