# 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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r04-proof-0100/)

Mathematical proof · qf-density13-proof-horn-rule-removal · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r04-proof-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r04-proof-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r04-proof-foundations/validation-report.json)