Question 4884: Inclusive Natural Successor Window

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0008-04884 · q-20261008-advance-20261011-r01-natural-number-0009

Original question and givens

N includes 0. n=7, k=8. List the complete ordered window beginning at n and applying the successor k times.

G001 — N includes 0. n=7, k=8. 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: natural_zero_convention · Uses: G001

S002 · implies

Before

initial n=7;successor count k=8

After

[7]

Conditions: none

Include the initial point;even k=0 has one item.Do not apply a successor before starting the list.

Step identifiers and dependencies

Rule: successor_window_initial · Uses: G001, S001

S003 · calculation

Before

successor step1 := sum(7, 1)

After

successor step1 = 8

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

successor step2 := sum(8, 1)

After

successor step2 = 9

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

successor step3 := sum(9, 1)

After

successor step3 = 10

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

successor step4 := sum(10, 1)

After

successor step4 = 11

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

successor step5 := sum(11, 1)

After

successor step5 = 12

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

successor step6 := sum(12, 1)

After

successor step6 = 13

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

successor step7 := sum(13, 1)

After

successor step7 = 14

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

successor step8 := sum(14, 1)

After

successor step8 = 15

Conditions: none

Add1 to the preceding natural at each step.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

[7, 8, 9, 10, 11, 12, 13, 14, 15]

After

8 successors and 9 total entries

Conditions: none

Applying k successors produces k+1 entries,retaining the full order and initial point.

Step identifiers and dependencies

Rule: successor_window_length · Uses: G001, S010

Reverse checks

S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Recompute successor step8 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute successor step7 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute successor step6 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute successor step5 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute successor step4 with exact rational arithmetic, not rounded decimals.
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

[7, 8, 9, 10, 11, 12, 13, 14, 15]

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: 88d8b041b427219d09bab8baf5071406e05c446d6fbde69fdce91e1acc4994c4
candidate record SHA-256: 05856a72f181fc237a1c5af28135b5b853f7ca5d9df090be493d54f4d7decdaa
semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
Lean: NOT_RUN · human review: NOT_PERFORMED

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