What to Learn After a Verified Proof

The options

The order shown is not a ranking of preference.

1. Option 1

   Proof reuse

   Applies verified components to neighboring propositions in the same system to examine the reach achieved, but explores the role of the key assumptions and human explainability less.

2. Option 2

   Find explanatory structure

   Reconstructs intermediate concepts and auxiliary arguments to find cues a person can use on unfamiliar problems, but may leave few new verification results without gaining explanations.

3. Option 3

   Counterexample boundary exploration

   Investigates counterexamples and the ranges where things hold in separate problems that vary one assumption at a time, but postpones the generative principles of the original proof and the extension to neighboring propositions.

The situation as it stands

If no allocation is made by the deadline, the reserved joint computing time shifts to other projects, and the researchers return to individual projects and lose the time to attend to the joint argument together.

Yeo-ul Joint Mathematics Laboratory, a fictional organization

1. Premise 1As a synthetic premise of this case, the laboratory has obtained machine-verified proofs of propositions about finite relational structures, verified under explicit axioms and rules of inference. The research team has not yet turned the key structures that make the proofs hold into explanations usable on new problems.
2. Premise 2The principal investigator has the authority to allocate the joint researchers' next research-term time and the reserved computing resources. The resources are at a scale that can concentrate on a single task, and changing direction after allocation means abandoning some of the computation and collaboration schedules already prepared.
3. Premise 3The proof-building team wants to extend the range of relevant propositions to confirm the usefulness of existing work, and the explanation team wants to reconstruct the key arguments. The boundary-exploration team asks first about the necessity of each assumption, but all three teams request the same joint research time.

Where it holdsAssign the joint researchers' next research-term time and computing resources to one lead task.

What can be lost

• If reuse is delayed, the proof-building team loses the chance to confirm the research value of the components it has built.
• If the explanation team does not secure time, it leaves a gap between the ability to read proofs and the ability to produce new inferences.
• If the boundary-exploration team does not investigate the role of each assumption, it gains no basis for setting the scope of follow-up tasks.

What is still unknown

1. Unknown 1It is not known whether the long dependency relations of the present proofs can be reconstructed into a small number of concepts usable by a person.
2. Unknown 2It is unconfirmed whether the neighboring propositions gained through reuse open new inference paths or merely repeat existing constructions.
3. Unknown 3If no counterexample is found in a problem with weakened assumptions, whether insufficient exploration can be distinguished from a wider range of validity remains unknown.

Signals to decide again

• The explanation team checks every two weeks for cases where explanation alone was used to construct a new auxiliary argument on an unfamiliar problem.
• The verification team records weekly the assumptions added at reuse time and the amount of modification made to existing components.
• The boundary team reports every two weeks on counterexample candidates for each weakened assumption and on the ranges left unexplored.