The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] 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. Ordered candidates=[-1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
G001 — The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] 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. Ordered candidates=[-1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
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.
Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.
The table value must be EXACTLY Boolean,not0/1.At ORIGINAL zero-based label i,ONLY False gives a counterexample.Do not relabel or infer False from missing data.
The table value must be EXACTLY Boolean,not0/1.At ORIGINAL zero-based label i,ONLY False gives a counterexample.Do not relabel or infer False from missing data.
AFTER traversing the whole GIVEN finite domain,no failures means its universal statement is true,including the empty domain.This assigned finite model does not prove arbitrary infinite statements.
Guard membership in Fin(n)BEFORE reading the predicate.An outside candidate is a legal No case with outside-domain value,NOT False;never use negative indexing.
Guard membership in Fin(n)BEFORE reading the predicate.An outside candidate is a legal No case with outside-domain value,NOT False;never use negative indexing.
Guard membership in Fin(n)BEFORE reading the predicate.An outside candidate is a legal No case with outside-domain value,NOT False;never use negative indexing.
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.
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.