Question 3551: Least Counterexample in a Finite Domain

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0006-03551 · q-20261003-advance-20261004-r02-counterexample-0026

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. 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=∅ and P(x) means x≥0. 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: finite_counterexample_domain · Uses: G001

S002 · implies

Before

[]

After

all finite-domain failures=∅

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 dependencies

Rule: complete_finite_failure_inventory · Uses: G001, S001

S003 · implies

Before

∅

After

does not exist

Conditions: none

Select the numerically least legal counterexample,not the first input row;explicitly report nonexistence if there is none.

Step identifiers and dependencies

Rule: least_numeric_failure · Uses: G001, S002

S004 · implies

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.

Step identifiers and dependencies

Rule: 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

none

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: e5fecf96a06c3eeaf3252a3ec40f088f9c138c0444ac616b3bf79a2b782e9ace
candidate record SHA-256: 9849f1edaddbb28fb6a3730ed56ff498f7238bb85883a5545d627af7e90061d5
semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
Lean: NOT_RUN · human review: NOT_PERFORMED

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