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 |
|----:|:---------------------|----------:|----------:|---------:|
|  10 | Anderson County      |  32990000 |  32990000 |        0 |
|  11 | Clinton City         |   4962000 |   4962000 |        0 |
|  12 | Oak Ridge City       |  23105000 |  23105000 |        0 |
|  20 | Bedford County       |  52444000 |  52444000 |        0 |
|  30 | Benton County        |  12652000 |  12652000 |        0 |
|  40 | Bledsoe County       |  12061000 |  12061000 |        0 |
|  50 | Blount County        |  48556000 |  48556000 |        0 |
|  51 | Alcoa City           |   9395000 |   9395000 |        0 |
|  52 | Maryville City       |  23744000 |  23744000 |        0 |
|  60 | Bradley County       |  51614000 |  51614000 |        0 |
|  61 | Cleveland City       |  30278000 |  30278000 |        0 |
|  70 | Campbell County      |  29936000 |  29936000 |        0 |
|  80 | Cannon County        |  12115000 |  12115000 |        0 |
|  90 | Carroll County       |   2041000 |   2041000 |        0 |
|  92 | H Rock-Bruceton SSD  |   4008000 |   4008000 |        0 |
|  93 | Huntingdon SSD       |   7531000 |   7531000 |        0 |
|  94 | McKenzie SSD         |   7435000 |   7435000 |        0 |
|  95 | South Carroll Co SSD |   2098000 |   2098000 |        0 |
|  97 | West Carroll Co SSD  |   5554000 |   5554000 |        0 |
| 100 | Carter County        |  29517000 |  29517000 |        0 |
| 101 | Elizabethton City    |  14314000 |  14314000 |        0 |
| 110 | Cheatham County      |  33299000 |  33299000 |        0 |
| 120 | Chester County       |  18162000 |  18162000 |        0 |
| 130 | Claiborne County     |  24506000 |  24506000 |        0 |
| 140 | Clay County          |   7171000 |   7171000 |        0 |
| 150 | Cocke County         |  26073000 |  26073000 |        0 |
| 151 | Newport City         |   3709000 |   3709000 |        0 |
| 160 | Coffee County        |  22862000 |  22862000 |        0 |
| 161 | Manchester City      |   7853000 |   7853000 |        0 |
| 162 | Tullahoma City       |  17709000 |  17709000 |        0 |
| 170 | Crockett County      |  12908000 |  12908000 |        0 |
| 171 | Alamo City           |   4088000 |   4088000 |        0 |
| 172 | Bells City           |   2608000 |   2608000 |        0 |
| 180 | Cumberland County    |  34511000 |  34511000 |        0 |
| 190 | Davidson County      | 300980000 | 316687000 | 15707000 |
| 200 | Decatur County       |   9249000 |   9249000 |        0 |
| 210 | DeKalb County        |  17023000 |  17023000 |        0 |
| 220 | Dickson County       |  41130000 |  41130000 |        0 |
| 230 | Dyer County          |  21370000 |  21370000 |        0 |
| 231 | Dyersburg City       |  13637000 |  13637000 |        0 |
| 240 | Fayette County       |  16821000 |  16821000 |        0 |
| 250 | Fentress County      |  12693000 |  12693000 |        0 |
| 260 | Franklin County      |  26768000 |  26768000 |        0 |
| 271 | Humboldt City        |   7084000 |   7084000 |        0 |
| 272 | Milan SSD            |  12004000 |  12004000 |        0 |
| 273 | Trenton SSD          |   7981000 |   7981000 |        0 |
| 274 | Bradford SSD         |   3713000 |   3713000 |        0 |
| 275 | Gibson County SSD    |  23062000 |  23062000 |        0 |
| 280 | Giles County         |  18886000 |  18886000 |        0 |
| 290 | Grainger County      |  21546000 |  21546000 |        0 |
| 300 | Greene County        |  33440000 |  33440000 |        0 |
| 301 | Greeneville City     |  15660000 |  15660000 |        0 |
| 310 | Grundy County        |  12357000 |  12357000 |        0 |
| 320 | Hamblen County       |  56527000 |  56527000 |        0 |
| 330 | Hamilton County      | 173759000 | 173759000 |        0 |
| 340 | Hancock County       |   7168000 |   7168000 |        0 |
| 350 | Hardeman County      |  20836000 |  20836000 |        0 |
| 360 | Hardin County        |  16612000 |  16612000 |        0 |
| 370 | Hawkins County       |  37361000 |  37361000 |        0 |
| 371 | Rogersville City     |   3736000 |   3736000 |        0 |
| 380 | Haywood County       |  16479000 |  16479000 |        0 |
| 390 | Henderson County     |  23490000 |  23490000 |        0 |
| 391 | Lexington City       |   5049000 |   5049000 |        0 |
| 400 | Henry County         |  16383000 |  16383000 |        0 |
| 401 | Paris SSD            |   8874000 |   8874000 |        0 |
| 410 | Hickman County       |  22301000 |  22301000 |        0 |
| 420 | Houston County       |   8865000 |   8865000 |        0 |
| 430 | Humphreys County     |  15410000 |  15410000 |        0 |
| 440 | Jackson County       |   9941000 |   9941000 |        0 |
| 450 | Jefferson County     |  38406000 |  38406000 |        0 |
| 460 | Johnson County       |  13352000 |  13352000 |        0 |
| 470 | Knox County          | 236373000 | 236373000 |        0 |
| 480 | Lake County          |   5146000 |   5146000 |        0 |
| 490 | Lauderdale County    |  24476000 |  24476000 |        0 |
| 500 | Lawrence County      |  38602000 |  38602000 |        0 |
| 510 | Lewis County         |  10095000 |  10095000 |        0 |
| 520 | Lincoln County       |  21793000 |  21793000 |        0 |
| 521 | Fayetteville City    |   7798000 |   7798000 |        0 |
| 530 | Loudon County        |  21692000 |  21692000 |        0 |
| 531 | Lenoir City          |  11357000 |  11357000 |        0 |
| 540 | McMinn County        |  26804000 |  26804000 |        0 |
| 541 | Athens City          |   8636000 |   8636000 |        0 |
| 542 | Etowah City          |   2008000 |   2008000 |        0 |
| 550 | McNairy County       |  24695000 |  24695000 |        0 |
| 560 | Macon County         |  24985000 |  24985000 |        0 |
| 570 | Madison County       |  51637000 |  51637000 |        0 |
| 580 | Marion County        |  20690000 |  20690000 |        0 |
| 581 | Richard City SSD     |   1422000 |   1422000 |        0 |
| 590 | Marshall County      |  29638000 |  29638000 |        0 |
| 600 | Maury County         |  59242000 |  59242000 |        0 |
| 610 | Meigs County         |  11108000 |  11108000 |        0 |
| 620 | Monroe County        |  29189000 |  29189000 |        0 |
| 621 | Sweetwater City      |   8149000 |   8149000 |        0 |
| 630 | Montgomery County    | 199145000 | 199145000 |        0 |
| 640 | Moore County         |   4873000 |   4873000 |        0 |
| 650 | Morgan County        |  18377000 |  18377000 |        0 |
| 660 | Obion County         |  18088000 |  18088000 |        0 |
| 661 | Union City           |   8691000 |   8691000 |        0 |
| 670 | Overton County       |  19211000 |  19211000 |        0 |
| 680 | Perry County         |   6770000 |   6770000 |        0 |
| 690 | Pickett County       |   3903000 |   3903000 |        0 |
| 700 | Polk County          |  13857000 |  13857000 |        0 |
| 710 | Putnam County        |  56224000 |  56224000 |        0 |
| 720 | Rhea County          |  24321000 |  24321000 |        0 |
| 721 | Dayton City          |   4917000 |   4917000 |        0 |
| 730 | Roane County         |  32229000 |  32229000 |        0 |
| 740 | Robertson County     |  71121000 |  71121000 |        0 |
| 750 | Rutherford County    | 232612000 | 232612000 |        0 |
| 751 | Murfreesboro City    |  46795000 |  46795000 |        0 |
| 760 | Scott County         |  18135000 |  18135000 |        0 |
| 761 | Oneida SSD           |   7753000 |   7753000 |        0 |
| 770 | Sequatchie County    |  13073000 |  13073000 |        0 |
| 780 | Sevier County        |  43278000 |  48175000 |  4897000 |
| 792 | Shelby County        | 650329000 | 650329000 |        0 |
| 793 | Arlington City       |  23198000 |  23198000 |        0 |
| 794 | Bartlett City        |  44528000 |  44528000 |        0 |
| 795 | Collierville City    |  45303000 |  45303000 |        0 |
| 796 | Germantown City      |  29632000 |  29632000 |        0 |
| 797 | Lakeland City        |   9726000 |   9726000 |        0 |
| 798 | Millington City      |  13313000 |  13313000 |        0 |
| 800 | Smith County         |  17543000 |  17543000 |        0 |
| 810 | Stewart County       |  12337000 |  12337000 |        0 |
| 820 | Sullivan County      |  39669000 |  39669000 |        0 |
| 821 | Bristol City         |  16885000 |  16885000 |        0 |
| 822 | Kingsport City       |  32369000 |  32369000 |        0 |
| 830 | Sumner County        | 153854000 | 153854000 |        0 |
| 840 | Tipton County        |  61993000 |  61993000 |        0 |
| 850 | Trousdale County     |   7930000 |   7930000 |        0 |
| 860 | Unicoi County        |  13128000 |  13128000 |        0 |
| 870 | Union County         |  31275000 |  31275000 |        0 |
| 880 | Van Buren County     |   5062000 |   5062000 |        0 |
| 890 | Warren County        |  36436000 |  36436000 |        0 |
| 900 | Washington County    |  36511000 |  36511000 |        0 |
| 901 | Johnson City         |  34407000 |  34407000 |        0 |
| 910 | Wayne County         |  14410000 |  14410000 |        0 |
| 920 | Weakley County       |  22665000 |  22665000 |        0 |
| 930 | White County         |  22848000 |  22848000 |        0 |
| 940 | Williamson County    | 139768000 | 139768000 |        0 |
| 941 | Franklin SSD         |  14845000 |  14845000 |        0 |
| 950 | Wilson County        |  85901000 |  85901000 |        0 |
| 951 | Lebanon SSD          |  18390000 |  18390000 |        0 |



label: E=D-C
labeled_formulas: E=D-C
output: {"Arithmetic-Relationship": ["E=D-C"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_667105___xlsx_63ec4976752dad4b600f15615073b09b613d160a.xlsx___line_337