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R0007-04301 · q-20261008-advance-20261010-r07-quantifier-0051
The explicitly finite integer domain is U=[] and P(x) means x≥0. Ordered candidates=[100, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
G001 — The explicitly finite integer domain is U=[] and P(x) means x≥0. Ordered candidates=[100, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.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
complete finite U=[]
After
universal on empty domain=True;existential on empty domain=False
Conditions: none
Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain.
Rule: finite_quantifier_scope · Uses: G001
Before
[]
After
T=[];F=[];T∪F=U and T∩F=∅
Conditions: none
The true and false parts are disjoint and cover the deduplicated domain;unlisted integers are not counterexamples.
Rule: complete_truth_partition · Uses: G001, S001
Before
candidate=100;inside=No;predicate if inside=null
After
No
Conditions: none
Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain candidates.
Rule: ordered_counterexample_candidate · Uses: G001, S002
Before
candidate=101;inside=No;predicate if inside=null
After
No
Conditions: none
Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain candidates.
Rule: ordered_counterexample_candidate · Uses: G001, S003
Before
candidate=102;inside=No;predicate if inside=null
After
No
Conditions: none
Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain candidates.
Rule: ordered_counterexample_candidate · 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.[No, No, No]
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: 646e021f5dc3de36fdb76d811864070a4fb0f7f0bf549f992b036d79f2d4f244candidate record SHA-256: 6dd300ea603c288f87362f116be7ce5e31662380dcceeb71aa14e59d49ea6339semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED