# Question 6227: Typed MP Goals and Minimum Depths [Return to catalogue page 32](https://ocme.evemisslab.com/en/questions/page/32/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261009-advance-20261013-r04-proof-foundations/#q-20261009-advance-20261013-r04-proof-0027) 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. R0011-06227 · q-20261009-advance-20261013-r04-proof-0027 ## Original question and givens 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). - G001 — 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). ## 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-proof](https://ocme.evemisslab.com/en/mko/mko-proof/) ## Step-by-step derivation S001 · impliesBefore n=4;given facts=[0];original indexed rules=[[0, 0, 1], [1, 1, 2]] After use only given facts and single-premise MP;not-derived does not mean False Conditions: none Atoms use0-based indices and a rule i→j requires an already derived i.A cycle without facts cannot generate conclusions.Not-derived is not False,and this finite check is not general logical completeness. Step identifiers and dependenciesRule: given_unary_MP_scope · Uses: G001 S002 · impliesBefore phase=base;given facts=[0] After phase=base;active_rules=[0,1] Conditions: none Start from all given facts,retaining original rule-row indices.Atoms with no derived premises remain not-derived. Step identifiers and dependenciesRule: horn_closure_start · Uses: G001, S001 S003 · impliesBefore phase=base;given fact=0 After phase=base;derive atom=0;depth=0;basis=given Conditions: none A supplied fact has depth0 and needs no MP antecedent;other atoms are not silently made facts. Step identifiers and dependenciesRule: horn_given_fact · Uses: G001, S002 S004 · calculationBefore base MP depth for atom1 := sum(0, 1) After base MP depth for atom1 = 1 Conditions: none One MP application adds1 to the already known antecedent depth. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore phase=base;rule[0]=0→1;antecedent depth=0 After phase=base;derive atom=1;depth=1;from=0;rule=0 Conditions: none An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order. Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S004 S006 · calculationBefore base MP depth for atom2 := sum(1, 1) After base MP depth for atom2 = 2 Conditions: none One MP application adds1 to the already known antecedent depth. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore phase=base;rule[1]=1→2;antecedent depth=1 After phase=base;derive atom=2;depth=2;from=1;rule=1 Conditions: none An already derived antecedent and this given rule yield the target for the first time.Expand the queue in nondecreasing depth order. Step identifiers and dependenciesRule: horn_MP_derivation · Uses: G001, S006 S008 · impliesBefore phase=base;edge closure/depth audits=[[0, 0, 1, 0, 1, Yes], [1, 1, 2, 1, 2, Yes]] After phase=base;depths=[0,1,2,null] Conditions: none Every derived atom has a displayed chain of that length.The reached set is closed under every given edge and satisfies the depth bounds,so any shorter chain would force a smaller label.These labels are shortest;null means not-derived in this rule model,not False. Step identifiers and dependenciesRule: horn_closed_depth_certificate · Uses: G001, S007 S009 · impliesBefore query atom=0;inside Fin(4)=Yes After derived=Yes;depth/status=0 Conditions: none Check typing before derivability;do not access the array outside its type,and do not classify a not-derived atom as a false proposition. Step identifiers and dependenciesRule: horn_goal_type_and_derivability · Uses: G001, S008 S010 · impliesBefore query atom=101;inside Fin(4)=No After derived=No;depth/status=type mismatch Conditions: none Check typing before derivability;do not access the array outside its type,and do not classify a not-derived atom as a false proposition. Step identifiers and dependenciesRule: horn_goal_type_and_derivability · Uses: G001, S009 S011 · impliesBefore query atom=102;inside Fin(4)=No After derived=No;depth/status=type mismatch Conditions: none Check typing before derivability;do not access the array outside its type,and do not classify a not-derived atom as a false proposition. Step identifiers and dependenciesRule: horn_goal_type_and_derivability · 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 — Recompute base MP depth for atom2 with exact rational arithmetic, not rounded decimals. - 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 — Recompute base MP depth for atom1 with exact rational arithmetic, not rounded decimals. - 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 ([Yes, No, No], [Yes, No, No], [0, ill-typed, ill-typed]) 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: 8b03b33303456231a1aae057d9822476df804c23fdffdf8c1b2b91f6c1b3b477 - candidate record SHA-256: 6341eb51741f5dbfa8b08fa77809cdc9a551954d9b03a2e6e6001ccd344051aa - semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80 - question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r04-proof-0027.json)