# Question 2404: Every Constraint at One Declared Point [Return to catalogue page 13](https://ocme.evemisslab.com/en/questions/page/13/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r06-linear-programming-foundations/#q-20261013-advance-20261021-r06-linear-programming-0004) 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. R0003-02404 · q-20261013-advance-20261021-r06-linear-programming-0004 ## Original question and givens Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?). - G001 — Use the complete explicitly supplied finite candidate grid, integer constraints A*x+B*y≤c and linear objective u*x+v*y. Feasibility precedes objective comparison; an optimum among these candidates is not a general continuous-domain LP optimum or dual certificate. point=[3, 3], constraints [A,B,c]=[[-1, 0, 0], [0, -1, 0], [1, 1, 6]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (point,every [A,B,c,A*x+B*y,satisfies?] row,all constraints hold?). ## 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-linear-programming](https://ocme.evemisslab.com/en/mko/mko-linear-programming/) ## Step-by-step derivation S001 · impliesBefore explicit finite candidate grid and linear rows a·x+b·y≤c After keep candidate-grid optimization separate from continuous LP Conditions: none Audit only the listed finite candidates;without a complete-vertex or full-feasible-domain certificate,this is not a general continuous-LP optimum claim. Step identifiers and dependenciesRule: finite_LP_scope · Uses: G001 S002 · calculationBefore point row0 a·x := mul(-1, 3) After point row0 a·x = -3 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore point row0 b·y := mul(0, 3) After point row0 b·y = 0 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore point row0 LHS := sum(-3, 0) After point row0 LHS = -3 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore row0: -3≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S004 S006 · calculationBefore point row1 a·x := mul(0, 3) After point row1 a·x = 0 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore point row1 b·y := mul(-1, 3) After point row1 b·y = -3 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore point row1 LHS := sum(0, -3) After point row1 LHS = -3 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore row1: -3≤0 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S008 S010 · calculationBefore point row2 a·x := mul(1, 3) After point row2 a·x = 3 Conditions: none Multiply the first coordinate by its row coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore point row2 b·y := mul(1, 3) After point row2 b·y = 3 Conditions: none Compute the second coordinate in the same declared order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore point row2 LHS := sum(3, 3) After point row2 LHS = 6 Conditions: none Add before comparing the complete row inequality. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore row2: 6≤6 After satisfied=Yes Conditions: none Equality is allowed and negative coefficients remain in the products;do not arbitrarily reverse the inequality. Step identifiers and dependenciesRule: inclusive_constraint · Uses: G001, S012 S014 · impliesBefore [Yes, Yes, Yes] After feasible for all supplied rows=Yes Conditions: none A point must pass every supplied constraint,not just one row. Step identifiers and dependenciesRule: constraint_conjunction · Uses: G001, S013 ## Reverse checks - 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 — Recompute point row2 LHS with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute point row2 b·y with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute point row2 a·x with exact rational arithmetic, not rounded decimals. - 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 — Recompute point row1 LHS with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute point row1 b·y with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute point row1 a·x 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 point row0 LHS with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute point row0 b·y with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute point row0 a·x with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([3, 3], [[-1, 0, 0, -3, Yes], [0, -1, 0, -3, Yes], [1, 1, 6, 6, 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: fcf30cda2bb89423f8c30fc05a8b23192758e8b78165d667e0eab803b1b8a509 - candidate record SHA-256: 6cc10db857d0f041419cbab9c46c12688e0ece4fdd4ea5dedfd9f2cc9dc25e62 - semantic review receipt SHA-256: 5dbc70ebd5f74b23384c93b63a2b1d6519e206c79f22ae16a6c2d7483cea783f - question-first witness SHA-256: cf121c8dd2fec13aa118b04b049ee39494f1d6d9ecdc13514cafb78c0fc021fa - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r06-linear-programming-0004.json)