ML/AI Theory Rigorous Proof Writer

---
name: proof-writer
description: "Writes rigorous mathematical proofs for ML/AI theory. Use when asked to prove a theorem, lemma, proposition, or corollary, fill in missing proof steps, formalize a proof sketch, /u8865/u5168/u8bc1/u660e, /u5199/u8bc1/u660e, /u8bc1/u660e/u67d0/u4e2a/u547d/u9898, or determine whether a claimed proof c"
---
# Proof Write: Rigorous Theorem / Lemma Drafting

Write a mathematically honest proof package, not a polished fake proof.

## Constants

- DEFAULT_PROOF_DOC = `PROOF_PACKAGE.md` in project root
- STATUS = `PROVABLE AS STATED | PROVABLE AFTER WEAKENING / EXTRA ASSUMPTION | NOT CURRENTLY JUSTIFIED`

## Context: $ARGUMENTS

## Goal

Produce exactly one of:
1. a complete proof of the original claim
2. a corrected claim plus a proof of the corrected claim
3. a blockage report explaining why the claim is not currently justified

## Inputs

Extract and normalize:
- exact theorem / lemma / proposition / corollary statement
- explicit assumptions
- notation and definitions
- any user-provided proof sketch, partial proof, or intended strategy
- nearby lemmas or claims in local notes, appendix files, or theorem drafts if the request points to them
- desired output style if specified: concise, appendix-ready, or full-detail

If notation or assumptions are ambiguous, state the exact interpretation you are using before proving anything.

## Workflow

### Step 1: Gather Proof Context
Determine the target proof file with this priority:
1. a file path explicitly specified by the user
2. a proof draft already referenced in local notes or theorem files
3. `PROOF_PACKAGE.md` in project root as the default target

Read the relevant local context:
- the chosen target proof file, if it already exists
- theorem notes, appendix drafts, or files explicitly mentioned by the user

Extract:
- exact claim
- assumptions
- notation
- proof sketch or partial proof
- nearby lemmas that the draft may depend on

### Step 2: Normalize the Claim
Restate:
- the exact claim being proved
- all assumptions, separately from conclusions
- all symbols used in the claim

Identify:
- hidden assumptions
- undefined notation
- scope ambiguities
- whether the available sketch proves the full claim or only a weaker variant

Preserve the user's original theorem statement unless a change is explicitly required.
If you use a stronger normalization or cleaner internal formulation only to make the proof easier, keep that as an internal proof device rather than silently replacing the original claim.

### Step 3: Feasibility Triage
Before writing a proof, classify the claim into exactly one status:
- `PROVABLE AS STATED`
- `PROVABLE AFTER WEAKENING / EXTRA ASSUMPTION`
- `NOT CURRENTLY JUSTIFIED`

Check explicitly:
- does the conclusion actually follow from the listed assumptions?
- is any cited theorem being used outside its conditions?
- is the claim stronger than what the available argument supports?
- is there an obvious counterexample, boundary case, or quantifier failure?

If the claim is not provable as stated, do NOT fabricate a proof.
Do NOT silently strengthen assumptions or narrow the theorem's scope just to make the proof work.

### Step 4: Build a Dependency Map
Choose a proof strategy, for example:
- direct
- contradiction
- induction
- construction
- reduction to a known result
- coupling / probabilistic argument
- optimization inequality chaining

Then write a dependency map:
- main claim
- required intermediate lemmas
- named theorems or inequalities that will be cited
- which assumptions each nontrivial step depends on
- boundary cases that must be handled separately

If one step is substantial, isolate it as a lemma instead of burying it in one sentence.

### Step 5: Write the Proof Document
Write to the chosen target proof file.

If the target proof file already exists:
- read it first
- update the relevant claim section
- do not blindly duplicate prior content

If the user does not specify a target, default to `PROOF_PACKAGE.md` in project root.

Do NOT write directly into paper sections or appendix `.tex` files unless the user explicitly asks for that target.

The proof package must include:
- exact claim
- explicit assumptions
- proof status
- announced strategy
- dependency map
- numbered major steps
- justification for every nontrivial implication

Mathematical rigor requirements:
- never use "clearly", "obviously", "it can be shown", "by standard arguments", or "similarly" to hide a gap
- define every constant and symbol before use
- check quantifier order carefully
- handle degenerate and boundary cases explicitly, or state why they are excluded
- if invoking a standard fact, state its name and why its assumptions are satisfied here
- use `$...$` for inline math and `$$...$$` for display equations
- never write math in plain text
- if the proof uses an equivalent normalization that is stronger in appearance than the user's original theorem statement, label it explicitly as a proof device and keep the original claim separate

### Step 6: Final Verification
Before finishing the target proof file, verify:
- the theorem statement exactly matches what was actually shown
- every assumption used is stated
- every nontrivial implication is justified
- every inequality direction is correct
- every cited result is applicable under the stated assumptions
- edge cases are handled or explicitly excluded
- no hidden dependence on an unproved lemma remains

If a key step still cannot be justified, downgrade the status and write a blockage report instead of forcing a proof.

## Required File Structure

Write the target proof file using this structure:

```md
# Proof Package

## Claim
[exact statement]

## Status
PROVABLE AS STATED / PROVABLE AFTER WEAKENING / NOT CURRENTLY JUSTIFIED

## Assumptions
- ...

## Notation
- ...

## Proof Strategy
[chosen approach and why]

## Dependency Map
1. Main claim depends on ...
2. Lemma A depends on ...
3. Step k uses ...

## Proof
Step 1. ...
Step 2. ...
...
Therefore the claim follows. ∎

## Corrections or Missing Assumptions
- [only if needed]

## Open Risks
- [remaining fragile points, if any]
```

## Output Modes

### If the claim is provable as stated
Write the full file structure above with a complete proof.

### If the original claim is too strong
Write:
- why the original statement is not justified
- the corrected claim
- the minimal extra assumption if one exists
- a proof of the corrected claim

### If the proof cannot be completed honestly
Write:
- `Status: NOT CURRENTLY JUSTIFIED`
- the exact blocker: missing lemma, invalid implication, hidden assumption, or counterexample direction
- what extra assumption, lemma, or derivation would be needed to finish the proof
- a corrected weaker statement if one is available

## Chat Response

After writing the target proof file, respond briefly with:
- status
- whether the original claim survived unchanged
- what file was updated

## Key Rules

- Never fabricate a missing proof step.
- Prefer weakening the claim over overclaiming.
- Separate assumptions, derived facts, heuristics, and conjectures.
- Preserve the user's original theorem statement unless you explicitly mark a corrected claim or an internal normalization.
- If the statement is false as written, say so explicitly and give a counterexample or repaired statement.
- If uncertainty remains, mark it explicitly in `Open Risks`; do not hide it inside polished prose.
- Correctness matters more than brevity.