# Question 9686: Legal Finite Trace Prefix and Post-Halt Continuation Guard [Return to catalogue page 49](https://ocme.evemisslab.com/en/questions/page/49/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r04-computational-model-foundations/#q-20261013-advance-20261019-r04-computational-model-0036) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0018-09686 · q-20261013-advance-20261019-r04-computational-model-0036 ## Original question and givens Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5], [3, 4], [2, 1]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. trace=[0, 1, 2, 5, 5]. Give (trace, per-state type mask, nonempty?, first state in Init?, per-adjacent-pair raw Step flags, continued after a halt?, legal prefix?). A legal prefix need not end at Halt. - G001 — Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5], [3, 4], [2, 1]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. trace=[0, 1, 2, 5, 5]. Give (trace, per-state type mask, nonempty?, first state in Init?, per-adjacent-pair raw Step flags, continued after a halt?, legal prefix?). A legal prefix need not end at Halt. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-computational-model](https://ocme.evemisslab.com/en/mko/mko-computational-model/) ## Step-by-step derivation S001 · impliesBefore n=6;raw Init=[0];raw Halt=[5];raw Step=[[0, 1], [1, 2], [2, 5], [3, 4], [2, 1]] After Init=[0];Halt=[5];canonical raw Step=[[0, 1], [1, 2], [2, 1], [2, 5], [3, 4]] Conditions: none Type the complete finite data and deduplicate directed edges.Retain raw outgoing halt edges as data,but execution stops at halt.Stuck is not successful halt,and one prefix proves no general program termination. Step identifiers and dependenciesRule: finite_transition_normalization · Uses: G001 S002 · impliesBefore trace position=0;raw state=0;domain Fin(6) After typed=Yes Conditions: none Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index. Step identifiers and dependenciesRule: trace_each_state_type_guard · Uses: G001, S001 S003 · impliesBefore trace position=1;raw state=1;domain Fin(6) After typed=Yes Conditions: none Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index. Step identifiers and dependenciesRule: trace_each_state_type_guard · Uses: G001, S002 S004 · impliesBefore trace position=2;raw state=2;domain Fin(6) After typed=Yes Conditions: none Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index. Step identifiers and dependenciesRule: trace_each_state_type_guard · Uses: G001, S003 S005 · impliesBefore trace position=3;raw state=5;domain Fin(6) After typed=Yes Conditions: none Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index. Step identifiers and dependenciesRule: trace_each_state_type_guard · Uses: G001, S004 S006 · impliesBefore trace position=4;raw state=5;domain Fin(6) After typed=Yes Conditions: none Check type and domain at every original trace position.Outside or Boolean data is not a legal integer index. Step identifiers and dependenciesRule: trace_each_state_type_guard · Uses: G001, S005 S007 · impliesBefore trace pair index=0;pair=[0, 1];both typed=Yes After raw Step edge=Yes;continues from Halt=No Conditions: none Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution. Step identifiers and dependenciesRule: trace_edge_and_halt_continuation_separate · Uses: G001, S006 S008 · impliesBefore trace pair index=1;pair=[1, 2];both typed=Yes After raw Step edge=Yes;continues from Halt=No Conditions: none Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution. Step identifiers and dependenciesRule: trace_edge_and_halt_continuation_separate · Uses: G001, S007 S009 · impliesBefore trace pair index=2;pair=[2, 5];both typed=Yes After raw Step edge=Yes;continues from Halt=No Conditions: none Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution. Step identifiers and dependenciesRule: trace_edge_and_halt_continuation_separate · Uses: G001, S008 S010 · impliesBefore trace pair index=3;pair=[5, 5];both typed=Yes After raw Step edge=No;continues from Halt=Yes Conditions: none Raw edge existence and the ban on continuing after Halt are separate.Even a listed raw halt edge cannot legally extend execution. Step identifiers and dependenciesRule: trace_edge_and_halt_continuation_separate · Uses: G001, S009 S011 · impliesBefore nonempty=Yes;initial in Init=Yes;state masks=[Yes, Yes, Yes, Yes, Yes];edge flags=[Yes, Yes, Yes, No];continued after halt=Yes After legal finite prefix=No Conditions: none A legal prefix is nonempty,typed,starts in Init,follows each edge and never continues after halt.It need not yet reach Halt and does not prove all branches terminate. Step identifiers and dependenciesRule: finite_trace_prefix_not_termination · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([0, 1, 2, 5, 5], [Yes, Yes, Yes, Yes, Yes], Yes, Yes, [Yes, Yes, Yes, No], Yes, No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a5270e31665fde797323ee99db5deddb2f2f612438a1a8280f5a181919a66581 - candidate record SHA-256: 144b8891024d7dfc072489030c2b5045d567bd1c435893f303bd20bd447b56d5 - semantic review receipt SHA-256: bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c - question-first witness SHA-256: fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r04-computational-model-0036.json)