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 |
|:------------|----------:|-----------:|----------:|----:|-----------:|----:|----:|-----------:|----------:|-----:|
| ALEXANDER   |   794.442 |   5493.69  |  2016.27  |   0 |   171.451  |   0 |   0 |   18.3679  |  3287.6   |  308 |
| BOND        |  1485.58  |   4436.51  |  2190.5   |   0 |   447.311  |   0 |   0 |   54.4996  |  1744.2   |  260 |
| BOONE       |  3404.09  |   7397.12  |  2996.59  |   0 |   787.337  |   0 |   0 |  158.809   |  3454.38  |  684 |
| BROWN       |   484.239 |    923.863 |   465.416 |   0 |    98.2146 |   0 |   0 |   31.2868  |   328.945 |   79 |
| BUREAU      |  3070.89  |   8246.68  |  3610.55  |   0 |   394.599  |   0 |   0 |  237.898   |  4003.63  |  901 |
| CALHOUN     |   471.644 |   1742.41  |  1015.69  |   0 |    35.8288 |   0 |   0 |    9.59668 |   681.296 |  102 |
| CARROLL     |  1534.48  |   4898.74  |  1822.69  |   0 |   195.791  |   0 |   0 |  183.211   |  2697.06  |  479 |
| CASS        |  1283.17  |   3558.24  |  1858.23  |   0 |   104.399  |   0 |   0 |  198.659   |  1396.95  |  247 |
| CHAMPAIGN   | 10808.9   |  40216.5   | 16652.1   |   0 |  5563.83   |   0 |   0 |  930.059   | 17070.5   | 2089 |
| CHRISTIAN   |  2840.11  |   8498.39  |  4335.84  |   0 |   554.219  |   0 |   0 |  250.31    |  3358.03  |  685 |
| CLARK       |  1350.09  |   5613.88  |  2656.49  |   0 |   226.63   |   0 |   0 |  100.239   |  2630.52  |  454 |
| CLAY        |  1081.56  |   4987.51  |  1956.11  |   0 |   212.203  |   0 |   0 |  127.979   |  2691.22  |  415 |
| CLINTON     |  3323.97  |  10751.9   |  6499.8   |   0 |  1239.08   |   0 |   0 |  232.411   |  2780.61  |  509 |
| COLES       |  3638.81  |  13231.9   |  6001.16  |   0 |  1363.89   |   0 |   0 |  190.052   |  5676.84  |  956 |
| CRAWFORD    |  1897.71  |   8949.02  |  3738.4   |   0 |   294.48   |   0 |   0 |   57.4171  |  4858.72  |  465 |
| CUMBERLAND  |   995.713 |   3596.66  |  1594.81  |   0 |   177.435  |   0 |   0 |   46.0424  |  1778.37  |  288 |
| DEKALB      |  7235.07  |  16804.1   |  6848.77  |   0 |  3837.12   |   0 |   0 |  532.691   |  5585.49  |  961 |
| DEWITT      |  1574.8   |   4738.02  |  1726.55  |   0 |   147.255  |   0 |   0 |   81.0945  |  2783.12  |  390 |
| DOUGLAS     |  1564.58  |   5688.3   |  1895.93  |   0 |   314.792  |   0 |   0 |   51.3682  |  3426.21  |  392 |
| DU PAGE     | 49703.3   | 131187     | 43752.5   |   0 | 16781.5    |   0 |   0 | 6102.84    | 64550.3   | 7372 |
| EDGAR       |  1593.17  |   6939.46  |  3177.93  |   0 |   231.761  |   0 |   0 |  162.51    |  3367.26  |  435 |
| EDWARDS     |   694.771 |   2684.87  |   886.886 |   0 |    53.9847 |   0 |   0 |   60.6064  |  1683.39  |  230 |
| EFFINGHAM   |  2830.65  |   8345.75  |  3462.32  |   0 |   560.413  |   0 |   0 |   83.8627  |  4239.15  |  989 |
| FAYETTE     |  1632.43  |   4917.65  |  2363.63  |   0 |   286.138  |   0 |   0 |   69.3616  |  2198.52  |  454 |
| FORD        |  1369.42  |   4030.17  |  1684.66  |   0 |   240.013  |   0 |   0 |   46.6228  |  2058.87  |  239 |
| FRANKLIN    |  4088.6   |  32221.6   | 10977.6   |   0 |   721.494  |   0 |   0 |  237.5     | 20285     | 1721 |
| FULTON      |  3255.14  |   7603.44  |  3650.58  |   0 |   448.469  |   0 |   0 |  168.489   |  3335.9   |  685 |
| GALLATIN    |   573.054 |   3355.73  |  1110.56  |   0 |    64.576  |   0 |   0 |   31.8176  |  2148.78  |  205 |
| GREENE      |  1480.47  |   4013.95  |  1937.9   |   0 |   127.12   |   0 |   0 |   84.2445  |  1864.68  |  250 |
| GRUNDY      |  3755.33  |   8308.07  |  3779.1   |   0 |   837.144  |   0 |   0 |  140.192   |  3551.64  |  551 |
| HAMILTON    |   644.938 |   4690.17  |  1908.82  |   0 |    55.9991 |   0 |   0 |   19.653   |  2705.69  |  298 |
| HANCOCK     |  1702.33  |   5824.65  |  2312.81  |   0 |   355.674  |   0 |   0 |  173.278   |  2982.88  |  523 |
| HARDIN      |   388.121 |   2304.92  |  1227.67  |   0 |    32.6156 |   0 |   0 |    1.38201 |  1043.26  |  142 |
| HENDERSON   |   868.064 |   2549.22  |  1086.47  |   0 |    24.9969 |   0 |   0 |   27.563   |  1410.19  |  206 |
| HENRY       |  4527.52  |  12755     |  6101.73  |   0 |   622.153  |   0 |   0 |  184.328   |  5846.83  | 1105 |
| IROQUOIS    |  2647.25  |   9535.15  |  3571.89  |   0 |   333.249  |   0 |   0 |  159.653   |  5470.35  |  728 |
| JACKSON     |  5239.33  |  26402.7   |  9735.12  |   0 |  4041.64   |   0 |   0 |  248.189   | 12377.8   | 1394 |
| JASPER      |   936.954 |   3200.11  |  1097.02  |   0 |    87.5773 |   0 |   0 |   41.9938  |  1973.52  |  284 |
| JEFFERSON   |  3518.39  |  17401.9   |  6701.79  |   0 |   716.429  |   0 |   0 |  484.99    |  9498.67  | 1120 |
| JERSEY      |  2317.02  |   6967.61  |  3474.72  |   0 |   421.64   |   0 |   0 |   65.3585  |  3005.9   |  437 |
| JO DAVIESS  |  2193.7   |   5446.81  |  2236.74  |   0 |   322.771  |   0 |   0 |  251.545   |  2635.75  |  642 |
| JOHNSON     |  1175.56  |   8086.58  |  3002.13  |   0 |   345.889  |   0 |   0 |  205.502   |  4533.06  |  486 |
| KANE        | 24646.4   |  56268.4   | 21452.8   |   0 |  8731.4    |   0 |   0 | 2165.95    | 23918.3   | 3860 |
| KANKAKEE    |  8082.38  |  31221.2   | 12261.8   |   0 |  1850.09   |   0 |   0 |  619.706   | 16489.5   | 2408 |
| KENDALL     |  5828.13  |  15399.9   |  6009.02  |   0 |  2418.4    |   0 |   0 |  292.322   |  6680.18  |  792 |
| KNOX        |  4527.47  |  15860.6   |  7062.03  |   0 |   798.054  |   0 |   0 |  236.202   |  7764.26  | 1407 |
| LA SALLE    |  9367.53  |  27199.5   | 12864.7   |   0 |  1740.64   |   0 |   0 |  558.879   | 12035.3   | 2463 |
| LAWRENCE    |  1464.91  |   4290.37  |  2161.32  |   0 |   253.446  |   0 |   0 |   27.1292  |  1848.48  |  372 |
| LEE         |  2431.84  |   7841.07  |  3490.01  |   0 |   549.856  |   0 |   0 |   93.0489  |  3708.15  |  784 |
| LIVINGSTON  |  2734.82  |   6624.93  |  3353.91  |   0 |   467.144  |   0 |   0 |  342.22    |  2461.66  |  661 |
| LOGAN       |  2451.39  |   5656.09  |  2484.64  |   0 |   256.657  |   0 |   0 |  328.478   |  2586.31  |  439 |
| MCDONOUGH   |  2333.19  |  10827.1   |  4645.17  |   0 |  1760.34   |   0 |   0 |  204.448   |  4217.15  |  876 |
| MCHENRY     | 20002.1   |  51341.9   | 17491.8   |   0 |  4740.15   |   0 |   0 | 2033.43    | 27076.6   | 3678 |
| MCLEAN      | 11420     |  28283.9   | 13475.1   |   0 |  4443.43   |   0 |   0 | 1110.32    |  9255.13  | 1814 |
| MACON       |  9094.96  |  28335.8   | 13928.1   |   0 |  1514.29   |   0 |   0 |  432.44    | 12461     | 2474 |
| MACOUPIN    |  4654.75  |  13091.1   |  6873.81  |   0 |   835.683  |   0 |   0 |  213.977   |  5167.68  |  850 |
| MARION      |  4036.74  |  16459.6   |  6772.9   |   0 |   609.216  |   0 |   0 |  152.827   |  8924.68  | 1155 |
| MARSHALL    |  1109.41  |   2925.27  |  1456.81  |   0 |   238.829  |   0 |   0 |  114.533   |  1115.1   |  241 |
| MASON       |  1499.61  |   3732.13  |  1690.73  |   0 |   104.681  |   0 |   0 |   62.7844  |  1873.93  |  336 |
| MASSAC      |  1291.94  |   7075.31  |  3120.28  |   0 |   195.91   |   0 |   0 |  103.347   |  3655.78  |  489 |
| MENARD      |  1102.23  |   2616.91  |  1515.7   |   0 |   168.296  |   0 |   0 |   28.1327  |   904.782 |  195 |
| MERCER      |  1774.1   |   5190.34  |  2208.06  |   0 |   241.173  |   0 |   0 |   28.7429  |  2712.36  |  407 |
| MONROE      |  2824.3   |   8044.09  |  3640.96  |   0 |   682.756  |   0 |   0 |  410.876   |  3309.5   |  455 |
| MONTGOMERY  |  2320.89  |   6990.9   |  3766.09  |   0 |   423.911  |   0 |   0 |  123.487   |  2677.41  |  525 |
| MORGAN      |  3073.24  |   8454.87  |  4748.32  |   0 |   572.906  |   0 |   0 |  246.603   |  2887.04  |  628 |
| MOULTRIE    |  1061.82  |   2607.7   |  1548.35  |   0 |   206.802  |   0 |   0 |   28.7104  |   823.833 |  295 |
| OGLE        |  4841.98  |  11357.1   |  4827.87  |   0 |  1166.55   |   0 |   0 |  270.741   |  5091.92  | 1001 |
| PEORIA      | 13926.1   |  34379     | 14822.3   |   0 |  2900.1    |   0 |   0 |  958.829   | 15697.8   | 2892 |
| PERRY       |  1712.77  |  10051.3   |  4286.16  |   0 |   464.697  |   0 |   0 |   87.0647  |  5213.43  |  596 |
| PIATT       |  1575.52  |   3093.33  |  1336.39  |   0 |   237.52   |   0 |   0 |   68.0702  |  1451.35  |  272 |
| PIKE        |  1328.9   |   4312.22  |  2214.79  |   0 |   312.75   |   0 |   0 |  127.29    |  1657.39  |  365 |
| POPE        |   428.838 |   2380.78  |   981.777 |   0 |    38.92   |   0 |   0 |   16.7302  |  1343.35  |  141 |
| PUTNAM      |   616.022 |   1353.91  |   648.201 |   0 |    78.9858 |   0 |   0 |   43.2244  |   583.498 |  180 |
| RANDOLPH    |  2957.61  |   9884.67  |  4769.29  |   0 |   493.985  |   0 |   0 |  403.752   |  4217.64  |  651 |
| RICHLAND    |  1328.65  |   4841.27  |  1981.53  |   0 |   238.48   |   0 |   0 |  128.464   |  2492.79  |  378 |
| SAINT CLAIR | 26336     | 132314     | 68511.3   |   0 | 19015      |   0 |   0 | 2207.26    | 42580.5   | 5053 |
| SALINE      |  2245.1   |  14532     |  5009.85  |   0 |   435.784  |   0 |   0 |   95.5205  |  8990.87  |  902 |
| SCHUYLER    |   618.381 |   2215.54  |  1422.16  |   0 |   114.841  |   0 |   0 |   21.9002  |   656.637 |  204 |
| SCOTT       |   467.08  |   1106.55  |   626.997 |   0 |    23.0832 |   0 |   0 |   37.5504  |   418.916 |   96 |
| SHELBY      |  2212.5   |   6405.94  |  3296.38  |   0 |   285.568  |   0 |   0 |  231.924   |  2592.07  |  540 |
| STARK       |   470.326 |   1334.58  |   320.452 |   0 |    59.4532 |   0 |   0 |    6.05166 |   948.627 |  155 |
| STEPHENSON  |  4145.2   |  14054.3   |  4838.84  |   0 |   640.574  |   0 |   0 |  320.577   |  8254.27  | 1506 |
| TAZEWELL    | 11974.1   |  25605.8   | 11675.9   |   0 |  2391.65   |   0 |   0 |  754.239   | 10784.1   | 2188 |
| UNION       |  1651.21  |  10942.1   |  4278.94  |   0 |   358.157  |   0 |   0 |   65.7901  |  6239.18  |  643 |
| WABASH      |  1462.09  |   3829.26  |  1771.59  |   0 |   237.79   |   0 |   0 |  112.554   |  1707.32  |  334 |
| WARREN      |  1276.33  |   5087.33  |  1872.74  |   0 |   194.71   |   0 |   0 |   48.9956  |  2970.89  |  469 |
| WASHINGTON  |  1471.6   |   4632.47  |  2245.14  |   0 |   457.356  |   0 |   0 |   91.2923  |  1838.67  |  314 |
| WAYNE       |  1173.62  |   4746.28  |  2348.58  |   0 |   145.579  |   0 |   0 |   71.0596  |  2181.07  |  469 |
| WHITE       |  1560.29  |   6130.73  |  2598.76  |   0 |   212.241  |   0 |   0 |  114.776   |  3204.95  |  475 |
| WHITESIDE   |  5594.19  |  15662.4   |  6596.26  |   0 |   799.844  |   0 |   0 |  354.941   |  7911.31  | 1609 |
| WILLIAMSON  |  5597.79  |  52574.6   | 16467.6   |   0 |  2320.38   |   0 |   0 |  402.606   | 33383.9   | 2767 |
| WINNEBAGO   | 22385.9   |  62356.4   | 24988.4   |   0 |  4786.65   |   0 |   0 | 1342.59    | 31238.8   | 5680 |
| WOODFORD    |  3353.07  |   5597.63  |  2719.1   |   0 |   503.046  |   0 |   0 |  174.723   |  2200.77  |  496 |



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*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)*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)*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)*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)*I
G=(E*F)*J
G=(E*F)*K
H=(E*F)*I
H=(E*F)*J
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119114___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_178