Question 31097: Restriction Can Remove an Original Counterexample

Original question and givens

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

• G001 — The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

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

source roles = [qf-density17-counter-restriction-witness, counter-restriction-witness, domain, Fin_predicate_table]

derived = [[fail, subset, restricted_fail, original_witness, restricted_witness, restricted_universal]]

source roles = [5, 0, No]

derived = [[0, No], Yes]

source roles = [5, 1, No]

derived = [[1, No], Yes]

source roles = [5, 2, No]

derived = [[2, No], Yes]

source roles = [5, 3, No]

derived = [[3, No], Yes]

source roles = [5, 4, Yes]

derived = [[4, Yes], No]

source roles = [5, [No, No, No, No, Yes]]

derived = [[[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No]

source roles = [5, [No, No, No, No, Yes], [4, 2, 0], -1, [0, 1, 2, 3]]

derived = [[0, 2, 4], [0, 2], No, No, No]

Reverse checks

• S008 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S006 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S005 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.

Result

([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, 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.