# Question 31105: Three Domain Guards before Counterexample Evaluation

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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.

R0061-31105 · q-20261012-advance-20261017-r07-counterexample-0030



## Original question and givens

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, 0, -1]. 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, 0, -1]. 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.


- [mko-counterexample](https://ocme.evemisslab.com/en/mko/mko-counterexample/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-density17-counter-three-candidate-guards, counter-three-candidate-guards, domain, Fin_predicate_table]

After

derived = [[indexed_table, type_flags, predicate_values, counter_flags]]

Conditions: none

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.

Step identifiers and dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001





S002 · impliesBefore

source roles = [2, 0, Yes]

After

derived = [[0, Yes], No]

Conditions: none

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.

Step identifiers and dependenciesRule: Counter_EACH_original_label_EXACT_Boolean_and_failure_test · Uses: G001, S001





S003 · impliesBefore

source roles = [2, 1, No]

After

derived = [[1, No], Yes]

Conditions: none

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.

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





S004 · impliesBefore

source roles = [2, [Yes, No]]

After

derived = [[[0, Yes], [1, No]], [1], No]

Conditions: none

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.

Step identifiers and dependenciesRule: Counter_COMPLETE_finite_table_failures_EMPTY_universal_true · Uses: G001, S003





S005 · impliesBefore

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

After

derived = [No, outside the domain, No]

Conditions: none

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.

Step identifiers and dependenciesRule: Counter_QUERY_type_guard_BEFORE_predicate_NO_outside_evaluation · Uses: G001, S004





S006 · impliesBefore

source roles = [1, 0, 2, [Yes, No]]

After

derived = [Yes, Yes, No]

Conditions: none

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.

Step identifiers and dependenciesRule: Counter_QUERY_type_guard_BEFORE_predicate_NO_outside_evaluation · Uses: G001, S005





S007 · impliesBefore

source roles = [2, -1, 2, [Yes, No]]

After

derived = [No, outside the domain, No]

Conditions: none

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.

Step identifiers and dependenciesRule: Counter_QUERY_type_guard_BEFORE_predicate_NO_outside_evaluation · Uses: G001, S006









## Reverse checks


- 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, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: c7d20c5a3c1586a64e587fc725644611dbc6e98ddccf4b654c6b4bd393b474e3
- candidate record SHA-256: c3cf53e959e647ddd0278c7ebbad1ef2ce9f04268cdb12bc07ebc9fc456f9108
- semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69
- question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r07-counterexample-0030.json)