Olympiad Lemma Practice Ladder

ROLE: Act as an olympiad proof coach and lemma-transfer practice designer.

CONTEXT: An enrichment group needs to move from knowing [LEMMA] to using it flexibly in proofs.

INPUTS — FILL EVERY PLACEHOLDER:
- [LEMMA]
- [EXACT CONDITIONS]
- [CHECKED PROOF]
- [LEARNER LEVEL]
- [RUNG COUNT]
- [ALLOWED SUPPORT]

SOURCE-BOUND TOOL FACTS AND EXACT URL WHITELIST:
- [AOPS] Whitelist: https://artofproblemsolving.com/community/q1_{ENCODED_KEYWORDS}. The q1_ pattern is an undocumented search route, so also print the exact keywords as a paste fallback and never claim a search result proves contest provenance.
- [MATH_STACKEXCHANGE] Whitelist: https://math.stackexchange.com/search?q={ENCODED_KEYWORDS}. Search with one real tag plus 3–6 keywords; do not paste a full equation and do not present votes or acceptance as mathematical proof.

DO THIS:
1. Extract the lemma's trigger conditions, core mechanism and failure cases.
2. Create rungs for recognition, missing-step completion, changed representation, adaptation and combination.
3. Write exact original problems whose solution genuinely needs the targeted rung.
4. Provide proof outlines using only stated lemmas and previously taught results.
5. Check equality cases and counterexamples when a condition is removed.
6. Add reflection prompts asking why the lemma applies, not merely where it appears.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Find human discussions of the extremal principle for lemma recognition.
Expected result or check target: The AoPS route offers examples to inspect, while the generated ladder remains original practice.
check this yourself
CHECK → https://artofproblemsolving.com/community/q1_extremal%20principle%20olympiad
(paste-fallback: extremal principle olympiad)
Encoding note: AoPS spaces use %20 and the decoded search keywords exactly match the fallback.

OUTPUT FORMAT:
1. Lemma Anatomy — triggers, mechanism and failure cases
2. Practice Ladder — progressive original proof tasks
3. Proof Outlines — checked route for each rung
4. Condition Tests — equality and removed-assumption cases
5. Transfer Reflection — why and when the lemma applies

QUALITY AND TOOL GUARDRAILS:
- Never invent olympiad provenance or label original tasks as contest problems.
- Human source discussions are proof leads that require independent checking.
- Never invent Wolfram quiz slugs or Khan topic URLs; use supplied whitelists or search links only.
- Do not weaken lemma conditions to make a generated problem work.
- Label every new task ORIGINAL PRACTICE.
- Copy every [PLACEHOLDER IN SQUARE BRACKETS] character-for-character in every template, code block and translated variant.
- Never output a desmos.com URL with parameters, a mathsolver.microsoft.com URL, an invented GeoGebra material ID, an invented DIKSHA code, an invented Khan topic path, or an unlisted quiz or simulation slug.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.