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R0008-04843 · q-20261008-advance-20261011-r01-natural-number-0043
N includes 0. The ordered values are [17, 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 [17, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.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.
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.
Rule: natural_zero_convention · Uses: G001
Before
natural predecessor0 := sub(17, 1)
After
natural predecessor0 = 16
Conditions: none
Subtracting1 from a positive natural remains in N.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
n=17
After
natural predecessor=16
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.
Rule: partial_predecessor_zero_boundary · Uses: G001, S002
Before
natural predecessor1 := sub(2, 1)
After
natural predecessor1 = 1
Conditions: none
Subtracting1 from a positive natural remains in N.
Rule: exact_rational_arithmetic · Uses: G001, S003
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.
Rule: partial_predecessor_zero_boundary · Uses: G001, S004
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.(16, 1)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4271e508a6db766d36883fdfb62d1472badfa4d8a7c122223092cd66d4bd615ccandidate record SHA-256: 2be1b6b79b2165dc4e1f76fd6196f4c8c2b0c10a0edef9397dcaec084ba4fe42semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED