Task: Arithmetic-Relationship (source dataset: Auto-Relate)

You are given a table with numerical values in multiple columns. Your task is to analyze the data and identify semantic relationships, expressed in arithmetic relationships, that exist between columns.

These relationships may involve any combination of the following operations: addition (+), subtraction (−), multiplication (*), and division (/). The expressions can involve two or more columns and may be nested (e.g., B = (A + C) / D).

Please only identify arithmetic relationships that hold semantically based on the data, and do not return relationships that are likely spurious, e.g., (A = B * 0), where A is an all 0 column. Please only return minimal atomic relationships, e.g., if A = B + C and D = E + F, then there is no need to return A = B + C + E + F - D.

No explanation, return all the formula(s) in JSON format: {"Arithmetic-Relationship": [<list-of-formulas>]}. If no relationship is found, return {"Arithmetic-Relationship": []}.

Input Table:
| A          |         B |        C |        D |   E |          F |   G |   H |          I |        J |    K |
|:-----------|----------:|---------:|---------:|----:|-----------:|----:|----:|-----------:|---------:|-----:|
| BOONE      |  1823.35  | 12217.3  |  7077.22 |   0 |  1859.95   |   0 |   0 |   68.8765  |  3211.24 |  583 |
| BRAXTON    |  1566.5   | 11620    |  5043.61 |   0 |   258.96   |   0 |   0 |   94.657   |  6222.81 |  686 |
| BROOKE     |  2557.65  |  6984.39 |  3249.41 |   0 |   173.208  |   0 |   0 |  140.768   |  3421    |  505 |
| CALHOUN    |   695.253 |  4377.14 |  2509.87 |   0 |   161.505  |   0 |   0 |    4.95264 |  1700.81 |  247 |
| CLAY       |   730.246 |  5647.19 |  3153.34 |   0 |    91.3758 |   0 |   0 |   48.1929  |  2354.28 |  314 |
| DODDRIDGE  |   608.975 |  5990.67 |  2011.08 |   0 |   137.734  |   0 |   0 |    6.05489 |  3835.8  |  401 |
| FAYETTE    |  4732.47  | 40343    | 18698.5  |   0 |   707.196  |   0 |   0 |  182.239   | 20755    | 1830 |
| GILMER     |   486.66  |  3564.01 |  1612    |   0 |   126.743  |   0 |   0 |    5.11043 |  1820.15 |  270 |
| GRANT      |  1168.73  |  6530.66 |  2878.59 |   0 |  1662.78   |   0 |   0 |   22.7568  |  1966.54 |  378 |
| GREENBRIER |  3216.31  | 20939.2  | 10915.9  |   0 |   439.688  |   0 |   0 |  188.955   |  9394.7  | 1175 |
| HAMPSHIRE  |  2370.16  | 11841.2  |  5938.36 |   0 |   205.553  |   0 |   0 |   28.6034  |  5668.63 |  630 |
| HANCOCK    |  3115.29  |  9220.33 |  4319.45 |   0 |   296.455  |   0 |   0 |  140.542   |  4463.89 |  766 |
| HARDY      |  1333     |  5678.12 |  3611.51 |   0 |    23.012  |   0 |   0 |   98.3237  |  1945.27 |  427 |
| JACKSON    |  2352.62  |  9660.25 |  5398.02 |   0 |   323.657  |   0 |   0 |   81.5596  |  3857.02 |  669 |
| JEFFERSON  |  5072.88  | 29367.1  | 11056.8  |   0 |  1614.77   |   0 |   0 |  187.584   | 16508    | 1398 |
| KANAWHA    | 17223.9   | 83394.8  | 40784.6  |   0 |  9381.4    |   0 |   0 | 1193.64    | 32035.2  | 5288 |
| LEWIS      |  1789.31  | 12879    |  5886.08 |   0 |   197.064  |   0 |   0 |   69.9637  |  6725.86 |  741 |
| LINCOLN    |  1419.48  | 11960.2  |  6834.22 |   0 |   200.497  |   0 |   0 |   46.4692  |  4879    |  599 |
| LOGAN      |  2446.7   | 41880.4  | 13438.1  |   0 | 20734.2    |   0 |   0 |   82.7743  |  7625.33 |  984 |
| MC DOWELL  |  1491.59  | 12212.1  |  7235    |   0 |   142.913  |   0 |   0 |   39.4837  |  4794.67 |  554 |
| MARION     |  5922.48  | 37062.5  | 16470.6  |   0 |  2007.51   |   0 |   0 |  562.145   | 18022.3  | 2058 |
| MARSHALL   |  3279.35  |  8922.42 |  4952.17 |   0 |   351.533  |   0 |   0 |  171.644   |  3447.07 |  789 |
| MASON      |  2394.08  | 12102.1  |  7244.97 |   0 |   363.344  |   0 |   0 |  117.92    |  4375.9  |  571 |
| MERCER     |  6015.09  | 43534.8  | 24419.1  |   0 |  1013.17   |   0 |   0 |  235.275   | 17867.3  | 2017 |
| MINERAL    |  3145.18  | 14647.7  |  7818.09 |   0 |   282.51   |   0 |   0 |  322.127   |  6225.02 | 1067 |
| MINGO      |  1617.41  | 14418    |  8867.78 |   0 |   121.622  |   0 |   0 |  105.338   |  5323.26 |  718 |
| MONONGALIA |  6326.93  | 28111.9  | 14302.3  |   0 |  3994.17   |   0 |   0 |  422.76    |  9392.62 | 1475 |
| MONROE     |  1363.35  |  8484.33 |  4450.7  |   0 |   160.841  |   0 |   0 |   30.8044  |  3841.99 |  459 |
| MORGAN     |  1464.61  | 11869.4  |  5648.89 |   0 |   337.065  |   0 |   0 |  160.958   |  5722.45 |  596 |
| NICHOLAS   |  2029.25  | 17473    |  9583.54 |   0 |   314.806  |   0 |   0 |   65.3946  |  7509.27 |  948 |
| OHIO       |  4261.13  | 14137    |  7022.81 |   0 |   654.804  |   0 |   0 |  444.452   |  6014.9  | 1164 |
| PENDLETON  |   834.165 |  4181.96 |  1842.38 |   0 |   179.137  |   0 |   0 |   80.2228  |  2080.23 |  299 |
| PLEASANTS  |   683.395 |  3582.01 |  1383.75 |   0 |   103.383  |   0 |   0 |   35.6156  |  2059.26 |  131 |
| POCAHONTAS |  1028.48  |  4457.36 |  2366.48 |   0 |    87.2779 |   0 |   0 |    6.37581 |  1997.23 |  280 |
| PRESTON    |  2857.05  | 17258    |  9321.81 |   0 |  1331.86   |   0 |   0 |  126.973   |  6477.4  |  861 |
| PUTNAM     |  5101.61  | 22189.7  | 11089    |   0 |  1066.69   |   0 |   0 |  159.772   |  9874.21 | 1425 |
| RANDOLPH   |  2726.85  | 19583.1  |  8301.78 |   0 |  1846.76   |   0 |   0 |  387.458   |  9047.1  | 1090 |
| RITCHIE    |  1040.09  |  7116.1  |  3308.33 |   0 |   511.93   |   0 |   0 |   67.8078  |  3228.04 |  404 |
| ROANE      |  1428.91  |  8044.96 |  4606.23 |   0 |   190.461  |   0 |   0 |   54.1255  |  3194.15 |  490 |
| SUMMERS    |  1386.24  | 10981.7  |  4889.66 |   0 |   129.856  |   0 |   0 |   51.1459  |  5911.03 |  583 |
| TUCKER     |   612.908 |  4127.24 |  1988.03 |   0 |    45.4713 |   0 |   0 |   42.2187  |  2051.52 |  271 |
| TYLER      |   853.618 |  4830.3  |  2195.88 |   0 |    89.4505 |   0 |   0 |   19.4335  |  2525.54 |  270 |
| UPSHUR     |  2238.8   | 17335.4  |  6804.14 |   0 |  1523.52   |   0 |   0 |   75.1284  |  8932.59 | 1009 |
| WEBSTER    |   903.002 |  6468.14 |  3325.05 |   0 |   336.071  |   0 |   0 |    8.96678 |  2798.05 |  353 |
| WETZEL     |  1378.34  |  7502.39 |  3428.53 |   0 |   152.639  |   0 |   0 |  143.795   |  3777.43 |  492 |
| WIRT       |   515.934 |  2907.28 |  1237.9  |   0 |   122.001  |   0 |   0 |   17.8506  |  1529.53 |  184 |
| WOOD       |  8557.22  | 35119.1  | 19003.5  |   0 |  3559.51   |   0 |   0 |  353.822   | 12202.3  | 2375 |
| WYOMING    |  2052.56  | 15364.5  |  7080.38 |   0 |   138.38   |   0 |   0 |   44.3181  |  8101.43 |  688 |



labeled_formulas: G=B*E
H=B*E
H=B*G
G=C*E
H=C*E
H=C*G
G=D*E
H=D*E
H=D*G
G=E*F
H=E*F
G=E*H
G=E*I
G=E*J
G=E*K
H=E*I
H=E*J
H=E*K
H=F*G
H=G*I
H=G*J
H=G*K
G=(B+C)*E
H=(B+C)*E
H=(B+C)*G
G=(B+D)*E
H=(B+D)*E
H=(B+D)*G
G=(B*E)*F
H=(B*E)*F
G=(B*E)*H
G=(B*E)*I
G=(B*E)*J
G=(B*E)*K
H=(B*E)*I
H=(B*E)*J
H=(B*E)*K
H=(B+F)*G
H=(B*G)*I
H=(B*G)*J
H=(B*G)*K
G=(C+D)*E
H=(C+D)*E
H=(C+D)*G
G=(C*E)*F
H=(C*E)*F
G=(C*E)*H
G=(C*E)*I
G=(C*E)*J
G=(C*E)*K
H=(C*E)*I
H=(C*E)*J
H=(C*E)*K
H=(C+F)*G
H=(C*G)*I
H=(C*G)*J
H=(C*G)*K
G=(D*E)*F
H=(D*E)*F
G=(D*E)*H
G=(D*E)*I
G=(D*E)*J
G=(D*E)*K
H=(D*E)*I
H=(D*E)*J
H=(D*E)*K
H=(D+F)*G
H=(D*G)*I
H=(D*G)*J
H=(D*G)*K
G=(E*F)*H
G=(E*F)*I
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119149___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_213