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 |
|:-------------|----------:|----------:|----------:|----------:|----------:|----------:|----------:|----:|----:|-----------:|
| Adair        |   5.67123 |   5.89041 |   0       |  14.2822  |   3.72877 |   3.96164 |  20.0712  |   0 |   0 |   53.6055  |
| Allen        |   4.09863 |  84.137   |   9.16712 |  12.1123  |  75.4685  |   0       |   0       |   0 |   0 |  184.984   |
| Ballard      |  13.4466  |   8.92055 |  95.7534  | 104.986   |   0       |   0       |   0       |   0 |   0 |  223.107   |
| Barren       |  84.137   |   4.09863 |   4.68767 |  14.4438  |   3.72877 |   9.16712 |  75.4685  |   0 |   0 |  195.732   |
| Bath         |   5.82192 |   6.93699 |  12.0959  |   5.05205 |  11.3123  |  19.9014  |  10.8959  |   0 |   0 |   72.0164  |
| Bell         |  64.5425  |   3.47397 |  13.1014  |  11.8685  |   3.29041 |  10.8849  |   0       |   0 |   0 |  107.162   |
| Boone        |  29.5205  |  19.5726  |   3.60548 |  76.674   |   0       |   0       |   0       |   0 |   0 |  129.373   |
| Boyd         |  18.7863  |  14.1205  |  29.2767  |  14.9397  |   0       |   0       |   0       |   0 |   0 |   77.1233  |
| Boyle        |  36.9973  |   5.89041 |   2.28767 |   9.9726  |  41.4321  |  16.5616  |  16.811   |   0 |   0 |  129.953   |
| Bracken      |   4.56438 |  11.3041  |   8.44918 |   2.47123 |   4.78082 |   0       |   0       |   0 |   0 |   31.5697  |
| Breathitt    |  49.3233  |   1.11507 |   2       |  21.5014  |  10.1151  |  17.9397  |   1.01096 |   0 |   0 |  103.005   |
| Breckinridge |   2.16986 |  11.5836  |   3.15342 | 164.132   |   5.75616 |  26.8986  |   0       |   0 |   0 |  213.693   |
| Bullitt      |  30.6192  | 164.132   | 791.8     |  27.0274  |  27.0274  |   0       |   0       |   0 |   0 | 1040.61    |
| Butler       |  15.2904  |   4.68767 |  11.5836  |  27.5671  |  15.3205  |  26.8986  |  75.4685  |   0 |   0 |  176.816   |
| Caldwell     |  48.0438  | 110.699   |  27.2164  | 148.436   |   8.8411  |  17.1178  |  19.9671  |   0 |   0 |  380.321   |
| Calloway     |  92.337   |  95.7534  |  48.3452  |  17.1178  |   0       |   0       |   0       |   0 |   0 |  253.553   |
| Campbell     | 101.088   |  76.674   |   2.47123 |   0       |   0       |   0       |   0       |   0 |   0 |  180.233   |
| Carlisle     |   8.92055 |  13.4466  |  95.7534  |  22.3507  | 104.986   |   0       |   0       |   0 |   0 |  245.458   |
| Carroll      |   9.05479 |  19.5726  |   7.64384 |   5.31781 |   8.83836 |   0       |   0       |   0 |   0 |   50.4274  |
| Carter       |  14.1205  |  18.7863  |   3.59452 |  29.2767  |  14.9397  |   5       |  10.8959  |   0 |   0 |   96.6137  |
| Christian    | 110.699   |  48.0438  | 148.436   |  15.3205  |   9.56986 |  17.1178  |   0       |   0 |   0 |  349.186   |
| Clark        |  53.2548  |   6.93699 |   4.90959 | 196.234   |  28.0986  |  11.3123  |   7.50959 |   0 |   0 |  308.256   |
| Clinton      |   3.7726  |   0       |   3.96164 |   1.53699 |   0       |   0       |   0       |   0 |   0 |    9.27123 |
| Crittenden   |  27.2164  |  48.0438  |  21.6466  |   8.8411  |   8.50685 |  19.9671  |   0       |   0 |   0 |  134.222   |
| Cumberland   |   0       |   5.67123 |   3.7726  |   3.72877 |   9.16712 |   3.96164 |   0       |   0 |   0 |   26.3014  |
| Daviess      | 135.427   |   3.15342 |  78.0712  |  16.8055  |  26.8986  |   0       |   0       |   0 |   0 |  260.356   |
| Edmonson     |   4.68767 |  84.137   |  15.2904  |  11.5836  |  14.4438  |  75.4685  |   0       |   0 |   0 |  205.611   |
| Elliott      |   3.59452 |  14.1205  |  14.9397  |   4.0411  |  10.8959  |   0       |   0       |   0 |   0 |   47.5918  |
| Estill       |   4.90959 |  53.2548  |   3.14795 |   2       |  28.0986  |   7.50959 |   0       |   0 |   0 |   98.9205  |
| Fayette      | 196.234   |   6.93699 |  53.2548  | 142.74    |  28.0986  |  13.4685  |  14.1726  |   0 |   0 |  454.905   |
| Fleming      |  12.0959  |   5.82192 |   5       |   8.44918 |  19.9014  |   4.78082 |  10.8959  |   0 |   0 |   66.9451  |
| Floyd        |  26.9973  |  35.7178  |   1.11507 |  21.5014  |   7.25205 |  92.5808  |   0       |   0 |   0 |  185.164   |
| Franklin     |  14.4329  |   8.13425 |   7.64384 |   5.31781 |  13.4685  |  24.9863  |  14.1726  |   0 |   0 |   88.1562  |
| Fulton       |   9.02192 |  22.3507  |   0       |   0       |   0       |   0       |   0       |   0 |   0 |   31.3726  |
| Gallatin     |  19.5726  |  29.5205  |   9.05479 |   3.60548 |   5.31781 |   0       |   0       |   0 |   0 |   67.0712  |
| Garrard      |   2.28767 |  36.9973  | 142.74    |   9.9726  |  28.0986  |  16.5616  |  17.7781  |   0 |   0 |  254.436   |
| Graves       |  95.7534  |  13.4466  |  92.337   |   8.92055 |  22.3507  |  48.3452  | 104.986   |   0 |   0 |  386.14    |
| Grayson      |  11.5836  |   2.16986 |  15.2904  |   4.68767 | 164.132   |  14.4438  |  26.8986  |   0 |   0 |  239.205   |
| Green        |  14.2822  |   5.67123 |  14.4438  |  16.611   |   3.72877 |  20.0712  |   0       |   0 |   0 |   74.8082  |
| Greenup      |  29.2767  |  18.7863  |  14.1205  |   5       |   0       |   0       |   0       |   0 |   0 |   67.1836  |
| Hancock      |   3.15342 |   2.16986 | 135.427   |  26.8986  |   0       |   0       |   0       |   0 |   0 |  167.649   |
| Harlan       |  13.1014  |  64.5425  |   3.29041 |  14.7863  |  17.9397  |   0       |   0       |   0 |   0 |  113.66    |
| Henderson    |  78.0712  | 135.427   |  16.8055  |   8.50685 |  19.9671  |   0       |   0       |   0 |   0 |  258.778   |
| Henry        |   7.64384 |   9.05479 |  14.4329  |  43.6603  |   5.31781 |  24.9863  |   8.83836 |   0 |   0 |  113.934   |
| Hickman      |  22.3507  |   8.92055 |   9.02192 |  95.7534  |   0       |   0       |   0       |   0 |   0 |  136.047   |
| Hopkins      | 148.436   |  48.0438  | 110.699   |  16.8055  |  15.3205  |  19.9671  |   0       |   0 |   0 |  359.271   |
| Jefferson    | 791.8     |  30.6192  | 164.132   |  43.6603  |  24.9863  |  27.0274  |   0       |   0 |   0 | 1082.22    |
| Jessamine    | 142.74    | 196.234   |   2.28767 |  28.0986  |  16.5616  |  14.1726  |   0       |   0 |   0 |  400.094   |
| Johnson      |  35.7178  |  26.9973  |  14.9397  |  21.5014  |   7.25205 |   4.0411  |   0       |   0 |   0 |  110.449   |
| Kenton       |  76.674   |  29.5205  | 101.088   |   3.60548 |   2.47123 |   0       |   0       |   0 |   0 |  213.359   |
| Knott        |   1.11507 |  49.3233  |  26.9973  |  14.7863  |  21.5014  |  17.9397  |  92.5808  |   0 |   0 |  224.244   |
| Knox         |  11.8685  |  64.5425  |   3.47397 |   8.3863  |  10.8849  |   0       |   0       |   0 |   0 |   99.1562  |
| Larue        |  16.611   |  14.2822  | 164.132   |  14.4438  |  41.4321  |  27.0274  |  20.0712  |   0 |   0 |  297.999   |
| Lawrence     |  14.9397  |  18.7863  |  14.1205  |   3.59452 |  35.7178  |   7.25205 |   4.0411  |   0 |   0 |   98.4521  |
| Lee          |   2       |  49.3233  |   4.90959 |   3.14795 |  10.1151  |   7.50959 |   1.01096 |   0 |   0 |   78.0164  |
| Leslie       |   3.29041 |  64.5425  |   3.47397 |  13.1014  |  17.9397  |   0       |   0       |   0 |   0 |  102.348   |
| Letcher      |  14.7863  |  13.1014  |   1.11507 |  17.9397  |  92.5808  |   0       |   0       |   0 |   0 |  139.523   |
| Lewis        |   5       |  14.1205  |  12.0959  |  29.2767  |   8.44918 |  10.8959  |   0       |   0 |   0 |   79.8382  |
| Lincoln      |   9.9726  |  36.9973  |   5.89041 |   2.28767 |  54.3836  |  17.7781  |   0       |   0 |   0 |  127.31    |
| Livingston   |  21.6466  |  27.2164  |   8.8411  |  48.3452  | 104.986   |   0       |   0       |   0 |   0 |  211.036   |
| Logan        |  27.5671  |  15.2904  |  15.3205  |  12.1123  |   9.56986 |  75.4685  |   0       |   0 |   0 |  155.329   |
| Lyon         |   8.8411  |  48.0438  |  27.2164  |  21.6466  |  48.3452  |  17.1178  |   0       |   0 |   0 |  171.211   |
| Magoffin     |  21.5014  |  49.3233  |  26.9973  |  35.7178  |   1.11507 |   4.0411  |   1.01096 |   0 |   0 |  139.707   |
| Marion       |  41.4321  |  36.9973  |   5.89041 |  16.611   |  27.0274  |  20.0712  |  16.811   |   0 |   0 |  164.84    |
| Marshall     |  48.3452  |  92.337   |  95.7534  |  21.6466  |   8.8411  | 104.986   |  17.1178  |   0 |   0 |  389.027   |
| Martin       |   7.25205 |  26.9973  |  35.7178  |  14.9397  |  92.5808  |   0       |   0       |   0 |   0 |  177.488   |
| Mason        |   8.44918 |   4.56438 |  12.0959  |   5       |   4.78082 |   0       |   0       |   0 |   0 |   34.8903  |
| McCracken    | 104.986   |  13.4466  |   8.92055 |  95.7534  |  21.6466  |  48.3452  |   0       |   0 |   0 |  293.099   |
| McCreary     |   3.07123 |   8.3863  |  54.3836  |   1.53699 |  10.8849  |   0       |   0       |   0 |   0 |   78.263   |
| McLean       |  16.8055  | 135.427   |  78.0712  | 148.436   |  15.3205  |  26.8986  |  19.9671  |   0 |   0 |  440.926   |
| Meade        |   5.75616 |   2.16986 | 164.132   |   0       |   0       |   0       |   0       |   0 |   0 |  172.058   |
| Menifee      |   5.05205 |   5.82192 |  11.3123  |   4.0411  |   7.50959 |  10.8959  |   1.01096 |   0 |   0 |   45.6438  |
| Mercer       |  16.5616  |   8.13425 |  36.9973  |   2.28767 | 142.74    |  16.811   |  14.1726  |   0 |   0 |  237.704   |
| Metcalfe     |   3.72877 |   5.67123 |  84.137   |   0       |  14.2822  |  14.4438  |   9.16712 |   0 |   0 |  131.43    |
| Monroe       |   9.16712 |   4.09863 |  84.137   |   0       |   3.72877 |   0       |   0       |   0 |   0 |  101.132   |
| Montgomery   |  11.3123  |   5.82192 |   6.93699 |  53.2548  |   5.05205 |  19.9014  |   7.50959 |   0 |   0 |  109.789   |
| Nicholas     |  19.9014  |   5.82192 |   6.93699 |  12.0959  |  11.3041  |  11.3123  |   4.78082 |   0 |   0 |   72.1534  |
| Oldham       |  43.6603  |   7.64384 | 791.8     |  24.9863  |   8.83836 |   0       |   0       |   0 |   0 |  876.929   |
| Owen         |   5.31781 |   9.05479 |  14.4329  |  19.5726  |   3.60548 |   7.64384 |  13.4685  |   0 |   0 |   73.0959  |
| Owsley       |  10.1151  |  49.3233  |   3.47397 |   3.14795 |   2       |  17.9397  |   0       |   0 |   0 |   86       |
| Pendleton    |   2.47123 |   4.56438 | 101.088   |   3.60548 |  11.3041  |  76.674   |   0       |   0 |   0 |  199.707   |
| Pike         |  92.5808  |  26.9973  |   1.11507 |  14.7863  |   7.25205 |   0       |   0       |   0 |   0 |  142.732   |
| Powell       |   7.50959 |  53.2548  |   4.90959 |   2       |   5.05205 |  11.3123  |   1.01096 |   0 |   0 |   85.0493  |
| Robertson    |   4.78082 |   4.56438 |  12.0959  |  11.3041  |   8.44918 |  19.9014  |   0       |   0 |   0 |   61.0958  |
| Rockcastle   |  17.7781  |   2.28767 |   3.14795 |   8.3863  |   9.9726  |  28.0986  |  54.3836  |   0 |   0 |  124.055   |
| Russell      |   3.96164 |   5.67123 |   5.89041 |   3.7726  |   0       |  54.3836  |   1.53699 |   0 |   0 |   75.2164  |
| Shelby       |  24.9863  |   8.13425 |  14.4329  |   7.64384 | 791.8     |  43.6603  |  27.0274  |   0 |   0 |  917.685   |
| Simpson      |  12.1123  |   4.09863 |  27.5671  |  75.4685  |   0       |   0       |   0       |   0 |   0 |  119.247   |
| Spencer      |  27.0274  |   8.13425 |  30.6192  | 791.8     |  27.0274  |  24.9863  |   0       |   0 |   0 |  909.595   |
| Taylor       |  20.0712  |   5.67123 |   5.89041 |  14.2822  |  16.611   |  41.4321  |   0       |   0 |   0 |  103.958   |
| Todd         |   9.56986 | 110.699   |  27.5671  |  15.3205  |   0       |   0       |   0       |   0 |   0 |  163.156   |
| Trigg        |  17.1178  |  48.0438  |  92.337   | 110.699   |   8.8411  |  48.3452  |   0       |   0 |   0 |  325.384   |
| Trimble      |   8.83836 |   9.05479 |   7.64384 |  43.6603  |   0       |   0       |   0       |   0 |   0 |   69.1973  |
| Union        |   8.50685 |  27.2164  |  78.0712  |  19.9671  |   0       |   0       |   0       |   0 |   0 |  133.762   |
| Warren       |  75.4685  |   4.09863 |  84.137   |  15.2904  |   4.68767 |  27.5671  |  12.1123  |   0 |   0 |  223.362   |
| Washington   |  16.811   |   8.13425 |  36.9973  |  41.4321  |  16.5616  |  27.0274  |   0       |   0 |   0 |  146.964   |
| Wayne        |   1.53699 |   3.7726  |   3.07123 |  54.3836  |   3.96164 |   0       |   0       |   0 |   0 |   66.726   |
| Webster      |  19.9671  |  48.0438  |  27.2164  |  78.0712  | 148.436   |  16.8055  |   8.50685 |   0 |   0 |  347.047   |
| Whitley      |  10.8849  |  64.5425  |  11.8685  |   8.3863  |   3.07123 |   0       |   0       |   0 |   0 |   98.7534  |
| Wolfe        |   1.01096 |  49.3233  |   2       |  21.5014  |   5.05205 |   4.0411  |   7.50959 |   0 |   0 |   90.4384  |
| Woodford     |  14.1726  |   8.13425 | 196.234   |  14.4329  | 142.74    |  16.5616  |  13.4685  |   0 |   0 |  405.744   |



labeled_formulas: J=B*I
J=C*I
J=D*I
J=E*I
J=F*I
J=G*I
J=H*I
J=I*K
J=(B+C)*I
J=(B+D)*I
J=(B+E)*I
J=(B+F)*I
J=(B+G)*I
J=(B-H)*I
J=(B*I)*K
J=(C+D)*I
J=(C+E)*I
J=(C+F)*I
J=(C+G)*I
J=(C+H)*I
J=(C*I)*K
J=(D+E)*I
J=(D+F)*I
J=(D+G)*I
J=(D+H)*I
J=(D*I)*K
J=(E+F)*I
J=(E-G)*I
J=(E+H)*I
J=(E*I)*K
J=(F+G)*I
J=(F+H)*I
J=(F*I)*K
J=(G+H)*I
J=(G*I)*K
J=(H*I)*K
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_216714___xlsx_e79043a777d3a93607c3a554d9f7d3717e2ba898.xlsx___line_717