AI RNG: Practical Systems That Ship
A proof that succeeds usually succeeds because its structure is right. The algebra and the words matter, but the real engine is the dependency graph: which lemmas are needed, where hypotheses are used, and how subclaims assemble into the main claim.
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A proof outline makes that dependency graph explicit. AI can help you build outlines quickly, but you must keep the outline honest. The outline is not a story about what you hope is true. It is a plan of proof obligations that you can actually discharge.
Write the theorem in obligations form
Before outlining, translate the statement into obligations.
- What must exist
- What must be shown for all objects
- What conditions must be preserved under the transformations you plan to use
This step prevents the most common outline failure: an outline that never actually touches the hardest obligation.
Identify the bottleneck lemma
Most proofs have one bottleneck: the step that is not routine. Find it early.
Ask:
- Which step would be impossible without a named theorem
- Which step is sensitive to the exact hypotheses
- Which step would break if the domain changed slightly
Once you know the bottleneck, the outline becomes a path that delivers you to that lemma and then uses it correctly.
Build a dependency outline with explicit hypothesis usage
A good outline names lemmas and notes where each hypothesis is used. This prevents silent assumption creep, especially when AI suggests “standard” steps that are not actually justified.
A simple dependency table is enough:
| Lemma | Statement role | Hypotheses used | Output |
|---|---|---|---|
| L1 | Reduction | H1, H2 | Equivalent problem in simpler form |
| L2 | Core estimate | H2, H3 | Key inequality or identity |
| L3 | Structural step | H1 | Existence or uniqueness |
| Main | Assembly | H1, H2, H3 | Final conclusion |
If AI proposes lemmas, ask it to fill this table. Then you verify each “hypotheses used” entry.
Keep lemmas small enough to prove independently
Outlines fail when lemmas are vague. A lemma should be a statement you can prove without guessing what “should be true.”
A useful lemma size test:
- Can you prove it on a page or less
- Can you state it without referencing the entire main theorem
- Can you test it on small cases or examples
If the lemma is too big, split it until each piece is a clear obligation.
A concrete example: how an outline prevents wasted work
Suppose your target is a classical existence-style result: show that an object with an optimality property exists, or that a maximum is attained under compactness-like hypotheses. A weak outline might jump straight into calculations. A strong outline isolates the structural steps:
- Reduce the problem to a statement about a particular set being nonempty and closed
- Use the relevant compactness principle to guarantee attainment
- Verify the candidate really satisfies the original conditions
AI can help here by proposing candidate lemmas, but the outline only becomes real when you add the verification hooks. For each lemma, you should be able to point to the exact hypothesis that makes it true.
Use AI to propose alternate lemma decompositions
Sometimes you get stuck because your chosen decomposition is wrong, not because the theorem is hard. AI can help by suggesting alternative decompositions.
Good prompts:
- Propose three different lemma decompositions for this theorem, each with a different main bottleneck.
- For each decomposition, state the required background theorems.
- For each lemma, list the exact hypotheses used.
Then you pick the decomposition that matches what you can realistically prove and what the hypotheses actually allow.
Outline-to-proof execution: one lemma at a time
Once the outline is solid, execute it in a disciplined order.
- Prove the reduction lemmas first
- Prove the bottleneck lemma with full verification
- Prove structural lemmas that depend on the bottleneck
- Assemble the main proof and ensure each lemma is invoked with its hypotheses satisfied
This modular approach also makes it easy to collaborate. Different people can prove different lemmas, and the dependency graph tells you how the pieces fit.
Common outline failure modes and how to catch them
| Failure mode | What it looks like | Fix |
|---|---|---|
| Hidden hypothesis | A lemma assumes extra structure | Add the hypothesis explicitly or replace the lemma |
| Non sequitur lemma | Lemma is true but irrelevant | Restate the main obligation and ensure the lemma advances it |
| Oversized lemma | Lemma is as hard as the theorem | Split into two or three smaller claims with clear roles |
| Circular dependency | L2 depends on L3 which depends on L2 | Reorder or replace one lemma so the chain becomes acyclic |
| Proof by vibe | Steps described without proof obligations | Rewrite each step as a statement you could prove or cite |
Why outlines make proofs readable
Even after the proof is complete, the outline remains valuable.
- It becomes the proof’s navigation system
- It makes assumptions visible
- It helps reviewers verify correctness quickly
- It supports future reuse of lemmas in other arguments
A good outline is not extra work. It is the scaffolding that keeps the proof stable while you build it.
Keep Exploring AI Systems for Engineering Outcomes
• AI Proof Writing Workflow That Stays Correct
https://ai-rng.com/ai-proof-writing-workflow-that-stays-correct/
• How to Check a Proof for Hidden Assumptions
https://ai-rng.com/how-to-check-a-proof-for-hidden-assumptions/
• Formalizing Mathematics with AI Assistance
https://ai-rng.com/formalizing-mathematics-with-ai-assistance/
• AI for Building Counterexamples
https://ai-rng.com/ai-for-building-counterexamples/
• The Proof Autopsy: Finding the One Step That Breaks Everything
https://ai-rng.com/the-proof-autopsy-finding-the-one-step-that-breaks-everything/
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