Finite-Domain Counterexamples and Explicit Predicate Extensions
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261012-advance-20261017-r07-counterexample-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([], [], Yes, does-not-exist)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, does-not-exist).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
2. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
3. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
4. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
5. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
6. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
7. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
8. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
9. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
10. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
11. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
12. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
13. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
14. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
15. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
16. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
17. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
18. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
19. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
20. Complete Failure Set and Least Counterexample
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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
21. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
22. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
23. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
24. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
25. Complete Failure Set and Least Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).
View answer and explanation
([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1)
Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate
26. Three Domain Guards before Counterexample Evaluation
The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] 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).
View answer and explanation
([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
27. Three Domain Guards before Counterexample Evaluation
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, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
28. Three Domain Guards before Counterexample Evaluation
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=[0, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
29. Three Domain Guards before Counterexample Evaluation
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, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
30. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
31. Three Domain Guards before Counterexample Evaluation
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=[0, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
32. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
33. Three Domain Guards before Counterexample Evaluation
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, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
34. Three Domain Guards before Counterexample Evaluation
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=[0, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
35. Three Domain Guards before Counterexample Evaluation
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, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
36. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
37. Three Domain Guards before Counterexample Evaluation
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=[0, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
38. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
39. Three Domain Guards before Counterexample Evaluation
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, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
40. Three Domain Guards before Counterexample Evaluation
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=[0, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
41. Three Domain Guards before Counterexample Evaluation
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, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
42. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
43. Three Domain Guards before Counterexample Evaluation
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=[0, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
44. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
45. Three Domain Guards before Counterexample Evaluation
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, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
46. Three Domain Guards before Counterexample Evaluation
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=[0, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
47. Three Domain Guards before Counterexample Evaluation
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, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
48. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
49. Three Domain Guards before Counterexample Evaluation
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=[0, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
50. Three Domain Guards before Counterexample Evaluation
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).
View answer and explanation
([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes])
Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes]).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate
51. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] 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=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([], [], [], No, No, Yes)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], No, No, Yes).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
52. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, 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. Restrict to subset=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 2], [], [], Yes, No, Yes)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], Yes, No, Yes).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
53. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, 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. Restrict to subset=[0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 2], [0], [0], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0], [0], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
54. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, 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. Restrict to subset=[1], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 2], [1], [], Yes, No, Yes)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [1], [], Yes, No, Yes).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
55. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, 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=[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?).
View answer and explanation
([], [0, 2], [], No, No, Yes)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 2], [], No, No, Yes).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
56. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
57. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
58. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, 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. 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?).
View answer and explanation
([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
59. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([2, 3, 4], [0, 2, 4], [2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [0, 2, 4], [2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
60. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, 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. 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?).
View answer and explanation
([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
61. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([1, 3, 4], [0, 2, 4], [4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [0, 2, 4], [4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
62. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
63. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([3, 4], [0, 2, 4], [4], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [0, 2, 4], [4], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
64. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
65. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, 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. 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?).
View answer and explanation
([1, 2, 4], [0, 2, 4], [2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 2, 4], [2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
66. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
67. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, 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. 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?).
View answer and explanation
([2, 4], [0, 2, 4], [2, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [0, 2, 4], [2, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
68. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 1, 4], [0, 2, 4], [0, 4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [0, 2, 4], [0, 4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
69. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([1, 4], [0, 2, 4], [4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 4], [4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
70. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
71. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, 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. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([4], [0, 2, 4], [4], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [0, 2, 4], [4], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
72. Restriction Can Remove an Original Counterexample
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?).
View answer and explanation
([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
73. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, 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=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([1, 2, 3], [0, 2, 4], [2], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 2, 4], [2], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
74. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, 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?).
View answer and explanation
([0, 2, 3], [0, 2, 4], [0, 2], No, No, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 4], [0, 2], No, No, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
75. Restriction Can Remove an Original Counterexample
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, 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=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).
View answer and explanation
([2, 3], [0, 2, 4], [2], Yes, Yes, No)
Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [0, 2, 4], [2], Yes, Yes, No).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate
76. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] 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. Append the explicitly given Boolean value true at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([], [[0, Yes]], [], Yes, Yes, 0)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes]], [], Yes, Yes, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
77. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] 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. Append the explicitly given Boolean value false at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([], [[0, No]], [0], Yes, No, 0)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, No]], [0], Yes, No, 0).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
78. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[true, true, true, true, true, true, true, 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. Append the explicitly given Boolean value false at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
79. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[false, false, false, false, false, false, false, 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. Append the explicitly given Boolean value true at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
80. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
81. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
82. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
83. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
84. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
85. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
86. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
87. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
88. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
89. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
90. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
91. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
92. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
93. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
94. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
95. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
96. One Declared New Predicate Value and Universal Truth
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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
97. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
98. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
99. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, 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. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
100. One Declared New Predicate Value and Universal Truth
The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, 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. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).
View answer and explanation
([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5)
Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5).
Reviewed step-by-step derivation
Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate
Canonical question JSONL · English JSONL · Validation report