Question 2418: Every Constraint at One Declared Point

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=[2, 1], 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=[2, 1], 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

Step-by-step derivation

explicit finite candidate grid and linear rows a·x+b·y≤c

keep candidate-grid optimization separate from continuous LP

point row0 a·x := mul(-1, 2)

point row0 a·x = -2

point row0 b·y := mul(0, 1)

point row0 b·y = 0

point row0 LHS := sum(-2, 0)

point row0 LHS = -2

row0: -2≤0

satisfied=Yes

point row1 a·x := mul(0, 2)

point row1 a·x = 0

point row1 b·y := mul(-1, 1)

point row1 b·y = -1

point row1 LHS := sum(0, -1)

point row1 LHS = -1

row1: -1≤0

satisfied=Yes

point row2 a·x := mul(1, 2)

point row2 a·x = 2

point row2 b·y := mul(1, 1)

point row2 b·y = 1

point row2 LHS := sum(2, 1)

point row2 LHS = 3

row2: 3≤6

satisfied=Yes

[Yes, Yes, Yes]

feasible for all supplied rows=Yes

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

([2, 1], [[-1, 0, 0, -2, Yes], [0, -1, 0, -1, Yes], [1, 1, 6, 3, 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.