Question 4838: Partial Natural Predecessors

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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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R0008-04838 · q-20261008-advance-20261011-r01-natural-number-0038

Original question and givens

N includes 0. The ordered values are [12, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.

G001 — N includes 0. The ordered values are [12, 2]. 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

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 · calculation

Before

natural predecessor0 := sub(12, 1)

After

natural predecessor0 = 11

Conditions: none

Subtracting1 from a positive natural remains in N.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

n=12

After

natural predecessor=11

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 dependencies

Rule: partial_predecessor_zero_boundary · Uses: G001, S002

S004 · calculation

Before

natural predecessor1 := sub(2, 1)

After

natural predecessor1 = 1

Conditions: none

Subtracting1 from a positive natural remains in N.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

n=2

After

natural predecessor=1

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 dependencies

Rule: 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

(11, 1)

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: 71efd260e8eae6bc5b6edec2ef6298990ed39b23880551ecc2f311b709f11459
candidate record SHA-256: 4711e1ae839ed57fc3cb7067c8bf91a15c83ff01f3faafcd8b94e56edc5ef49f
semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289
question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67
Lean: NOT_RUN · human review: NOT_PERFORMED

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