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      |  2187.19 |  12638.1  |  4979.44 |   0 |   179.877 |   0 |   0 |   75.1763 |  7403.59 |   707 |
| ALLEN      |  9090.49 |  24932.2  | 12227.6  |   0 |  1579.98  |   0 |   0 |  547.124  | 10577.5  |  1743 |
| ASHLAND    |  3852.05 |  14364.9  |  6291.39 |   0 |   549.963 |   0 |   0 |  442.7    |  7080.83 |  1109 |
| ASHTABULA  |  9056.39 |  36466.4  | 14696.8  |   0 |  1363.73  |   0 |   0 |  891.77   | 19514.1  |  2997 |
| ATHENS     |  4039.64 |  21954.7  |  9401.89 |   0 |  1514.06  |   0 |   0 |  201.893  | 10836.9  |  1197 |
| AUGLAIZE   |  4296.75 |   9578.69 |  4577.75 |   0 |   855.428 |   0 |   0 |  170.727  |  3974.78 |   724 |
| BELMONT    |  5941.94 |  19983.3  | 11188.6  |   0 |   827.796 |   0 |   0 |  387.678  |  7579.17 |  1696 |
| BROWN      |  3865.35 |  17797.1  |  7058.32 |   0 |   651.776 |   0 |   0 |   81.068  | 10006    |  1112 |
| BUTLER     | 26680.7  |  88491    | 37536    |   0 |  6957.77  |   0 |   0 | 1872.9    | 42124.4  |  5003 |
| CARROLL    |  2540.04 |   9241.87 |  3671.22 |   0 |   468.355 |   0 |   0 |   74.9037 |  5027.4  |   666 |
| CHAMPAIGN  |  3268.84 |   9612.41 |  4745.75 |   0 |   699.526 |   0 |   0 |   79.3881 |  4087.75 |   586 |
| CLARK      | 14099.1  |  47942.6  | 24445    |   0 |  2400.41  |   0 |   0 |  910.849  | 20186.3  |  2984 |
| CLERMONT   | 15309.3  |  54760.1  | 21467.1  |   0 |  3589.25  |   0 |   0 |  552.673  | 29151.1  |  3572 |
| CLINTON    |  3971.41 |  14443.7  |  6316.08 |   0 |   840.704 |   0 |   0 |  174.098  |  7112.82 |   704 |
| COLUMBIANA |  9202.81 |  39060    | 17924.8  |   0 |  1863.57  |   0 |   0 |  430.265  | 18841.4  |  3108 |
| COSHOCTON  |  3027.79 |  10756.1  |  5794.38 |   0 |   418.727 |   0 |   0 |  100.778  |  4442.2  |   893 |
| CRAWFORD   |  3908.25 |  13389.2  |  6567.14 |   0 |   485.62  |   0 |   0 |  320.856  |  6015.59 |  1044 |
| DARKE      |  4110.47 |  14424.1  |  6713.85 |   0 |   579.67  |   0 |   0 |  192.658  |  6937.93 |   888 |
| DEFIANCE   |  3219.74 |   8981.15 |  5009.12 |   0 |   598.717 |   0 |   0 |  107.642  |  3265.67 |   589 |
| DELAWARE   | 11498.3  |  21580.5  | 10320.1  |   0 |  2151.75  |   0 |   0 |  759.388  |  8349.29 |  1382 |
| ERIE       |  7631.6  |  32021.8  | 14365.8  |   0 |  1225.7   |   0 |   0 |  324.62   | 16105.7  |  2166 |
| FAIRFIELD  | 12711    |  41336.7  | 17331.4  |   0 |  3005.07  |   0 |   0 | 1080.81   | 19919.4  |  2925 |
| FAYETTE    |  2213.29 |  12844.5  |  4596.77 |   0 |   358.812 |   0 |   0 |  103.223  |  7785.66 |   618 |
| FULTON     |  3058.62 |   7607.44 |  3687.21 |   0 |   608.755 |   0 |   0 |   92.7471 |  3218.72 |   603 |
| GALLIA     |  2509.2  |  14626.3  |  6930.07 |   0 |   620.666 |   0 |   0 |   55.5075 |  7020.1  |   750 |
| GEAUGA     |  7532.15 |  15463.1  |  6792.12 |   0 |   911.803 |   0 |   0 |  719.292  |  7039.84 |  1305 |
| GREENE     | 14836    |  71350.7  | 40047.2  |   0 |  8660.15  |   0 |   0 | 1573.09   | 21070.2  |  2736 |
| GUERNSEY   |  3311.28 |  17261.7  |  8181.52 |   0 |   530.165 |   0 |   0 |   86.7621 |  8463.26 |  1274 |
| HANCOCK    |  5602.11 |  13254.5  |  6486.36 |   0 |  1469.92  |   0 |   0 |  182.453  |  5115.76 |   912 |
| HARDIN     |  2243.86 |   6509.8  |  3498.75 |   0 |   607.806 |   0 |   0 |  116.055  |  2287.18 |   463 |
| HARRISON   |  1804.8  |   5827.36 |  2658.48 |   0 |   290.269 |   0 |   0 |   48.8323 |  2829.78 |   424 |
| HENRY      |  2360.59 |   5170.3  |  2493.97 |   0 |   371.12  |   0 |   0 |   94.1594 |  2211.05 |   334 |
| HIGHLAND   |  3485.47 |  17246.2  |  6734.04 |   0 |   540.4   |   0 |   0 |  149.857  |  9821.92 |   979 |
| HOCKING    |  2816.88 |  11575.1  |  4875.89 |   0 |   315.505 |   0 |   0 |   33.9576 |  6349.76 |   737 |
| HOLMES     |  1694.21 |  10775.4  |  3226.22 |   0 |   152.234 |   0 |   0 |   12.8534 |  7384.1  |   408 |
| HURON      |  4865.1  |  15512    |  6341.68 |   0 |   742.334 |   0 |   0 |  345.085  |  8082.87 |  1199 |
| JACKSON    |  3061.69 |  20939.2  |  8121.36 |   0 |   545.582 |   0 |   0 |  215.464  | 12056.8  |   991 |
| JEFFERSON  |  6613.37 |  27605.1  | 11134.7  |   0 |   952.343 |   0 |   0 |  505.404  | 15012.7  |  1733 |
| KNOX       |  4513.52 |  14393.3  |  6826.05 |   0 |   898.24  |   0 |   0 |  454.183  |  6214.8  |  1080 |
| LAKE       | 19376.9  |  60856.5  | 23555.8  |   0 |  4837.88  |   0 |   0 | 1536.39   | 30926.4  |  4818 |
| LAWRENCE   |  5523.98 |  36196.9  | 16238.4  |   0 |   915.399 |   0 |   0 |  283.169  | 18760    |  2051 |
| LICKING    | 13529.6  |  44357.7  | 21960.7  |   0 |  2611.66  |   0 |   0 |  734.318  | 19051.1  |  3379 |
| LOGAN      |  4179.12 |   9936.97 |  4622.01 |   0 |   650.234 |   0 |   0 |  177.357  |  4487.37 |   663 |
| LORAIN     | 25046    |  79653.1  | 33092.2  |   0 |  4731.65  |   0 |   0 | 1454.93   | 40374.3  |  6004 |
| LUCAS      | 30578.2  |  86050.2  | 39243.9  |   0 |  7657.24  |   0 |   0 | 2120.95   | 37028.1  |  5740 |
| MADISON    |  3984.73 |   9664.36 |  4713.84 |   0 |   664.129 |   0 |   0 |  182.292  |  4104.11 |   577 |
| MAHONING   | 21691.2  |  74152.3  | 32670.8  |   0 |  3300.93  |   0 |   0 | 1576.88   | 36603.7  |  6059 |
| MARION     |  5589.5  |  15042.1  |  7509.54 |   0 |   774.511 |   0 |   0 |  258.881  |  6499.2  |  1227 |
| MEDINA     | 13858.2  |  35497.8  | 16321.3  |   0 |  2805.49  |   0 |   0 | 1027.6    | 15343.3  |  2169 |
| MEIGS      |  2381.27 |   9289.82 |  4469.1  |   0 |   272.309 |   0 |   0 |   70.6437 |  4477.77 |   607 |
| MERCER     |  2901.14 |   9034.74 |  4383.45 |   0 |   618.174 |   0 |   0 |  148.782  |  3884.34 |   646 |
| MIAMI      |  8378    |  26836.8  | 12561.1  |   0 |  1672.41  |   0 |   0 |  522.901  | 12080.4  |  1640 |
| MONROE     |  1722.54 |   5581.64 |  2800.37 |   0 |   132.438 |   0 |   0 |   25.4817 |  2623.35 |   456 |
| MORGAN     |  1157.94 |   6204.66 |  3142.67 |   0 |   173.621 |   0 |   0 |  111.939  |  2776.43 |   447 |
| MORROW     |  3229.89 |   9534.25 |  4314.92 |   0 |   541.323 |   0 |   0 |  106.122  |  4571.88 |   777 |
| MUSKINGUM  |  6337.84 |  22622.1  | 11381.4  |   0 |  1502.86  |   0 |   0 |  294.817  |  9442.98 |  1996 |
| NOBLE      |  1318.73 |   3442.54 |  1831.41 |   0 |   126.655 |   0 |   0 |   22.4063 |  1462.07 |   305 |
| OTTAWA     |  3730.56 |  11183.1  |  5227.32 |   0 |   574.134 |   0 |   0 |  349.966  |  5031.69 |   898 |
| PAULDING   |  1631.86 |   4140.15 |  1823.18 |   0 |   170.064 |   0 |   0 |   92.6784 |  2054.22 |   325 |
| PERRY      |  2878.49 |  10294.1  |  4718.35 |   0 |   361.128 |   0 |   0 |  143.147  |  5071.45 |   708 |
| PICKAWAY   |  4867.93 |  25025.5  |  9120.17 |   0 |   832.919 |   0 |   0 |  219.451  | 14853    |  1189 |
| PIKE       |  2361.27 |  17789.1  |  6488.07 |   0 |   283.958 |   0 |   0 |   74.9017 | 10942.2  |   819 |
| PORTAGE    | 12527.8  |  38795.9  | 16597.7  |   0 |  4040.92  |   0 |   0 |  762.504  | 17394.8  |  2915 |
| PREBLE     |  3624.6  |  13993.9  |  5458.15 |   0 |   621.565 |   0 |   0 |  143.815  |  7770.37 |   876 |
| PUTNAM     |  2078.86 |   4694.12 |  2446.17 |   0 |   565.274 |   0 |   0 |  222.376  |  1460.3  |   360 |
| RICHLAND   | 11633.1  |  36843.2  | 16459.2  |   0 |  2044.66  |   0 |   0 |  964.908  | 17374.4  |  3586 |
| SANDUSKY   |  4838.48 |  12868.2  |  6517.19 |   0 |   908.379 |   0 |   0 |  280.017  |  5162.6  |  1007 |
| SCIOTO     |  6442.6  |  36017    | 15985.2  |   0 |  1148.26  |   0 |   0 |  237.528  | 18646    |  2214 |
| SENECA     |  4435.57 |  10969.2  |  5171.62 |   0 |   915.723 |   0 |   0 |  395.94   |  4485.87 |  1010 |
| SHELBY     |  3702.72 |  10409.8  |  4350.55 |   0 |   694.642 |   0 |   0 |  199.406  |  5165.17 |   687 |
| STARK      | 30760.8  |  97262.9  | 41675.6  |   0 |  5655.91  |   0 |   0 | 2163.42   | 47768    |  7215 |
| SUMMIT     | 41985.3  | 146002    | 58683.8  |   0 | 10633.6   |   0 |   0 | 3927.57   | 72756.6  | 10091 |
| TRUMBULL   | 20355.2  |  66406.5  | 28601.5  |   0 |  3432.14  |   0 |   0 | 1654.53   | 32718.4  |  5776 |
| TUSCARAWAS |  7934.42 |  27483.9  | 12451.2  |   0 |  1260.52  |   0 |   0 |  443.717  | 13328.4  |  2387 |
| UNION      |  3992.49 |   8407.51 |  4573.73 |   0 |   801.196 |   0 |   0 |  141.192  |  2891.39 |   498 |
| VAN WERT   |  2544.1  |   6460.66 |  2881.37 |   0 |   500.847 |   0 |   0 |   50.2995 |  3028.15 |   442 |
| VINTON     |  1265.58 |   9396.58 |  3498.67 |   0 |   317.892 |   0 |   0 |   30.3066 |  5549.71 |   426 |
| WARREN     | 15319.4  |  40157    | 17837    |   0 |  4024.52  |   0 |   0 | 1113.83   | 17181.6  |  2245 |
| WASHINGTON |  6024.71 |  20636.6  | 10829.9  |   0 |   924.744 |   0 |   0 |  362.955  |  8519    |  1611 |
| WAYNE      |  8752.85 |  26615.3  | 10777.2  |   0 |  1114.44  |   0 |   0 |  832.735  | 13891    |  1835 |
| WILLIAMS   |  3214.49 |   8140.09 |  3771.26 |   0 |   515.104 |   0 |   0 |  124.532  |  3729.2  |   598 |
| WOOD       |  8717.08 |  18506    |  8872.23 |   0 |  2749.62  |   0 |   0 |  561.275  |  6322.87 |  1212 |
| WYANDOT    |  1595.71 |   4386.39 |  2134.63 |   0 |   282.093 |   0 |   0 |  143.574  |  1826.1  |   363 |



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
H=E+G
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
H=(B+E)*G
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
H=(C+E)*G
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
H=(D+E)*G
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
H=(E+F)*G
G=(E*F)*I
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_119135___xlsx_41733d0294b45d708bccba64327d3a7c1798edcb.xlsx___line_199