Given-Fact MP Depths and Sequential Trace Audits
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261009-advance-20261013-r04-proof-foundations · 2026-10-09 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
2. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
3. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(1)={0,…,0}, facts=[0], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
4. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 2, 3, 4, 5, 6, 7]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 3, 4, 5, 6, 7].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
5. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 1, 2, 3, 4, 5, 6]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 2, 3, 4, 5, 6].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
6. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 0, 1, 2, 3, 4, 5, 6]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 2, 3, 4, 5, 6].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
7. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived, 0, 1, 2, 3, 4, 5]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 1, 2, 3, 4, 5].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
8. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 0, 1, 2, 3, 4, 5]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 1, 2, 3, 4, 5].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
9. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 0, 1, 2, 3, 4, 5]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 1, 2, 3, 4, 5].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
10. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 2], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 0, 0, 1, 2, 3, 4, 5]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 0, 1, 2, 3, 4, 5].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
11. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived, not-derived, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, not-derived, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
12. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 2, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
13. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 1, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
14. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 0, 1, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
15. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived, 0, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
16. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 0, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
17. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 0, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
18. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 2, 3], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 0, 0, 0, 1, 2, 3, 4]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 0, 0, 1, 2, 3, 4].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
19. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived, not-derived, not-derived, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, not-derived, not-derived, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
20. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 2, 3, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 2, 3, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
21. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 1, 2, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 1, 2, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
22. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 1, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 0, 1, 2, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 0, 1, 2, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
23. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, not-derived, 0, 1, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, not-derived, 0, 1, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
24. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0, 2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[0, 1, 0, 1, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [0, 1, 0, 1, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
25. All Minimum Depths in a Given MP Protocol
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[1, 2, 4], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. List minimum derivation depths at every index, using not-derived where unreachable.
View answer and explanation
[not-derived, 0, 0, 1, 0, 1, 2, 3]
Multi-source traversal from facts finds shortest rule paths. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is [not-derived, 0, 0, 1, 0, 1, 2, 3].
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-min-depths · candidate
26. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[-1, 0, 1]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, No, No], [No, No, No], [ill-typed, ill-typed, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No], [No, No, No], [ill-typed, ill-typed, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
27. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, No, No], [Yes, No, No], [0, ill-typed, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, No, No], [0, ill-typed, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
28. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, No], [No, No, No], [not-derived, not-derived, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, No, No], [not-derived, not-derived, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
29. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, No], [Yes, No, No], [0, not-derived, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, No, No], [0, not-derived, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
30. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, Yes, No], [No, Yes, No], [ill-typed, 1, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No], [No, Yes, No], [ill-typed, 1, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
31. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, No], [No, Yes, No], [not-derived, 1, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, Yes, No], [not-derived, 1, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
32. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 102]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, No], [Yes, Yes, No], [0, 1, ill-typed])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, No], [0, 1, ill-typed]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
33. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, No, Yes], [No, No, No], [ill-typed, ill-typed, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, No], [ill-typed, ill-typed, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
34. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, No, Yes], [No, No, No], [not-derived, ill-typed, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, No], [not-derived, ill-typed, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
35. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, No, Yes], [Yes, No, No], [0, ill-typed, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, No], [0, ill-typed, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
36. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, Yes, Yes], [No, No, No], [ill-typed, not-derived, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, No], [ill-typed, not-derived, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
37. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [No, No, No], [not-derived, not-derived, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, No], [not-derived, not-derived, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
38. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [Yes, No, No], [0, not-derived, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, No], [0, not-derived, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
39. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, Yes, Yes], [No, Yes, No], [ill-typed, 1, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, No], [ill-typed, 1, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
40. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [No, Yes, No], [not-derived, 1, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, No], [not-derived, 1, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
41. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 3]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [Yes, Yes, No], [0, 1, not-derived])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, No], [0, 1, not-derived]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
42. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, No, Yes], [No, No, Yes], [ill-typed, ill-typed, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, Yes], [ill-typed, ill-typed, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
43. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, No, Yes], [No, No, Yes], [not-derived, ill-typed, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, Yes], [not-derived, ill-typed, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
44. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 101, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, No, Yes], [Yes, No, Yes], [0, ill-typed, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, Yes], [0, ill-typed, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
45. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, Yes, Yes], [No, No, Yes], [ill-typed, not-derived, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, Yes], [ill-typed, not-derived, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
46. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [No, No, Yes], [not-derived, not-derived, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, Yes], [not-derived, not-derived, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
47. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 3, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [Yes, No, Yes], [0, not-derived, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes], [0, not-derived, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
48. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[100, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([No, Yes, Yes], [No, Yes, Yes], [ill-typed, 1, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, Yes], [ill-typed, 1, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
49. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[3, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [No, Yes, Yes], [not-derived, 1, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes], [not-derived, 1, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
50. Typed MP Goals and Minimum Depths
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered queries=[0, 1, 2]. Give (goal-type mask, derivability mask, minimum depths or not-derived or ill-typed).
View answer and explanation
([Yes, Yes, Yes], [Yes, Yes, Yes], [0, 1, 2])
Check typing before reachability; not-derived does not mean the proposition is False. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes], [0, 1, 2]).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-goal-check · candidate
51. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(0)=∅, facts=[], rules=[], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([], none)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([], none).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
52. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[0, 1, 0]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
53. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, No], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
54. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
55. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, No], 2)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No], 2).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
56. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
57. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, Yes], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
58. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
59. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, Yes], none)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes], none).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
60. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, No, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
61. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, No, No], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No, No], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
62. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, No, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
63. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, No, No], 2)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No, No], 2).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
64. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, Yes, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
65. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, Yes, No], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes, No], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
66. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, Yes, No], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes, No], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
67. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1, 2]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, Yes, No], 3)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes, No], 3).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
68. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, No, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, No, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
69. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, No, Yes], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, No, Yes], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
70. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, No, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, No, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
71. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 2, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, No, Yes], 2)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, No, Yes], 2).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
72. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 2, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, No, Yes, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, No, Yes, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
73. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 2, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, No, Yes, Yes], 1)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, No, Yes, Yes], 1).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
74. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[2, 1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([No, Yes, Yes, Yes], 0)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([No, Yes, Yes, Yes], 0).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
75. Sequential Candidate Trace and First Rejected Row
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Ordered trace=[1, 1, 1, 1]. Each row may use only facts and earlier accepted rows for a single MP. Give (row-eligibility mask, first invalid row’s zero-based index or none). Continue auditing; invalid rows do not become premises.
View answer and explanation
([Yes, Yes, Yes, Yes], none)
Only accepted rows enlarge available premises, and an empty trace proves no goal. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is ([Yes, Yes, Yes, Yes], none).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-candidate-trace · candidate
76. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[], rules=[[0, 1], [1, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(not-derived, not-derived, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (not-derived, not-derived, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
77. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(1)={0,…,0}, facts=[0], rules=[[0, 0]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=0. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(0, 0, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (0, 0, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
78. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1], [0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 1, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 1, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
79. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 2, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 2, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
80. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[3], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 3, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 3, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
81. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 4, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 4, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
82. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 5, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 5, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
83. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 6]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 6, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 6, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
84. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 7]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, 7, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, 7, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
85. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(2)={0,…,1}, facts=[0], rules=[[0, 1]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[0], goal=1. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(1, not-derived, Yes)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (1, not-derived, Yes).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
86. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 2, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 2, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
87. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[3], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 3, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 3, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
88. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 4, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 4, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
89. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 5, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 5, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
90. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 6, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 6, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
91. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 6]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, 7, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, 7, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
92. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(3)={0,…,2}, facts=[0], rules=[[0, 1], [1, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[1], goal=2. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(2, not-derived, Yes)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (2, not-derived, Yes).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
93. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, 3, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 3, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
94. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[4], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, 4, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 4, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
95. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, 5, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 5, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
96. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(7)={0,…,6}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [0, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[6], goal=6. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, 6, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 6, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
97. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(8)={0,…,7}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [0, 5]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[7], goal=7. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, 7, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, 7, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
98. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(4)={0,…,3}, facts=[0], rules=[[0, 1], [1, 2], [2, 3]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[2], goal=3. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(3, not-derived, Yes)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (3, not-derived, Yes).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
99. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(5)={0,…,4}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[], goal=4. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(4, 4, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (4, 4, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
100. Goal Depths Before and After Removing Original Rules
Given facts and single-premise MP only: Fin(6)={0,…,5}, facts=[0], rules=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [0, 2]], where [i,j] means i→j. Facts have depth 0 and each MP adds one. Original rule-row indices start at 0; remove_indices=[5], goal=5. Give (depth before or not-derived, depth after or not-derived, did derivability change?).
View answer and explanation
(4, 5, No)
Validate all original rules, remove by original indices, and recompute each reachability result. A longer but still finite depth is not a change of derivability. Use only the declared rules; a cycle without a seed fact cannot derive itself, and this finite audit is not logical completeness. The exact result, retaining the requested order, is (4, 5, No).
Reviewed step-by-step derivation
Mathematical proof · qf-density13-proof-horn-rule-removal · candidate
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