# Question 9511: Inductiveness versus Merely Covering Reachable States [Return to catalogue page 48](https://ocme.evemisslab.com/en/questions/page/48/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261019-r05-algorithm-correctness-foundations/#q-20261013-advance-20261019-r05-algorithm-correctness-0011) 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. R0018-09511 · q-20261013-advance-20261019-r05-algorithm-correctness-0011 ## 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. Proposed invariant=[0, 1, 2, 5]. Give (invariant, contains Init?, every Step edge from an invariant nonhalt to outside the invariant, step-preserved?, inductive?, covers all reachable states?). Inspect all invariant states for preservation, including unreachable ones. - 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. Proposed invariant=[0, 1, 2, 5]. Give (invariant, contains Init?, every Step edge from an invariant nonhalt to outside the invariant, step-preserved?, inductive?, covers all reachable states?). Inspect all invariant states for preservation, including unreachable ones. ## 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-algorithm-correctness](https://ocme.evemisslab.com/en/mko/mko-algorithm-correctness/) ## 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 Init=[0] After zero-step reachable=[0] Conditions: none Every initial state is reachable in zero steps,including an initial halt.An empty Init has no reachable states. Step identifiers and dependenciesRule: zero_step_reachability · Uses: G001, S001 S003 · impliesBefore depth=0;state=0;halt=No;raw successors=[1] After expanded successors=[1];newly reached=[1] Conditions: none Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist. Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S002 S004 · impliesBefore completed frontier=[0];depth=0 After next frontier=[1];all reached=[0, 1] Conditions: none The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff. Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S003 S005 · impliesBefore depth=1;state=1;halt=No;raw successors=[2] After expanded successors=[2];newly reached=[2] Conditions: none Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist. Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S004 S006 · impliesBefore completed frontier=[1];depth=1 After next frontier=[2];all reached=[0, 1, 2] Conditions: none The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff. Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S005 S007 · impliesBefore depth=2;state=2;halt=No;raw successors=[1, 5] After expanded successors=[1, 5];newly reached=[5] Conditions: none Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist. Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S006 S008 · impliesBefore completed frontier=[2];depth=2 After next frontier=[5];all reached=[0, 1, 2, 5] Conditions: none The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff. Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S007 S009 · impliesBefore depth=3;state=5;halt=Yes;raw successors=[] After expanded successors=[];newly reached=[] Conditions: none Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist. Step identifiers and dependenciesRule: reachability_halt_truncates_expansion · Uses: G001, S008 S010 · impliesBefore completed frontier=[5];depth=3 After next frontier=[];all reached=[0, 1, 2, 5] Conditions: none The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff. Step identifiers and dependenciesRule: complete_finite_reachability_layer · Uses: G001, S009 S011 · impliesBefore candidate invariant=[0, 1, 2, 5];Init=[0] After contains every initial state=Yes Conditions: none An inductive invariant must first contain every initial state,not just one selected initial point. Step identifiers and dependenciesRule: invariant_initial_obligation · Uses: G001, S010 S012 · impliesBefore raw edge=[0, 1];source in invariant=Yes;source halt=No After preservation obligation active=Yes;violates invariant=No Conditions: none Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation. Step identifiers and dependenciesRule: invariant_all_states_not_only_reachable · Uses: G001, S011 S013 · impliesBefore raw edge=[1, 2];source in invariant=Yes;source halt=No After preservation obligation active=Yes;violates invariant=No Conditions: none Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation. Step identifiers and dependenciesRule: invariant_all_states_not_only_reachable · Uses: G001, S012 S014 · impliesBefore raw edge=[2, 1];source in invariant=Yes;source halt=No After preservation obligation active=Yes;violates invariant=No Conditions: none Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation. Step identifiers and dependenciesRule: invariant_all_states_not_only_reachable · Uses: G001, S013 S015 · impliesBefore raw edge=[2, 5];source in invariant=Yes;source halt=No After preservation obligation active=Yes;violates invariant=No Conditions: none Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation. Step identifiers and dependenciesRule: invariant_all_states_not_only_reachable · Uses: G001, S014 S016 · impliesBefore raw edge=[3, 4];source in invariant=No;source halt=No After preservation obligation active=No;violates invariant=No Conditions: none Check every raw edge from an invariant nonhalt,including unreachable sources.Covering the actual reachable states is a different,weaker observation. Step identifiers and dependenciesRule: invariant_all_states_not_only_reachable · Uses: G001, S015 S017 · impliesBefore bad preservation edges=[];initial obligation=Yes;reachable=[0, 1, 2, 5] After step preserved=Yes;inductive=Yes;covers reachable=Yes Conditions: none Initial validity plus one-step preservation gives inductiveness.Compute reachable-state coverage separately;true coverage does not replace preservation. Step identifiers and dependenciesRule: inductiveness_and_reachable_coverage_separate · Uses: G001, S016 ## Reverse checks - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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], Yes, [], Yes, Yes, Yes) 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: 5f67f49f360367ab13a5b03bd6232c96f04e4723621108246d0c9969b81f3830 - candidate record SHA-256: 79ea1ff302cd4c20ee5a633680d0a4f9358ae7ef5b1f03f4cbbac5991ce366df - 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-r05-algorithm-correctness-0011.json)