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R0011-06251 · q-20261009-advance-20261013-r04-proof-0001
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.
G001 — 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.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.
Before
n=0;given facts=[];original indexed rules=[]
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.
Rule: given_unary_MP_scope · Uses: G001
Before
phase=base;given facts=[]
After
phase=base;active_rules=[]
Conditions: none
Start from all given facts,retaining original rule-row indices.Atoms with no derived premises remain not-derived.
Rule: horn_closure_start · Uses: G001, S001
Before
phase=base;edge closure/depth audits=[]
After
phase=base;depths=[]
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.
Rule: horn_closed_depth_certificate · Uses: G001, S002
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.[]
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f9ecb4cf01d5bdf1f728f86565faaf66f991cc82845594d4026ed977a48d5363candidate record SHA-256: 0201dd0dc6e5113fd10edb551e94d222b8abac95ee76fe388a41e0905841f7a5semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED