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R0007-04322 · q-20261008-advance-20261010-r07-quantifier-0072
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, -3]. 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=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, -3]. 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=[-3, -2, -1, 1, 2, 3]
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
x=-3;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S001
Before
x=-2;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S002
Before
x=-1;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S003
Before
x=1;predicate=ge 0
After
P(x)=Yes
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S004
Before
x=2;predicate=ge 0
After
P(x)=Yes
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S005
Before
x=3;predicate=ge 0
After
P(x)=Yes
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S006
Before
[[-3, No], [-2, No], [-1, No], [1, Yes], [2, Yes], [3, Yes]]
After
T=[1, 2, 3];F=[-3, -2, -1];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, S007
Before
candidate=-1;inside=Yes;predicate if inside=No
After
Yes
Conditions: none
Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain candidates.
Rule: ordered_counterexample_candidate · Uses: G001, S008
Before
candidate=2;inside=Yes;predicate if inside=Yes
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, S009
Before
candidate=-3;inside=Yes;predicate if inside=No
After
Yes
Conditions: none
Check membership before predicate failure;preserve candidate order and repeated positions,and reject outside-domain candidates.
Rule: ordered_counterexample_candidate · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.[Yes, No, 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: 8268fe391928787b27559c700a3abe6f0a71b10c2d40b685f5d6f2a5e9f7a36ccandidate record SHA-256: 2f88a71c28f78cbf76a7b2e6392aa8912a67379c078c1377de3b3582c234d8acsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED