# Question 4832: Partial Natural Predecessors

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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.

R0008-04832 · q-20261008-advance-20261011-r01-natural-number-0032



## Original question and givens

N includes 0. The ordered values are [6, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.


- G001 — N includes 0. The ordered values are [6, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.





## 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 · calculationBefore

natural predecessor0 := sub(6, 1)

After

natural predecessor0 = 5

Conditions: none

Subtracting1 from a positive natural remains in N.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

n=6

After

natural predecessor=5

Conditions: none

Zero has no natural predecessor because every natural plus1 is positive;this is neither truncated pred(0)=0 nor a return of−1 as a natural.

Step identifiers and dependenciesRule: partial_predecessor_zero_boundary · Uses: G001, S002





S004 · calculationBefore

natural predecessor1 := sub(1, 1)

After

natural predecessor1 = 0

Conditions: none

Subtracting1 from a positive natural remains in N.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

n=1

After

natural predecessor=0

Conditions: none

Zero has no natural predecessor because every natural plus1 is positive;this is neither truncated pred(0)=0 nor a return of−1 as a natural.

Step identifiers and dependenciesRule: partial_predecessor_zero_boundary · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute natural predecessor1 with exact rational arithmetic, not rounded decimals.
- 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 — Recompute natural predecessor0 with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(5, 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: b5ed9fbc974868b5bc2a55b8755cfa668e8e5f306c1bb5c2174d868ec8faa7dc
- candidate record SHA-256: 3f542f41a1ba432d3371a02e4ae5e32327d5d41f4d15b0118ebb833cf6fcf5ed
- 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-0032.json)