# Question 4876: Inclusive Natural Successor Window

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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.

R0008-04876 · q-20261008-advance-20261011-r01-natural-number-0001



## Original question and givens

N includes 0. n=0, k=0. List the complete ordered window beginning at n and applying the successor k times.


- G001 — N includes 0. n=0, k=0. 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=0;successor count k=0

After

[0]

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 · impliesBefore

[0]

After

0 successors and 1 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, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

[0]

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: d1238350a71a47741b6985545a4325825fc14ffd690849e362dbfd46dd3f40b7
- candidate record SHA-256: 530ea54ef58239af31c2490ce50d7a1f85a1d21c589921a48da39d0e12c3ddca
- 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-0001.json)