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R0008-04686 · q-20261008-advance-20261011-r02-arithmetic-operations-0086
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=7, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=7, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.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
exact arithmetic in Q;N={0,1,2,...} when membership is askedAfter
do not truncate subtraction or rational division
Conditions: none
Compute exact results in Q before testing N-membership;do not clamp negative differences to0 or replace rational division by integer division.Division by0 remains undefined.
Rule: exact_rational_arithmetic_scope · Uses: G001
Before
Q difference := sub(7, 1)
After
Q difference = 6
Conditions: none
Subtract in Q,retaining a negative result.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Q quotient := div(7, 1)
After
Q quotient = 7
Conditions: none
Use exact division only with a nonzero denominator.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a=7;b=1;difference=6
After
(6, Yes, 7/1, Yes)
Conditions: none
Membership in N requires a nonnegative integer.An undefined quotient has no numeric N-membership verdict,so do not fill its verdict with Yes or No.
Rule: rational_result_natural_membership · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute Q quotient with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Q difference 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.(6, Yes, 7/1, Yes)
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: 3153c9d446da9e79b1131438035cad43e150795b6bb5fff255e7ccd1c867ab7acandidate record SHA-256: 17e939c1a224575439365e0f87e3426326a771703c623068219d31f3c014fbeesemantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED