Closed Truths, Open Bindings, and Candidate Reports
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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-20261008-advance-20261012-r03-proposition-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, False], [0, -2, 2])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [0, -2, 2]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
2. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-11, -2, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-11, -2, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
3. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, False], [-10, -3, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-10, -3, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
4. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-9, -4, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-9, -4, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
5. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-8, -5, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-8, -5, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
6. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, False], [-7, -6, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-7, -6, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
7. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-6, -7, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-6, -7, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
8. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-5, -8, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-5, -8, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
9. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, False], [-4, -9, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-4, -9, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
10. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-3, -10, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-3, -10, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
11. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [-2, -11, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-2, -11, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
12. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, False], [-1, -12, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-1, -12, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
13. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([True, True, True], [0, -13, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [0, -13, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
14. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [1, -14, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [1, -14, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
15. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, False], [2, -15, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [2, -15, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
16. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [3, -16, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [3, -16, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
17. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [4, -17, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [4, -17, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
18. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, False], [5, -18, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [5, -18, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
19. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [6, -19, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [6, -19, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
20. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [7, -20, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [7, -20, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
21. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, False], [8, -21, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [8, -21, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
22. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [9, -22, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [9, -22, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
23. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [10, -23, 0])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [10, -23, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
24. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, False], [11, -24, 1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [11, -24, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
25. Closed Comparison Truths and Signed Witnesses
In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).
View answer and explanation
([False, True, True], [12, -25, -1])
Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [12, -25, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-comparison-witnesses · candidate
26. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-2 with assignment=null (unbound). Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(No, open, not-applicable)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (No, open, not-applicable).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
27. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-1 with assignment=-12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -11)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -11).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
28. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤0 with assignment=-10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -10)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -10).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
29. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤1 with assignment=-8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -9)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -9).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
30. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤2 with assignment=-6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -8)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -8).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
31. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-2 with assignment=-9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -7)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -7).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
32. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-1 with assignment=-7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -6)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -6).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
33. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤0 with assignment=-5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -5)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -5).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
34. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤1 with assignment=-3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -4)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -4).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
35. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤2 with assignment=-1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -3)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -3).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
36. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-2 with assignment=-4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -2)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -2).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
37. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-1 with assignment=-2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, -1)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -1).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
38. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤0 with assignment=0. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, True, 0)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, 0).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
39. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤1 with assignment=2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 1)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 1).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
40. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤2 with assignment=4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 2)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 2).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
41. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-2 with assignment=1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 3)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 3).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
42. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-1 with assignment=3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 4)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 4).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
43. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤0 with assignment=5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 5)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 5).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
44. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤1 with assignment=7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 6)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 6).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
45. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤2 with assignment=9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 7)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 7).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
46. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-2 with assignment=6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 8)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 8).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
47. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤-1 with assignment=8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 9)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 9).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
48. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤0 with assignment=10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 10)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 10).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
49. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤1 with assignment=12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 11)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 11).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
50. Open-Sentence Binding and Truth Eligibility
In the classical integer context, consider x≤2 with assignment=14. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).
View answer and explanation
(Yes, False, 12)
Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 12).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-open-binding · candidate
51. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([True, False, False], [False, True, True], [0, -2, 2])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -2, 2]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
52. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [-11, -2, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-11, -2, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
53. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-10, -3, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-10, -3, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
54. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-9, -4, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-9, -4, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
55. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [-8, -5, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-8, -5, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
56. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-7, -6, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-7, -6, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
57. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-6, -7, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-6, -7, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
58. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [-5, -8, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-5, -8, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
59. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-4, -9, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-4, -9, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
60. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-3, -10, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-3, -10, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
61. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [-2, -11, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-2, -11, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
62. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [-1, -12, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-1, -12, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
63. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([True, False, False], [False, True, True], [0, -13, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -13, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
64. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [1, -14, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [1, -14, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
65. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [2, -15, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [2, -15, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
66. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [3, -16, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [3, -16, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
67. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [4, -17, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [4, -17, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
68. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [5, -18, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [5, -18, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
69. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [6, -19, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [6, -19, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
70. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [7, -20, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [7, -20, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
71. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [8, -21, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [8, -21, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
72. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [9, -22, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [9, -22, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
73. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, True], [True, True, False], [10, -23, 0])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [10, -23, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
74. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [11, -24, 1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [11, -24, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
75. Equality, Negation, and Signed Difference Witnesses
In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).
View answer and explanation
([False, False, False], [True, True, True], [12, -25, -1])
A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [12, -25, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-equality-negation-witnesses · candidate
76. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, False], [Yes, Yes, Yes], [0, -2, 2])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [0, -2, 2]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
77. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [Yes, No, No], [-11, -2, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-11, -2, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
78. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, False], [No, Yes, Yes], [-10, -3, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, Yes, Yes], [-10, -3, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
79. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [Yes, Yes, No], [-9, -4, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, No], [-9, -4, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
80. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [No, No, Yes], [-8, -5, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [-8, -5, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
81. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, False], [Yes, No, No], [-7, -6, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, No, No], [-7, -6, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
82. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [No, Yes, Yes], [-6, -7, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, Yes], [-6, -7, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
83. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [Yes, Yes, Yes], [-5, -8, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, Yes], [-5, -8, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
84. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, False], [No, No, Yes], [-4, -9, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, No, Yes], [-4, -9, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
85. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [Yes, No, No], [-3, -10, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-3, -10, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
86. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [No, Yes, No], [-2, -11, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, No], [-2, -11, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
87. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, False], [Yes, Yes, Yes], [-1, -12, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [-1, -12, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
88. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([True, True, True], [No, No, Yes], [0, -13, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [0, -13, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
89. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [No, No, Yes], [1, -14, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [1, -14, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
90. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, False], [Yes, Yes, No], [2, -15, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, Yes, No], [2, -15, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
91. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [No, Yes, Yes], [3, -16, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, Yes], [3, -16, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
92. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [Yes, No, No], [4, -17, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [4, -17, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
93. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, False], [No, No, Yes], [5, -18, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, No, Yes], [5, -18, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
94. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [Yes, Yes, No], [6, -19, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, No], [6, -19, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
95. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [No, Yes, No], [7, -20, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, No], [7, -20, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
96. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, False], [Yes, No, No], [8, -21, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, No, No], [8, -21, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
97. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [No, No, Yes], [9, -22, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [9, -22, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
98. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [Yes, Yes, Yes], [10, -23, 0])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, Yes], [10, -23, 0]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
99. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, False], [No, Yes, No], [11, -24, 1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, Yes, No], [11, -24, 1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
100. Candidate Truth Reports Checked Independently
In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).
View answer and explanation
([False, True, True], [Yes, No, No], [12, -25, -1])
Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [12, -25, -1]).
Reviewed step-by-step derivation
Proposition · qf-density12-proposition-report-defects · candidate
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