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R0011-06101 · q-20261009-advance-20261013-r02-integer-0001
Integer a=0, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.
G001 — Integer a=0, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.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
declared exact integer or rational-representation protocol
After
retain signs,ordered operands and original index order
Conditions: none
Check these finite data with exact integer arithmetic,without floating approximations or a claim to prove all integer-ring axioms.
Rule: integer_arithmetic_scope · Uses: G001
Before
divide a=0 by positive b=5
After
0=0·5+0
Conditions: none
The quotient and remainder satisfy the integer identity and0≤r<b;negative dividends still do not use a negative truncated remainder.
Rule: euclidean_division_witness · Uses: G001, S001
Before
0≤0<5;a=bq+r
After
unique pair(q,r)=(0,0)
Conditions: none
The difference of two legal remainders has absolute value below b but is an integer multiple of b,so it is0 and the quotient-remainder pair is unique.
Rule: bounded_euclidean_remainder_uniqueness · Uses: G001, S002
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 — 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.(0, 0)
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: 69daadd666e5d808b7367789c8057f25ea9aecb28362b0983d707f1af0a86421candidate record SHA-256: 546b1c88b271973af8064e1a63709bdae6ac2405b667f23c3a6cd1147fd522bdsemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED