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R0011-06401 · q-20261009-advance-20261013-r03-rational-number-0051
Exact rational representations have nonzero denominators; a=5/-2, b=0/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — Exact rational representations have nonzero denominators; a=5/-2, b=0/-3. Negative input denominators are allowed. Give (is b nonzero?, a/b or undefined, (a/b)b or undefined). Use exact reduced fractions with positive denominators, retaining /1 for integers.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
raw a=[5, -2];raw b=[0, -3]
After
a=-5/2;b=0/1
Conditions: none
Original denominators need only be nonzero;negative denominators are valid.Normalize to reduced positive-denominator fractions for exact arithmetic while retaining the original representations and sources.
Rule: nonzero_signed_rational_representation · Uses: G001
Before
divisor value b=0/1
After
quotient and reverse product are undefined
Conditions: none
Division is invalid when the divisor's value is0,even if its original denominator is negative;do not invent a quotient and certify it by multiplying back.
Rule: rational_zero_divisor_guard · Uses: G001, S001
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.(No, undefined, undefined)
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: 5aa64edef2fa3feee6a059543a1b6947a5b9f9c295dbaa0b4df76e5d9f6b030dcandidate record SHA-256: 7b48f8f064cf4100255440cd2a44097ef7956b141a05156468e6f1899101785fsemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED