# Question 3559: Least Counterexample in a Finite Domain [Return to catalogue page 18](https://ocme.evemisslab.com/en/questions/page/18/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r02-counterexample-foundations/#q-20261003-advance-20261004-r02-counterexample-0034) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0006-03559 · q-20261003-advance-20261004-r02-counterexample-0034 ## Original question and givens Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample. - G001 — Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample. ## Definition and scope 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. - [mko-counterexample](https://ocme.evemisslab.com/en/mko/mko-counterexample/) ## Step-by-step derivation S001 · impliesBefore U={-4, -3, -1, 1};predicate=ge -3 After 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. Step identifiers and dependenciesRule: finite_counterexample_domain · Uses: G001 S002 · impliesBefore x=-4;predicate=ge -3 After P(-4)=No Conditions: none Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility. Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S001 S003 · impliesBefore x=-3;predicate=ge -3 After P(-3)=Yes Conditions: none Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility. Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S002 S004 · impliesBefore x=-1;predicate=ge -3 After P(-1)=Yes Conditions: none Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility. Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S003 S005 · impliesBefore x=1;predicate=ge -3 After P(1)=Yes Conditions: none Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility. Step identifiers and dependenciesRule: explicit_predicate_value · Uses: G001, S004 S006 · impliesBefore [[-4, No], [-3, Yes], [-1, Yes], [1, Yes]] After all finite-domain failures={-4} Conditions: none Cover the entire declared finite domain;the universal statement on empty U is vacuously true and has no counterexample.This does not extend to larger or infinite domains. Step identifiers and dependenciesRule: complete_finite_failure_inventory · Uses: G001, S005 S007 · impliesBefore {-4} After -4 Conditions: none Select the numerically least legal counterexample,not the first input row;explicitly report nonexistence if there is none. Step identifiers and dependenciesRule: least_numeric_failure · Uses: G001, S006 S008 · impliesBefore 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. Step identifiers and dependenciesRule: finite_logic_scope · Uses: G001, S007 ## Reverse checks - 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. ## Result -4 The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 8ae61624ba9f350da6179d579e8bb645a403d9ed0d6555b90d83e293da93e672 - candidate record SHA-256: c9c77eeba04f52c80a53503602a4d81d8610d2945e55ee8d2b05c2042f7d5a72 - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r02-counterexample-0034.json)