# Question 4897: Inclusive Natural Successor Window [Return to catalogue page 25](https://ocme.evemisslab.com/en/questions/page/25/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261011-r01-natural-number-foundations/#q-20261008-advance-20261011-r01-natural-number-0022) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0008-04897 · q-20261008-advance-20261011-r01-natural-number-0022 ## Original question and givens N includes 0. n=20, k=3. List the complete ordered window beginning at n and applying the successor k times. - G001 — N includes 0. n=20, k=3. List the complete ordered window beginning at n and applying the successor k times. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-natural-number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) ## Step-by-step derivation S001 · impliesBefore N={0,1,2,...} After 0 belongs;negative and nonintegral rational values do not Conditions: none This convention includes0 in N;distinguish membership,rational values,partial predecessors and closed intervals. Step identifiers and dependenciesRule: natural_zero_convention · Uses: G001 S002 · impliesBefore initial n=20;successor count k=3 After [20] Conditions: none Include the initial point;even k=0 has one item.Do not apply a successor before starting the list. Step identifiers and dependenciesRule: successor_window_initial · Uses: G001, S001 S003 · calculationBefore successor step1 := sum(20, 1) After successor step1 = 21 Conditions: none Add1 to the preceding natural at each step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore successor step2 := sum(21, 1) After successor step2 = 22 Conditions: none Add1 to the preceding natural at each step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore successor step3 := sum(22, 1) After successor step3 = 23 Conditions: none Add1 to the preceding natural at each step. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore [20, 21, 22, 23] After 3 successors and 4 total entries Conditions: none Applying k successors produces k+1 entries,retaining the full order and initial point. Step identifiers and dependenciesRule: successor_window_length · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute successor step3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute successor step2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute successor step1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result [20, 21, 22, 23] The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2cd3c383fc98b7ff553bf18e54611faf4cb3196cfc9a024eb1e9ab3dbc5d7756 - candidate record SHA-256: 59b0090fe3a33d675f4a95517349b5a05b995120392c649a82c577fbc38ae298 - semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289 - question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r01-natural-number-0022.json)