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R0006-03548 · q-20261003-advance-20261004-r02-counterexample-0098
Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x≥11. The value c=10 is a valid counterexample to the original universal claim. After restricting to V={11, 12}⊆U, is c still a valid counterexample?
G001 — Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x≥11. The value c=10 is a valid counterexample to the original universal claim. After restricting to V={11, 12}⊆U, is c still a valid counterexample?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
U={10, 11, 12};predicate=ge 11After
a counterexample requires c∈U and P(c)=False
Conditions: none
A counterexample needs both domain membership and predicate failure;an outside integer does not refute this restricted universal statement even if its formula is false.
Rule: finite_counterexample_domain · Uses: G001
Before
x=10;predicate=ge 11
After
P(10)=No
Conditions: none
Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility.
Rule: explicit_predicate_value · Uses: G001, S001
Before
c=10;c∈U=Yes;P(c)=No
After
original valid witness=Yes
Conditions: none
Use conjunction,not just predicate failure or domain membership alone.
Rule: membership_and_predicate_failure · Uses: G001, S002
Before
V={11, 12}⊆U;original witness=10After
c∈V=No
Conditions: none
The old counterexample remains eligible only if it is still in V;losing this witness does not remove every other possible counterexample in V.
Rule: restricted_domain_witness · Uses: G001, S003
Before
the explicitly declared finite exercise
After
no inference about an unspecified infinite domain
Conditions: none
Complete enumeration decides this finite universal statement,but finite success or absence of counterexamples is not a universal proof over an infinite domain.
Rule: finite_logic_scope · 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
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: 1255dfd7ad23701426b1819087e43e7ce45267612a17d643c685139adb46ea0dcandidate record SHA-256: ed6441dabc2dfb174eae70f90df6def0939d320b399875ac8dff77441959a393semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED