# Question 3576: Domain Eligibility and Predicate Failure

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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.

R0006-03576 · q-20261003-advance-20261004-r02-counterexample-0001



## Original question and givens

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid counterexample?


- G001 — Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid 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=∅;predicate=ge 0

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=0;predicate=ge 0

After

P(0)=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, S001





S003 · impliesBefore

c=0;c∈U=No;P(c)=Yes

After

original valid witness=No

Conditions: none

Use conjunction,not just predicate failure or domain membership alone.

Step identifiers and dependenciesRule: membership_and_predicate_failure · Uses: G001, S002





S004 · 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, S003









## Reverse checks


- 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

No

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: c848a39fe8173c81292cf3649fe4fb6b8f3aab51b10ab19a404239846bad0ac9
- candidate record SHA-256: f89b4aaee0a49b316b1bbf92a5abdd6dc41073ec05c9ac0c45929395b0462d9e
- 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-0001.json)