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 |
|----:|:-----------------------------------|:---------|-----------------:|-----------------:|----:|----:|----:|----:|----:|
|   1 | Accomack                           | Eastern  |      1.02353e+06 |      1.02353e+06 |   0 |   0 |   0 |   0 |   0 |
|   3 | Albemarle                          | Piedmont | 446493           | 446493           |   0 |   0 |   0 |   0 |   0 |
|   5 | Alleghany/Covington                | Piedmont | 476175           | 476175           |   0 |   0 |   0 |   0 |   0 |
|   7 | Amelia                             | Central  | 179058           | 179058           |   0 |   0 |   0 |   0 |   0 |
|   9 | Amherst                            | Piedmont | 446248           | 446248           |   0 |   0 |   0 |   0 |   0 |
|  11 | Appomattox                         | Piedmont | 317112           | 317112           |   0 |   0 |   0 |   0 |   0 |
|  13 | Arlington                          | Northern | 271752           | 271752           |   0 |   0 |   0 |   0 |   0 |
|  15 | Augusta/ Staunton/ Waynesboro      | Piedmont |      1.14535e+06 |      1.14535e+06 |   0 |   0 |   0 |   0 |   0 |
|  17 | Bath                               | Piedmont |  62420.4         |  62420.4         |   0 |   0 |   0 |   0 |   0 |
|  19 | Bedford                            | Piedmont | 529004           | 529004           |   0 |   0 |   0 |   0 |   0 |
|  21 | Bland                              | Western  |  97743.5         |  97743.5         |   0 |   0 |   0 |   0 |   0 |
|  23 | Botetourt                          | Piedmont | 175949           | 175949           |   0 |   0 |   0 |   0 |   0 |
|  25 | Brunswick                          | Eastern  | 587579           | 587579           |   0 |   0 |   0 |   0 |   0 |
|  27 | Buchanan                           | Western  |      1.13324e+06 |      1.13324e+06 |   0 |   0 |   0 |   0 |   0 |
|  29 | Buckingham                         | Central  | 374539           | 374539           |   0 |   0 |   0 |   0 |   0 |
|  31 | Campbell                           | Piedmont | 911553           | 911553           |   0 |   0 |   0 |   0 |   0 |
|  33 | Caroline                           | Central  | 233162           | 233162           |   0 |   0 |   0 |   0 |   0 |
|  35 | Carroll                            | Western  | 808327           | 808327           |   0 |   0 |   0 |   0 |   0 |
|  36 | Charles City                       | Central  |  98915.3         |  98915.3         |   0 |   0 |   0 |   0 |   0 |
|  37 | Charlotte                          | Piedmont | 337357           | 337357           |   0 |   0 |   0 |   0 |   0 |
|  41 | Chesterfield/Colonial Heights      | Central  | 990699           | 990699           |   0 |   0 |   0 |   0 |   0 |
|  43 | Clarke                             | Northern |  45929.7         |  45929.7         |   0 |   0 |   0 |   0 |   0 |
|  45 | Craig                              | Piedmont |  88491.5         |  88491.5         |   0 |   0 |   0 |   0 |   0 |
|  47 | Culpeper                           | Northern | 280971           | 280971           |   0 |   0 |   0 |   0 |   0 |
|  49 | Cumberland                         | Central  | 195726           | 195726           |   0 |   0 |   0 |   0 |   0 |
|  51 | Dickenson                          | Western  | 649494           | 649494           |   0 |   0 |   0 |   0 |   0 |
|  53 | Dinwiddie                          | Eastern  | 427003           | 427003           |   0 |   0 |   0 |   0 |   0 |
|  57 | Essex                              | Central  | 217833           | 217833           |   0 |   0 |   0 |   0 |   0 |
|  59 | Fairfax Co./ City/ Falls Church    | Northern | 644833           | 644833           |   0 |   0 |   0 |   0 |   0 |
|  61 | Fauquier                           | Northern | 161062           | 161062           |   0 |   0 |   0 |   0 |   0 |
|  63 | Floyd                              | Western  | 252707           | 252707           |   0 |   0 |   0 |   0 |   0 |
|  65 | Fluvanna                           | Central  | 159887           | 159887           |   0 |   0 |   0 |   0 |   0 |
|  67 | Franklin Co.                       | Piedmont | 786490           | 786490           |   0 |   0 |   0 |   0 |   0 |
|  69 | Frederick                          | Northern | 272384           | 272384           |   0 |   0 |   0 |   0 |   0 |
|  71 | Giles                              | Western  | 348978           | 348978           |   0 |   0 |   0 |   0 |   0 |
|  73 | Gloucester                         | Eastern  | 290341           | 290341           |   0 |   0 |   0 |   0 |   0 |
|  75 | Goochland                          | Central  | 108840           | 108840           |   0 |   0 |   0 |   0 |   0 |
|  77 | Grayson                            | Western  | 533810           | 533810           |   0 |   0 |   0 |   0 |   0 |
|  79 | Greene                             | Northern | 135937           | 135937           |   0 |   0 |   0 |   0 |   0 |
|  81 | Greensville/ Emporia               | Eastern  | 385441           | 385441           |   0 |   0 |   0 |   0 |   0 |
|  83 | Halifax/ So. Boston                | Piedmont |      1.08458e+06 |      1.08458e+06 |   0 |   0 |   0 |   0 |   0 |
|  85 | Hanover                            | Central  | 305511           | 305511           |   0 |   0 |   0 |   0 |   0 |
|  87 | Henrico                            | Central  |      1.4783e+06  |      1.4783e+06  |   0 |   0 |   0 |   0 |   0 |
|  89 | Henry/Martinsville                 | Piedmont |      1.7003e+06  |      1.7003e+06  |   0 |   0 |   0 |   0 |   0 |
|  91 | Highland                           | Piedmont |  25566.7         |  25566.7         |   0 |   0 |   0 |   0 |   0 |
|  93 | Isle Of Wight                      | Eastern  | 265837           | 265837           |   0 |   0 |   0 |   0 |   0 |
|  95 | James City                         | Eastern  | 244951           | 244951           |   0 |   0 |   0 |   0 |   0 |
|  97 | King & Queen                       | Central  | 110291           | 110291           |   0 |   0 |   0 |   0 |   0 |
|  99 | King George                        | Northern | 184138           | 184138           |   0 |   0 |   0 |   0 |   0 |
| 101 | King William                       | Central  | 113512           | 113512           |   0 |   0 |   0 |   0 |   0 |
| 103 | Lancaster                          | Central  | 231825           | 231825           |   0 |   0 |   0 |   0 |   0 |
| 105 | Lee                                | Western  |      1.22462e+06 |      1.22462e+06 |   0 |   0 |   0 |   0 |   0 |
| 107 | Loudoun                            | Northern | 103472           | 103472           |   0 |   0 |   0 |   0 |   0 |
| 109 | Louisa                             | Northern | 459610           | 459610           |   0 |   0 |   0 |   0 |   0 |
| 111 | Lunenburg                          | Central  | 390635           | 390635           |   0 |   0 |   0 |   0 |   0 |
| 113 | Madison                            | Northern | 118529           | 118529           |   0 |   0 |   0 |   0 |   0 |
| 115 | Mathews                            | Eastern  | 103522           | 103522           |   0 |   0 |   0 |   0 |   0 |
| 117 | Mecklenburg                        | Piedmont | 832801           | 832801           |   0 |   0 |   0 |   0 |   0 |
| 119 | Middlesex                          | Central  | 157567           | 157567           |   0 |   0 |   0 |   0 |   0 |
| 121 | Montgomery                         | Western  | 798766           | 798766           |   0 |   0 |   0 |   0 |   0 |
| 125 | Nelson                             | Piedmont | 303460           | 303460           |   0 |   0 |   0 |   0 |   0 |
| 127 | New Kent                           | Central  |  58216.4         |  58216.4         |   0 |   0 |   0 |   0 |   0 |
| 131 | Northampton                        | Eastern  | 495345           | 495345           |   0 |   0 |   0 |   0 |   0 |
| 133 | Northumberland                     | Central  | 268968           | 268968           |   0 |   0 |   0 |   0 |   0 |
| 135 | Nottoway                           | Central  | 384581           | 384581           |   0 |   0 |   0 |   0 |   0 |
| 137 | Orange                             | Northern | 260301           | 260301           |   0 |   0 |   0 |   0 |   0 |
| 139 | Page                               | Northern | 424376           | 424376           |   0 |   0 |   0 |   0 |   0 |
| 141 | Patrick                            | Western  | 494469           | 494469           |   0 |   0 |   0 |   0 |   0 |
| 143 | Pittsylvania                       | Piedmont |      1.1861e+06  |      1.1861e+06  |   0 |   0 |   0 |   0 |   0 |
| 145 | Powhatan                           | Central  |  78545.5         |  78545.5         |   0 |   0 |   0 |   0 |   0 |
| 147 | Prince Edward                      | Central  | 394515           | 394515           |   0 |   0 |   0 |   0 |   0 |
| 149 | Prince George                      | Eastern  | 137281           | 137281           |   0 |   0 |   0 |   0 |   0 |
| 153 | Prince William                     | Northern | 392154           | 392154           |   0 |   0 |   0 |   0 |   0 |
| 155 | Pulaski                            | Western  | 862962           | 862962           |   0 |   0 |   0 |   0 |   0 |
| 157 | Rappahannock                       | Northern |  33514.3         |  33514.3         |   0 |   0 |   0 |   0 |   0 |
| 159 | Richmond Co.                       | Central  | 179529           | 179529           |   0 |   0 |   0 |   0 |   0 |
| 161 | Roanoke Co./ Salem                 | Piedmont | 538344           | 538344           |   0 |   0 |   0 |   0 |   0 |
| 163 | Rockbridge/ Buena Vista/ Lexington | Piedmont | 599692           | 599692           |   0 |   0 |   0 |   0 |   0 |
| 165 | Rockingham/ Harrisonburg           | Northern | 874656           | 874656           |   0 |   0 |   0 |   0 |   0 |
| 167 | Russell                            | Western  |      1.12899e+06 |      1.12899e+06 |   0 |   0 |   0 |   0 |   0 |
| 169 | Scott                              | Western  | 750387           | 750387           |   0 |   0 |   0 |   0 |   0 |
| 171 | Shenandoah                         | Northern | 466042           | 466042           |   0 |   0 |   0 |   0 |   0 |
| 173 | Smyth                              | Western  |      1.05357e+06 |      1.05357e+06 |   0 |   0 |   0 |   0 |   0 |
| 175 | Southampton                        | Eastern  | 287141           | 287141           |   0 |   0 |   0 |   0 |   0 |
| 177 | Spotsylvania                       | Northern | 330253           | 330253           |   0 |   0 |   0 |   0 |   0 |
| 179 | Stafford                           | Northern | 186594           | 186594           |   0 |   0 |   0 |   0 |   0 |
| 181 | Surry                              | Eastern  | 103327           | 103327           |   0 |   0 |   0 |   0 |   0 |
| 183 | Sussex                             | Eastern  | 208840           | 208840           |   0 |   0 |   0 |   0 |   0 |
| 185 | Tazewell                           | Western  |      1.29549e+06 |      1.29549e+06 |   0 |   0 |   0 |   0 |   0 |
| 187 | Warren                             | Northern | 284747           | 284747           |   0 |   0 |   0 |   0 |   0 |
| 191 | Washington                         | Western  | 942618           | 942618           |   0 |   0 |   0 |   0 |   0 |
| 193 | Westmoreland                       | Central  | 376831           | 376831           |   0 |   0 |   0 |   0 |   0 |
| 195 | Wise                               | Western  |      1.26157e+06 |      1.26157e+06 |   0 |   0 |   0 |   0 |   0 |
| 197 | Wythe                              | Western  | 578236           | 578236           |   0 |   0 |   0 |   0 |   0 |
| 199 | York/ Poquoson                     | Eastern  | 129965           | 129965           |   0 |   0 |   0 |   0 |   0 |
| 510 | Alexandria                         | Northern | 250359           | 250359           |   0 |   0 |   0 |   0 |   0 |
| 520 | Bristol                            | Western  | 287552           | 287552           |   0 |   0 |   0 |   0 |   0 |
| 540 | Charlottesville                    | Piedmont | 280872           | 280872           |   0 |   0 |   0 |   0 |   0 |
| 550 | Chesapeake                         | Eastern  | 834130           | 834130           |   0 |   0 |   0 |   0 |   0 |
| 590 | Danville                           | Piedmont |      1.1037e+06  |      1.1037e+06  |   0 |   0 |   0 |   0 |   0 |
| 620 | Franklin                           | Eastern  | 276285           | 276285           |   0 |   0 |   0 |   0 |   0 |
| 630 | Fredericksburg                     | Northern | 196584           | 196584           |   0 |   0 |   0 |   0 |   0 |
| 640 | Galax                              | Western  | 299233           | 299233           |   0 |   0 |   0 |   0 |   0 |
| 650 | Hampton                            | Eastern  | 830275           | 830275           |   0 |   0 |   0 |   0 |   0 |
| 670 | Hopewell                           | Central  | 332189           | 332189           |   0 |   0 |   0 |   0 |   0 |
| 680 | Lynchburg                          | Piedmont |      1.37485e+06 |      1.37485e+06 |   0 |   0 |   0 |   0 |   0 |
| 683 | Manassas                           | Northern |  47219.8         |  47219.8         |   0 |   0 |   0 |   0 |   0 |
| 685 | Manassas Park                      | Northern |  17387.4         |  17387.4         |   0 |   0 |   0 |   0 |   0 |
| 700 | Newport News                       | Eastern  |      1.44785e+06 |      1.44785e+06 |   0 |   0 |   0 |   0 |   0 |
| 710 | Norfolk                            | Eastern  |      1.85031e+06 |      1.85031e+06 |   0 |   0 |   0 |   0 |   0 |
| 720 | Norton                             | Western  | 152952           | 152952           |   0 |   0 |   0 |   0 |   0 |
| 730 | Petersburg                         | Central  | 688678           | 688678           |   0 |   0 |   0 |   0 |   0 |
| 740 | Portsmouth                         | Eastern  | 759588           | 759588           |   0 |   0 |   0 |   0 |   0 |
| 750 | Radford                            | Western  | 188264           | 188264           |   0 |   0 |   0 |   0 |   0 |
| 760 | Richmond                           | Central  |      2.38455e+06 |      2.38455e+06 |   0 |   0 |   0 |   0 |   0 |
| 770 | Roanoke                            | Piedmont |      1.50638e+06 |      1.50638e+06 |   0 |   0 |   0 |   0 |   0 |
| 800 | Suffolk                            | Eastern  | 782293           | 782293           |   0 |   0 |   0 |   0 |   0 |
| 810 | Virginia Beach                     | Eastern  |      1.00478e+06 |      1.00478e+06 |   0 |   0 |   0 |   0 |   0 |
| 830 | Williamsburg                       | Eastern  |  17502.8         |  17502.8         |   0 |   0 |   0 |   0 |   0 |
| 840 | Winchester                         | Northern | 188868           | 188868           |   0 |   0 |   0 |   0 |   0 |



label: J=H+I
labeled_formulas: J=H+I
output: {"Arithmetic-Relationship": ["J=H+I"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_155177___xlsx_626ce64c9360feed315a74c45b0da2ce9c633089.xlsx___line_422