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 |
|:-------------|----------:|-----------:|-----------:|----:|------------:|----:|----:|------------:|----------:|-----:|
| ADAMS        | 2528.65   | 15925.8    |  9581.99   |   0 |  516.025    |   0 |   0 | 217.002     | 5610.8    |  947 |
| ANTELOPE     |  715.103  |  3834.61   |  2334.83   |   0 |  130.47     |   0 |   0 |  12.6227    | 1356.69   |  243 |
| ARTHUR       |   48.3255 |    89.0646 |    29.7347 |   0 |    6.15385  |   0 |   0 |  34.9611    |   18.215  |    9 |
| BANNER       |   63.3682 |   349.037  |   292.704  |   0 |    4.25126  |   0 |   0 |   6.67559   |   45.4059 |   13 |
| BLAINE       |   44.4872 |   215.589  |   116.052  |   0 |    0.618551 |   0 |   0 |   0.974727  |   97.9444 |   20 |
| BOONE        |  588.971  |  3458.38   |  1971.38   |   0 |   74.0129   |   0 |   0 |  17.3106    | 1395.68   |  253 |
| BOX BUTTE    |  980.969  |  4395.27   |  1639.86   |   0 |  125.138    |   0 |   0 |  30.8821    | 2599.39   |  418 |
| BOYD         |  240.895  |  1391.74   |   543.748  |   0 |   21.7102   |   0 |   0 |   2.67071   |  823.61   |   98 |
| BROWN        |  298.573  |  3082.48   |  1925.76   |   0 |   32.2438   |   0 |   0 |   9.03638   | 1115.45   |  163 |
| BUFFALO      | 3086.77   | 17510.2    | 10559      |   0 |  996.448    |   0 |   0 | 342.597     | 5612.21   | 1089 |
| BURT         |  736.055  |  2789.81   |  1234.18   |   0 |  108.127    |   0 |   0 |  78.873     | 1368.64   |  174 |
| BUTLER       |  810.911  |  3559.92   |  2149.5    |   0 |  114.723    |   0 |   0 |   8.73841   | 1286.96   |  237 |
| CASS         | 2906.48   | 16081.7    | 10183.8    |   0 | 1051.34     |   0 |   0 | 438.939     | 4407.56   |  691 |
| CEDAR        |  775.899  |  3146.17   |  1529.95   |   0 |   32.6079   |   0 |   0 | 101.903     | 1481.71   |  246 |
| CHASE        |  357.593  |  1076.11   |   697.05   |   0 |    9.02944  |   0 |   0 |   9.15119   |  360.879  |  125 |
| CHERRY       |  394.513  |  2527.11   |  1319.59   |   0 |   57.8223   |   0 |   0 |  15.5817    | 1134.12   |  223 |
| CHEYENNE     |  659.845  |  4304.11   |  1738.75   |   0 |  238.74     |   0 |   0 |  72.7217    | 2253.9    |  289 |
| CLAY         |  638.713  |  2415.82   |  1590.95   |   0 |   60.6298   |   0 |   0 |  35.669     |  728.574  |  166 |
| COLFAX       |  824.098  |  1775.2    |   770.281  |   0 |   30.4358   |   0 |   0 |  33.5974    |  940.887  |  160 |
| CUMING       |  784.031  |  2266.32   |  1263.14   |   0 |   74.5246   |   0 |   0 |  31.6356    |  897.017  |  195 |
| CUSTER       | 1005.84   |  6348.11   |  4410.79   |   0 |   59.6304   |   0 |   0 |  47.5246    | 1830.16   |  405 |
| DAKOTA       | 1176.47   |  4501.58   |  2491.77   |   0 |  150.475    |   0 |   0 |  91.7513    | 1767.58   |  379 |
| DAWES        |  804.958  |  7048.76   |  2569      |   0 |  349.856    |   0 |   0 |  56.8818    | 4073.01   |  486 |
| DAWSON       | 1845.17   |  6914.49   |  3364.86   |   0 |  111.628    |   0 |   0 |  44.4809    | 3393.52   |  576 |
| DEUEL        |  242.389  |   841.422  |   418.302  |   0 |    2.29847  |   0 |   0 |  32.9108    |  387.911  |   65 |
| DIXON        |  530.282  |  1341.56   |   750.151  |   0 |   89.8741   |   0 |   0 |  18.6537    |  482.886  |  115 |
| DODGE        | 3412.8    | 14285.4    |  7726.26   |   0 |  427.739    |   0 |   0 | 213.765     | 5917.63   |  840 |
| DUNDY        |  236.787  |   814.745  |   557.925  |   0 |    3.80736  |   0 |   0 |   2.58231   |  250.43   |   62 |
| FILLMORE     |  561.787  |  4007.98   |  2414.69   |   0 |   81.9556   |   0 |   0 | 102.371     | 1408.96   |  298 |
| FRANKLIN     |  377.767  |  2467.26   |  1401.28   |   0 |   25.3778   |   0 |   0 |   6.08122   | 1034.53   |  187 |
| FRONTIER     |  287.87   |  1149.03   |   774.996  |   0 |   39.0853   |   0 |   0 |   8.19321   |  326.754  |   64 |
| FURNAS       |  518.431  |  2454.67   |  1587.63   |   0 |    8.93058  |   0 |   0 |  25.8822    |  832.232  |  186 |
| GAGE         | 1662.04   | 13728.5    |  8062.03   |   0 |  228.481    |   0 |   0 | 119.072     | 5318.9    |  835 |
| GARDEN       |  162.603  |  1093.35   |   568.564  |   0 |   39.5762   |   0 |   0 |  16.1284    |  469.08   |   81 |
| GARFIELD     |  183.292  |  1100.73   |   337.388  |   0 |    1.24211  |   0 |   0 |   2.19289   |  759.902  |  101 |
| GOSPER       |  199.906  |  1381.33   |   846.108  |   0 |    3.00427  |   0 |   0 |   8.11421   |  524.107  |  103 |
| GRANT        |   91.7191 |   193.93   |   118.685  |   0 |    0.286042 |   0 |   0 |   0         |   74.9586 |   21 |
| GREELEY      |  224.097  |  1562.65   |   866.704  |   0 |   21.7035   |   0 |   0 |  66.4922    |  607.754  |  125 |
| HAMILTON     |  870.224  |  4684.93   |  2625.48   |   0 |   77.2088   |   0 |   0 |  73.2903    | 1908.94   |  318 |
| HARLAN       |  351.334  |  2526.67   |  1698.83   |   0 |   48.0362   |   0 |   0 |   7.72335   |  772.073  |  184 |
| HAYES        |  113.513  |   637.937  |   510.08   |   0 |    2.14136  |   0 |   0 |   6.97473   |  118.741  |   17 |
| HITCHCOCK    |  246.873  |  1615.22   |   865.602  |   0 |   27.5774   |   0 |   0 |  13.6934    |  708.347  |  112 |
| HOLT         |  871.722  |  4583.57   |  2923.54   |   0 |  111.19     |   0 |   0 |  60.2901    | 1488.54   |  368 |
| HOOKER       |   80.797  |   283.233  |   120.496  |   0 |    2.29545  |   0 |   0 |   3.94881   |  156.493  |   30 |
| HOWARD       |  568.038  |  5448.78   |  3661.02   |   0 |   55.7998   |   0 |   0 |  14.4523    | 1717.51   |  324 |
| JEFFERSON    |  927.034  |  3985.77   |  2322.74   |   0 |   15.8525   |   0 |   0 |  34.2215    | 1612.95   |  273 |
| JOHNSON      |  517.644  |  1439.95   |   791.464  |   0 |   52.5073   |   0 |   0 |   5.29587   |  590.682  |   89 |
| KEARNEY      |  640.705  |  2130.97   |  1060.02   |   0 |   61.9424   |   0 |   0 |  17.6881    |  991.326  |  195 |
| KEITH        |  801.085  |  3146.5    |  1843.09   |   0 |   81.8031   |   0 |   0 |  14.0516    | 1207.56   |  290 |
| KIMBALL      |  455.906  |  2110.67   |   919.868  |   0 |   27.0817   |   0 |   0 |  24.5785    | 1139.14   |  159 |
| KNOX         |  781.146  |  4196.4    |  2550.27   |   0 |   34.4914   |   0 |   0 |  33.1171    | 1578.52   |  352 |
| LOGAN        |   86.985  |   161.644  |   105.081  |   0 |    0.556335 |   0 |   0 |   0.354069  |   55.653  |    9 |
| LOUP         |   57.8344 |   443.368  |   345.945  |   0 |    0.680474 |   0 |   0 |   1.23067   |   95.5121 |   23 |
| MC PHERSON   |   49.3009 |   293.619  |   126.586  |   0 |    3.90579  |   0 |   0 |   3.86579   |  159.262  |   14 |
| MADISON      | 2536.1    | 10253.3    |  5271.52   |   0 |  423.235    |   0 |   0 | 280.575     | 4277.95   |  795 |
| MERRICK      |  619.877  |  5334.1    |  3170.69   |   0 |   91.7566   |   0 |   0 |  33.7322    | 2037.92   |  364 |
| MORRILL      |  471.138  |  2622.42   |  1157.25   |   0 |    5.78875  |   0 |   0 |  12.0328    | 1447.35   |  160 |
| NEMAHA       |  638.595  |  2387.87   |  1227.44   |   0 |   96.029    |   0 |   0 | 147.192     |  917.209  |  140 |
| NUCKOLLS     |  458.021  |  2391.8    |  1314.03   |   0 |    9.54003  |   0 |   0 |  64.4586    | 1003.77   |  190 |
| OTOE         | 1802.73   |  7243.89   |  4090.36   |   0 |  152.159    |   0 |   0 |  83.0987    | 2918.27   |  445 |
| PAWNEE       |  284.192  |  1035.05   |   647.15   |   0 |   22.7497   |   0 |   0 |   9.31869   |  355.829  |   95 |
| PERKINS      |  231.158  |  1102.4    |   745.006  |   0 |   20.7056   |   0 |   0 |  23.7353    |  312.955  |   82 |
| PHELPS       |  840.247  |  4050.53   |  2702.71   |   0 |   38.8488   |   0 |   0 |  14.5939    | 1294.38   |  388 |
| PIERCE       |  589.394  |  1996.87   |  1177.24   |   0 |   51.4241   |   0 |   0 |  14.8866    |  753.325  |  190 |
| PLATTE       | 2377.11   |  8681.99   |  4859.33   |   0 |  244.521    |   0 |   0 | 113.707     | 3464.43   |  669 |
| POLK         |  458.599  |  2556.57   |  1443.39   |   0 |   64.2572   |   0 |   0 |  13.5616    | 1035.36   |  190 |
| RED WILLOW   |  904.771  |  5281.82   |  3624.16   |   0 |  113.994    |   0 |   0 |  91.5419    | 1452.13   |  373 |
| RICHARDSON   |  895.873  |  4236.34   |  2562.3    |   0 |   48.428    |   0 |   0 |  61.6149    | 1563.99   |  288 |
| ROCK         |  185.742  |   848.604  |   493.432  |   0 |    8.93503  |   0 |   0 |   8.03993   |  338.197  |   78 |
| SALINE       | 1311      |  4689.9    |  3009.21   |   0 |   67.677    |   0 |   0 |  26.7673    | 1586.24   |  357 |
| SAUNDERS     | 1883.43   |  8905.39   |  5154.36   |   0 |  467.824    |   0 |   0 |  48.3729    | 3234.83   |  527 |
| SCOTTS BLUFF | 2877.55   | 18563.4    | 10504.7    |   0 |  652.414    |   0 |   0 | 254.085     | 7152.21   | 1350 |
| SEWARD       | 1359.78   |  7009.8    |  3933.21   |   0 |  404.978    |   0 |   0 |  74.5892    | 2597.02   |  458 |
| SHERIDAN     |  473.925  |  4651.66   |  2249.82   |   0 |   53.4837   |   0 |   0 |  62.364     | 2285.99   |  251 |
| SHERMAN      |  329.854  |  2628.97   |  1689.09   |   0 |   61.2554   |   0 |   0 |  22.6014    |  856.029  |  143 |
| SIOUX        |  162.447  |   565.319  |   230.394  |   0 |   16.3404   |   0 |   0 |  19.0269    |  299.558  |   45 |
| STANTON      |  572.067  |  1725.28   |  1054.79   |   0 |   46.4435   |   0 |   0 |  34.3801    |  589.66   |  131 |
| THAYER       |  600.951  |  2277.07   |  1467.52   |   0 |    0        |   0 |   0 |  11.3927    |  798.159  |  174 |
| THOMAS       |   78.3994 |   966.905  |   271.34   |   0 |    0        |   0 |   0 |   2.05062   |  693.514  |   25 |
| THURSTON     |  547.569  |  1215.35   |   639.362  |   0 |   46.7661   |   0 |   0 |   1.37877   |  527.844  |   91 |
| VALLEY       |  534.997  |  3144.87   |  1502.66   |   0 |   44.2063   |   0 |   0 |   2.04442   | 1595.96   |  193 |
| WASHINGTON   | 1660.9    |  6145.93   |  3529.73   |   0 |  443.358    |   0 |   0 |  88.0753    | 2084.77   |  342 |
| WAYNE        |  776.926  |  2364.64   |  1113.76   |   0 |  176.979    |   0 |   0 |  11.2552    | 1062.64   |  192 |
| WEBSTER      |  364.325  |  2024.01   |  1354.76   |   0 |   18.8442   |   0 |   0 |  22.0762    |  628.326  |  127 |
| WHEELER      |   88.3592 |   488.364  |   197.97   |   0 |   13.4742   |   0 |   0 |   0.0964233 |  276.823  |   26 |
| YORK         | 1294.14   |  7960.43   |  4759.26   |   0 |  163.439    |   0 |   0 | 103.887     | 2933.84   |  619 |



labeled_formulas: H=B*E
H=B*G
H=C*E
H=C*G
H=D*E
H=D*G
H=E*F
H=E*I
H=E*J
H=E*K
H=F*G
H=G*I
H=G*J
H=G*K
H=(B+C)*E
H=(B+C)*G
H=(B+D)*E
H=(B+D)*G
H=(B*E)*F
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
H=(C+D)*E
H=(C+D)*G
H=(C*E)*F
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
H=(D*E)*F
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
H=(E*F)*I
H=(E*F)*J
H=(E*F)*K
H=(E*I)*J
H=(E*I)*K
H=(E*J)*K
H=(F*G)*I
H=(F*G)*J
H=(F*G)*K
H=(G*I)*J
H=(G*I)*K
H=(G*J)*K
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119129___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_193