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 |
|----:|:---------------------------------------|:---------|-----------------:|-----------------:|----:|-----------------:|-----------------:|-----------------:|-----------------:|-----------------:|
|   1 | Accomack                               | Eastern  |  33730           |  14930           |   0 |  87917           | 136577           | 119712           |  16865           | 136577           |
|   3 | Albemarle                              | Piedmont | 310101           |  16056           |   0 | 619069           | 945226           | 790176           | 155050           | 945226           |
|   5 | Alleghany/Covington                    | Piedmont |  30491           |  14092           |   0 |  89264           | 133847           | 118602           |  15245.5         | 133847           |
|   7 | Amelia                                 | Central  |  12464           |  10786           |   0 |  42617           |  65867           |  59635           |   6232           |  65867           |
|   9 | Amherst                                | Piedmont |  89110           |  20175           |   0 | 182867           | 292152           | 247597           |  44555           | 292152           |
|  11 | Appomattox                             | Piedmont |  41150           |      0           |   0 |  42804           |  83954           |  63379           |  20575           |  83954           |
|  13 | Arlington                              | Northern | 678571           | 698684           |   0 |      1.24253e+06 |      2.61978e+06 |      2.2805e+06  | 339286           |      2.61978e+06 |
|  15 | Augusta/ Staunton/ Waynesboro          | Piedmont | 354400           |  69324           |   0 | 567342           | 991066           | 813866           | 177200           | 991066           |
|  17 | Bath                                   | Piedmont |      0           |   2472           |   0 |      0           |   2472           |   2472           |      0           |   2472           |
|  19 | Bedford                                | Piedmont | 235024           |  16254           |   0 | 415575           | 666853           | 549341           | 117512           | 666853           |
|  21 | Bland                                  | Western  |      0           |      0           |   0 |  10308           |  10308           |  10308           |      0           |  10308           |
|  23 | Botetourt                              | Piedmont |  84511           |  30762           |   0 | 183857           | 299130           | 256874           |  42255.5         | 299130           |
|  25 | Brunswick                              | Eastern  |  15396           |      0           |   0 |  54000           |  69396           |  61698           |   7698           |  69396           |
|  27 | Buchanan                               | Western  |      0           |      0           |   0 |      0           |      0           |      0           |      0           |      0           |
|  29 | Buckingham                             | Central  |  15252           |      0           |   0 |  33404           |  48656           |  41030           |   7626           |  48656           |
|  31 | Campbell                               | Piedmont | 127879           |   3435           |   0 | 349019           | 480333           | 416394           |  63939.5         | 480333           |
|  33 | Caroline                               | Central  | 322609           |  30417           |   0 | 381250           | 734276           | 572972           | 161304           | 734276           |
|  35 | Carroll                                | Western  | 147489           |   6604           |   0 | 202141           | 356234           | 282490           |  73744.5         | 356234           |
|  36 | Charles City                           | Central  |   9670           |  45338           |   0 |  97159           | 152167           | 147332           |   4835           | 152167           |
|  37 | Charlotte                              | Piedmont |   7822           |      0           |   0 |  20621           |  28443           |  24532           |   3911           |  28443           |
|  41 | Chesterfield/Colonial Heights          | Central  |      2.15224e+06 | 188142           |   0 |      1.58058e+06 |      3.92096e+06 |      2.84484e+06 |      1.07612e+06 |      3.92096e+06 |
|  43 | Clarke                                 | Northern |  39874           |   9974           |   0 | 130515           | 180363           | 160426           |  19937           | 180363           |
|  45 | Craig                                  | Piedmont |   5139           |  89813           |   0 |  22213           | 117165           | 114596           |   2569.5         | 117165           |
|  47 | Culpeper                               | Northern | 352482           | 741965           |   0 | 604504           |      1.69895e+06 |      1.52271e+06 | 176241           |      1.69895e+06 |
|  49 | Cumberland                             | Central  |  54214           |      0           |   0 |  68478           | 122692           |  95585           |  27107           | 122692           |
|  51 | Dickenson                              | Western  |  14498           |  10935           |   0 |  10903           |  36336           |  29087           |   7249           |  36336           |
|  53 | Dinwiddie                              | Eastern  |  58389           |   1802           |   0 | 112380           | 172571           | 143376           |  29194.5         | 172571           |
|  57 | Essex                                  | Central  | 165644           |  18095           |   0 | 165217           | 348956           | 266134           |  82822           | 348956           |
|  59 | Fairfax Co./Fairfax City/ Falls Church | Northern |      5.65302e+06 |      2.03342e+06 |   0 |      1.75279e+07 |      2.52144e+07 |      2.23879e+07 |      2.82651e+06 |      2.52144e+07 |
|  61 | Fauquier                               | Northern | 228112           |  66936           |   0 | 426459           | 721507           | 607451           | 114056           | 721507           |
|  63 | Floyd                                  | Western  |  23968           |      0           |   0 |   8089           |  32057           |  20073           |  11984           |  32057           |
|  65 | Fluvanna                               | Central  |  45687           |  13922           |   0 | 134883           | 194492           | 171648           |  22843.5         | 194492           |
|  67 | Franklin Co.                           | Piedmont | 163196           |  19184           |   0 | 256508           | 438888           | 357290           |  81598           | 438888           |
|  69 | Frederick                              | Northern | 133105           |  19351           |   0 | 509083           | 661539           | 594986           |  66552.5         | 661539           |
|  71 | Giles                                  | Western  |  51713           |   6796           |   0 |  56894           | 115403           |  89546.5         |  25856.5         | 115403           |
|  73 | Gloucester                             | Eastern  | 164670           |  26300           |   0 | 230192           | 421162           | 338827           |  82335           | 421162           |
|  75 | Goochland                              | Central  |  38464           |      0           |   0 | 187818           | 226282           | 207050           |  19232           | 226282           |
|  77 | Grayson                                | Western  |  22511           |      0           |   0 |  30621           |  53132           |  41876.5         |  11255.5         |  53132           |
|  79 | Greene                                 | Northern |  64439           |   2191           |   0 |  45451           | 112081           |  79861.5         |  32219.5         | 112081           |
|  81 | Greensville/ Emporia                   | Eastern  |  51546           |      0           |   0 | 171044           | 222590           | 196817           |  25773           | 222590           |
|  83 | Halifax/ So. Boston                    | Piedmont |  60688           |  22059           |   0 |  66940           | 149687           | 119343           |  30344           | 149687           |
|  85 | Hanover                                | Central  | 245823           |  49808           |   0 | 648909           | 944540           | 821628           | 122912           | 944540           |
|  87 | Henrico                                | Central  |      3.5702e+06  | 226715           |   0 |      3.1795e+06  |      6.97642e+06 |      5.19132e+06 |      1.7851e+06  |      6.97642e+06 |
|  89 | Henry/Martinsville                     | Piedmont | 252763           |      0           |   0 | 315515           | 568278           | 441896           | 126382           | 568278           |
|  91 | Highland                               | Piedmont |      0           |      0           |   0 |      0           |      0           |      0           |      0           |      0           |
|  93 | Isle Of Wight                          | Eastern  |  46451           |      0           |   0 |  95901           | 142352           | 119126           |  23225.5         | 142352           |
|  95 | James City                             | Eastern  | 229027           | 141304           |   0 | 407433           | 777764           | 663250           | 114514           | 777764           |
|  97 | King & Queen                           | Central  |  10456           |   6568           |   0 |  40015           |  57039           |  51811           |   5228           |  57039           |
|  99 | King George                            | Northern | 198311           |  50257           |   0 | 292071           | 540639           | 441484           |  99155.5         | 540639           |
| 101 | King William                           | Central  |  37422           |  23797           |   0 | 137457           | 198676           | 179965           |  18711           | 198676           |
| 103 | Lancaster                              | Central  |  17390           |  14777           |   0 |  36700           |  68867           |  60172           |   8695           |  68867           |
| 105 | Lee                                    | Western  |  14854           |      0           |   0 |  12906           |  27760           |  20333           |   7427           |  27760           |
| 107 | Loudoun                                | Northern |      1.37137e+06 | 826157           |   0 |      1.29259e+06 |      3.49012e+06 |      2.80444e+06 | 685687           |      3.49012e+06 |
| 109 | Louisa                                 | Northern | 128085           |      0           |   0 |  26409           | 154494           |  90451.5         |  64042.5         | 154494           |
| 111 | Lunenburg                              | Central  |   1345           |      0           |   0 |  17903           |  19248           |  18575.5         |    672.5         |  19248           |
| 113 | Madison                                | Northern |  33225           |  38404           |   0 |  20463           |  92092           |  75479.5         |  16612.5         |  92092           |
| 115 | Mathews                                | Eastern  |  32983           |      0           |   0 |  38476           |  71459           |  54967.5         |  16491.5         |  71459           |
| 117 | Mecklenburg                            | Piedmont |  51656           |      0           |   0 |  76079           | 127735           | 101907           |  25828           | 127735           |
| 119 | Middlesex                              | Central  |  83350           |  53347           |   0 |  90153           | 226850           | 185175           |  41675           | 226850           |
| 121 | Montgomery                             | Western  | 315524           |  55645           |   0 | 479145           | 850314           | 692552           | 157762           | 850314           |
| 125 | Nelson                                 | Piedmont |  18652           |   1314           |   0 |  36928           |  56894           |  47568           |   9326           |  56894           |
| 127 | New Kent                               | Central  | 153009           |  41705           |   0 |  75201           | 269915           | 193410           |  76504.5         | 269915           |
| 131 | Northampton                            | Eastern  |  91035           |     99           |   0 | 158501           | 249635           | 204118           |  45517.5         | 249635           |
| 133 | Northumberland                         | Central  |   2643           |    618           |   0 |  24600           |  27861           |  26539.5         |   1321.5         |  27861           |
| 135 | Nottoway                               | Central  |  11228           |      0           |   0 |  90323           | 101551           |  95937           |   5614           | 101551           |
| 137 | Orange                                 | Northern | 128324           |  40849           |   0 | 227729           | 396902           | 332740           |  64162           | 396902           |
| 139 | Page                                   | Northern |      0           |      0           |   0 |  79272           |  79272           |  79272           |      0           |  79272           |
| 141 | Patrick                                | Western  |  32127           |      0           |   0 |  69345           | 101472           |  85408.5         |  16063.5         | 101472           |
| 143 | Pittsylvania                           | Piedmont |  93054           |  13712           |   0 | 195832           | 302598           | 256071           |  46527           | 302598           |
| 145 | Powhatan                               | Central  |  86506           |  29199           |   0 |  75335           | 191040           | 147787           |  43253           | 191040           |
| 147 | Prince Edward                          | Central  |  57503           |      0           |   0 |  78785           | 136288           | 107536           |  28751.5         | 136288           |
| 149 | Prince George                          | Eastern  |  59365           |      0           |   0 | 105921           | 165286           | 135604           |  29682.5         | 165286           |
| 153 | Prince William                         | Northern |      2.90916e+06 | 936695           |   0 |      2.73156e+06 |      6.57741e+06 |      5.12283e+06 |      1.45458e+06 |      6.57741e+06 |
| 155 | Pulaski                                | Western  | 123642           |   6381           |   0 | 135800           | 265823           | 204002           |  61821           | 265823           |
| 157 | Rappahannock                           | Northern |   6296           |      0           |   0 |  84604           |  90900           |  87752           |   3148           |  90900           |
| 159 | Richmond Co.                           | Central  |   7191           |      0           |   0 |  65003           |  72194           |  68598.5         |   3595.5         |  72194           |
| 161 | Roanoke Co./ Salem                     | Piedmont | 834636           | 110630           |   0 | 973958           |      1.91922e+06 |      1.50191e+06 | 417318           |      1.91922e+06 |
| 163 | Rockbridge/ Buena Vista/ Lexington     | Piedmont |  28624           |  12737           |   0 |  44467           |  85828           |  71516           |  14312           |  85828           |
| 165 | Rockingham/ Harrisonburg               | Northern | 373685           |  34803           |   0 | 432522           | 841010           | 654168           | 186842           | 841010           |
| 167 | Russell                                | Western  |  17349           |    845           |   0 |  63597           |  81791           |  73116.5         |   8674.5         |  81791           |
| 169 | Scott                                  | Western  |   9927           |      0           |   0 |   2844           |  12771           |   7807.5         |   4963.5         |  12771           |
| 171 | Shenandoah                             | Northern |  52512           |   1622           |   0 | 145807           | 199941           | 173685           |  26256           | 199941           |
| 173 | Smyth                                  | Western  |  68528           |   3905           |   0 |  74270           | 146703           | 112439           |  34264           | 146703           |
| 175 | Southampton                            | Eastern  |  30346           |   3382           |   0 |   8192           |  41920           |  26747           |  15173           |  41920           |
| 177 | Spotsylvania                           | Northern | 688458           |  55562           |   0 | 207023           | 951043           | 606814           | 344229           | 951043           |
| 179 | Stafford                               | Northern | 916576           | 276184           |   0 | 398798           |      1.59156e+06 |      1.13327e+06 | 458288           |      1.59156e+06 |
| 181 | Surry                                  | Eastern  |  15199           |    406           |   0 |  48930           |  64535           |  56935.5         |   7599.5         |  64535           |
| 183 | Sussex                                 | Eastern  |   5026           |   5878           |   0 | 103451           | 114355           | 111842           |   2513           | 114355           |
| 185 | Tazewell                               | Western  | 153101           |    282           |   0 | 122220           | 275603           | 199052           |  76550.5         | 275603           |
| 187 | Warren                                 | Northern | 270015           |   2460           |   0 | 419734           | 692209           | 557202           | 135008           | 692209           |
| 191 | Washington                             | Western  |  49731           |   1933           |   0 |  90240           | 141904           | 117038           |  24865.5         | 141904           |
| 193 | Westmoreland                           | Central  |  55992           |   3412           |   0 | 159438           | 218842           | 190846           |  27996           | 218842           |
| 195 | Wise                                   | Western  | 106034           |   3240           |   0 |  61301           | 170575           | 117558           |  53017           | 170575           |
| 197 | Wythe                                  | Western  |  76746           |  15188           |   0 | 144314           | 236248           | 197875           |  38373           | 236248           |
| 199 | York/ Poquoson                         | Eastern  | 173966           | 113336           |   0 | 271951           | 559253           | 472270           |  86983           | 559253           |
| 510 | Alexandria                             | Northern |      1.26879e+06 | 298646           |   0 |      1.80102e+06 |      3.36845e+06 |      2.73406e+06 | 634394           |      3.36845e+06 |
| 520 | Bristol                                | Western  | 288620           |  38183           |   0 | 341240           | 668043           | 523733           | 144310           | 668043           |
| 540 | Charlottesville                        | Piedmont | 595200           |  16138           |   0 | 661079           |      1.27242e+06 | 974817           | 297600           |      1.27242e+06 |
| 550 | Chesapeake                             | Eastern  |      1.81022e+06 |  47411           |   0 |      1.73881e+06 |      3.59644e+06 |      2.69133e+06 | 905109           |      3.59644e+06 |
| 590 | Danville                               | Piedmont | 300485           |  52272           |   0 | 578588           | 931345           | 781102           | 150242           | 931345           |
| 620 | Franklin                               | Eastern  |  90336           |      0           |   0 |  69723           | 160059           | 114891           |  45168           | 160059           |
| 630 | Fredericksburg                         | Northern | 660814           | 387038           |   0 | 285300           |      1.33315e+06 |      1.00274e+06 | 330407           |      1.33315e+06 |
| 640 | Galax                                  | Western  |  31515           |   2213           |   0 |  46216           |  79944           |  64186.5         |  15757.5         |  79944           |
| 650 | Hampton                                | Eastern  |      2.15596e+06 |      0           |   0 | 557307           |      2.71327e+06 |      1.63529e+06 |      1.07798e+06 |      2.71327e+06 |
| 670 | Hopewell                               | Central  | 361794           | 126502           |   0 | 212728           | 701024           | 520127           | 180897           | 701024           |
| 680 | Lynchburg                              | Piedmont | 506299           |  32952           |   0 | 836669           |      1.37592e+06 |      1.12277e+06 | 253150           |      1.37592e+06 |
| 683 | Manassas                               | Northern | 371946           |  58784           |   0 |      1.26408e+06 |      1.69481e+06 |      1.50883e+06 | 185973           |      1.69481e+06 |
| 685 | Manassas Park                          | Northern |  60277           |  46210           |   0 | 199841           | 306328           | 276190           |  30138.5         | 306328           |
| 700 | Newport News                           | Eastern  |      3.00817e+06 | 364796           |   0 |      2.58384e+06 |      5.9568e+06  |      4.45272e+06 |      1.50408e+06 |      5.9568e+06  |
| 710 | Norfolk                                | Eastern  |      3.20566e+06 |  34849           |   0 |      3.96673e+06 |      7.20724e+06 |      5.60441e+06 |      1.60283e+06 |      7.20724e+06 |
| 720 | Norton                                 | Western  |  17357           |      0           |   0 |  10257           |  27614           |  18935.5         |   8678.5         |  27614           |
| 730 | Petersburg                             | Central  | 473452           |  61590           |   0 | 484640           |      1.01968e+06 | 782956           | 236726           |      1.01968e+06 |
| 740 | Portsmouth                             | Eastern  |      1.68625e+06 | 324753           |   0 | 918935           |      2.92994e+06 |      2.08681e+06 | 843126           |      2.92994e+06 |
| 750 | Radford                                | Western  |  86579           |   7269           |   0 |  33115           | 126963           |  83673.5         |  43289.5         | 126963           |
| 760 | Richmond                               | Central  |      4.13518e+06 | 162845           |   0 |      3.06126e+06 |      7.35928e+06 |      5.29169e+06 |      2.06759e+06 |      7.35928e+06 |
| 770 | Roanoke                                | Piedmont |      2.04373e+06 | 406672           |   0 |      1.69833e+06 |      4.14873e+06 |      3.12686e+06 |      1.02187e+06 |      4.14873e+06 |
| 800 | Suffolk                                | Eastern  | 251198           |  73104           |   0 | 441852           | 766154           | 640555           | 125599           | 766154           |
| 810 | Virginia Beach                         | Eastern  |      1.83418e+06 |  55332           |   0 |      5.56925e+06 |      7.45876e+06 |      6.54167e+06 | 917090           |      7.45876e+06 |
| 830 | Williamsburg                           | Eastern  |  79178           |  35944           |   0 | 189053           | 304175           | 264586           |  39589           | 304175           |
| 840 | Winchester                             | Northern | 145707           |  34694           |   0 | 234398           | 414799           | 341946           |  72853.5         | 414799           |



label: K=I+J
labeled_formulas: K=I+J
output: {"Arithmetic-Relationship": ["K=I+J"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_518628___xlsx_3092d67527a3e1cad7a3b0bdd14ce703917ebf64.xlsx___line_232