# Question 29888: A Linear Constraint on a Complete Ordered Finite Point List [Return to catalogue page 150](https://ocme.evemisslab.com/en/questions/page/150/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261020-r04-optimization-problem-foundations/#q-20261013-advance-20261020-r04-optimization-problem-0088) 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. R0058-29888 · q-20261013-advance-20261020-r04-optimization-problem-0088 ## Original question and givens Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 3], [3, 0]], constraint [A,B,c]=[1, 1, 9] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points. - G001 — Use the complete explicitly declared finite candidate list and Boolean feasible mask, with at least one feasible candidate. Preserve original zero-based indices and every tie; no unlisted or continuous-domain optimum or general complexity claim is inferred. Original points=[[0, 0], [1, 2], [2, 3], [3, 0]], constraint [A,B,c]=[1, 1, 9] means A*x+B*y≤c, objective [u,v]=[0, 2] means u*x+v*y, direction=max. Give (constraint, objective, direction, every [original index,x,y,feasible?,objective value] row, best feasible value, all optimal indices). Preserve input point order and repeated points. ## 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-optimization-problem](https://ocme.evemisslab.com/en/mko/mko-optimization-problem/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-app20-optimization-linear-constraint-audit, optimization-linear-constraint-audit] After derived = [[constraint, objective, direction, ALLrows, best, ALLarg_indices]] Conditions: none Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention. Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001 S002 · calculationBefore Optimization ORIGINAL row0 constraint A times x := mul(1, 0) After Optimization ORIGINAL row0 constraint A times x = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Optimization ORIGINAL row0 constraint B times y := mul(1, 0) After Optimization ORIGINAL row0 constraint B times y = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Optimization ORIGINAL row0 constraint sum := sum(0, 0) After Optimization ORIGINAL row0 constraint sum = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Optimization ORIGINAL row0 objective u times x := mul(0, 0) After Optimization ORIGINAL row0 objective u times x = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Optimization ORIGINAL row0 objective v times y := mul(2, 0) After Optimization ORIGINAL row0 objective v times y = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Optimization ORIGINAL row0 objective sum := sum(0, 0) After Optimization ORIGINAL row0 objective sum = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [0, [0, 0], [1, 1, 9], [0, 2]] After derived = [0, 9, [0, 0, 0, Yes, 0]] Conditions: none At the SAME original row calculate A*x+B*y<=c INCLUSIVELY and u*x+v*y.Preserve original index and repeated points.Semantic-identity sorting/deduplication cannot change this raw question or its answer. Step identifiers and dependenciesRule: Optimization_EACH_original_point_INCLUSIVE_constraint_AND_objective · Uses: G001, S007 S009 · calculationBefore Optimization ORIGINAL row1 constraint A times x := mul(1, 1) After Optimization ORIGINAL row1 constraint A times x = 1 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Optimization ORIGINAL row1 constraint B times y := mul(1, 2) After Optimization ORIGINAL row1 constraint B times y = 2 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Optimization ORIGINAL row1 constraint sum := sum(1, 2) After Optimization ORIGINAL row1 constraint sum = 3 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Optimization ORIGINAL row1 objective u times x := mul(0, 1) After Optimization ORIGINAL row1 objective u times x = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Optimization ORIGINAL row1 objective v times y := mul(2, 2) After Optimization ORIGINAL row1 objective v times y = 4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Optimization ORIGINAL row1 objective sum := sum(0, 4) After Optimization ORIGINAL row1 objective sum = 4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [1, [1, 2], [1, 1, 9], [0, 2]] After derived = [3, 9, [1, 1, 2, Yes, 4]] Conditions: none At the SAME original row calculate A*x+B*y<=c INCLUSIVELY and u*x+v*y.Preserve original index and repeated points.Semantic-identity sorting/deduplication cannot change this raw question or its answer. Step identifiers and dependenciesRule: Optimization_EACH_original_point_INCLUSIVE_constraint_AND_objective · Uses: G001, S014 S016 · calculationBefore Optimization ORIGINAL row2 constraint A times x := mul(1, 2) After Optimization ORIGINAL row2 constraint A times x = 2 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Optimization ORIGINAL row2 constraint B times y := mul(1, 3) After Optimization ORIGINAL row2 constraint B times y = 3 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Optimization ORIGINAL row2 constraint sum := sum(2, 3) After Optimization ORIGINAL row2 constraint sum = 5 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Optimization ORIGINAL row2 objective u times x := mul(0, 2) After Optimization ORIGINAL row2 objective u times x = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Optimization ORIGINAL row2 objective v times y := mul(2, 3) After Optimization ORIGINAL row2 objective v times y = 6 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Optimization ORIGINAL row2 objective sum := sum(0, 6) After Optimization ORIGINAL row2 objective sum = 6 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore source roles = [2, [2, 3], [1, 1, 9], [0, 2]] After derived = [5, 9, [2, 2, 3, Yes, 6]] Conditions: none At the SAME original row calculate A*x+B*y<=c INCLUSIVELY and u*x+v*y.Preserve original index and repeated points.Semantic-identity sorting/deduplication cannot change this raw question or its answer. Step identifiers and dependenciesRule: Optimization_EACH_original_point_INCLUSIVE_constraint_AND_objective · Uses: G001, S021 S023 · calculationBefore Optimization ORIGINAL row3 constraint A times x := mul(1, 3) After Optimization ORIGINAL row3 constraint A times x = 3 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Optimization ORIGINAL row3 constraint B times y := mul(1, 0) After Optimization ORIGINAL row3 constraint B times y = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Optimization ORIGINAL row3 constraint sum := sum(3, 0) After Optimization ORIGINAL row3 constraint sum = 3 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Optimization ORIGINAL row3 objective u times x := mul(0, 3) After Optimization ORIGINAL row3 objective u times x = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Optimization ORIGINAL row3 objective v times y := mul(2, 0) After Optimization ORIGINAL row3 objective v times y = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Optimization ORIGINAL row3 objective sum := sum(0, 0) After Optimization ORIGINAL row3 objective sum = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore source roles = [3, [3, 0], [1, 1, 9], [0, 2]] After derived = [3, 9, [3, 3, 0, Yes, 0]] Conditions: none At the SAME original row calculate A*x+B*y<=c INCLUSIVELY and u*x+v*y.Preserve original index and repeated points.Semantic-identity sorting/deduplication cannot change this raw question or its answer. Step identifiers and dependenciesRule: Optimization_EACH_original_point_INCLUSIVE_constraint_AND_objective · Uses: G001, S028 S030 · impliesBefore source roles = [linear finite candidate list, [0, 4, 6, 0], [0, 1, 2, 3], max] After derived = [6, [2], [[0, 0, Yes, No], [1, 4, Yes, No], [2, 6, Yes, Yes], [3, 0, Yes, No]]] Conditions: none Check the corresponding bound for EVERY feasible ORIGINAL index and retain ALL equality indices.The optimal value differs from its arg set.Ignore better infeasible values and do not assume uniqueness.The conclusion is ONLY for this complete declared finite set. Step identifiers and dependenciesRule: Optimization_ALL_feasible_objectives_bound_AND_every_tied_index · Uses: G001, S029 S031 · impliesBefore source roles = [[[0, 0], [1, 2], [2, 3], [3, 0]], [1, 1, 9], [0, 2], max] After derived = [[[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 3, Yes, 6], [3, 3, 0, Yes, 0]], 6, [2]] Conditions: none All rows are evaluated,but optimize ONLY eligible ORIGINAL listed candidates.The whole half-plane was not declared as the candidate set;the finite best is not a continuous-domain global solution. Step identifiers and dependenciesRule: Optimization_FINITE_list_not_continuous_halfplane · Uses: G001, S030 ## Reverse checks - S031 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S030 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S029 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S028 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective v times y with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 objective u times x with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint sum with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint B times y with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Optimization ORIGINAL row3 constraint A times x with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S021 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective sum with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective v times y with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 objective u times x with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint B times y with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Optimization ORIGINAL row2 constraint A times x with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S014 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective sum with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective v times y with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 objective u times x with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint sum with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint B times y with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Optimization ORIGINAL row1 constraint A times x with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S007 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective v times y with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 objective u times x with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint sum with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint B times y with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Optimization ORIGINAL row0 constraint A times x with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. ## Result ([1, 1, 9], [0, 2], max, [[0, 0, 0, Yes, 0], [1, 1, 2, Yes, 4], [2, 2, 3, Yes, 6], [3, 3, 0, Yes, 0]], 6, [2]) 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: 8728675278b26f62f78f17276bcd70665e1a0e886c05057a6b9e103ca412fff4 - candidate record SHA-256: 5cbd413a577c2b8ade2de872421c74dd2306b230091953e5732a5b0af7c28501 - semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df - question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r04-optimization-problem-0088.json)