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 |
|:--------------------------|------:|-----------------:|-----------------:|---------:|----:|---------:|---------:|----:|---------:|---------:|
| Aberdeen 06-1             |  6001 |      2.70322e+07 | 789553           |  5029801 |   0 |  8091547 |  5107940 |   0 |  8013408 | 16104955 |
| Alcester-Hudson 61-1      | 61001 |      2.338e+06   |  74045.6         |   461068 |   0 |   670909 |   439604 |   0 |   692373 |  1363282 |
| Andes Central 11-1        | 11001 |      2.25814e+06 |  72001.6         |   355562 |   0 |   737509 |   372760 |   0 |   720311 |  1457820 |
| Arlington 38-1            | 38001 |      1.86653e+06 |  52252.7         |   508020 |   0 |   399117 |   514598 |   0 |   392539 |   791656 |
| Armour 21-1               | 21001 |      1.34488e+06 |  41084.6         |   198879 |   0 |   453017 |   188910 |   0 |   462986 |   916003 |
| Avon 04-1                 |  4001 |      1.70253e+06 |  28785.5         |   222832 |   0 |   614039 |   214395 |   0 |   622476 |  1236515 |
| Baltic 49-1               | 49001 |      3.25015e+06 |  41821.4         |   359935 |   0 |  1244228 |   390005 |   0 |  1214158 |  2458386 |
| Bennett County 03-1       |  3001 |      3.08136e+06 | 113483           |   233261 |   0 |  1250679 |   249257 |   0 |  1234683 |  2485362 |
| Beresford 61-2            | 61002 |      4.24537e+06 | 129467           |   715904 |   0 |  1342047 |   698405 |   0 |  1359546 |  2701593 |
| Bison 52-1                | 52001 |      1.05057e+06 |  61820.6         |   306548 |   0 |   187826 |   295034 |   0 |   199340 |   387166 |
| Bon Homme 04-2            |  4002 |      3.13289e+06 | 106097           |   558533 |   0 |   954864 |   545186 |   0 |   968211 |  1923075 |
| Bowdle 22-1               | 22001 | 881136           |  55940           |   260697 |   0 |   151901 |   257691 |   0 |   154907 |   306808 |
| Brandon Valley 49-2       | 49002 |      2.79782e+07 | 698681           |  4317201 |   0 |  9322556 |  4736667 |   0 |  8903090 | 18225646 |
| Bridgewater-Emery 30-3    | 30003 |      2.30697e+06 |  41381.8         |   451439 |   0 |   681354 |   448826 |   0 |   683967 |  1365321 |
| Britton - Hecla 45-4      | 45004 |      2.8557e+06  | 305679           |   989012 |   0 |   285999 |   979016 |   0 |   295995 |   581994 |
| Brookings 05-1            |  5001 |      2.00651e+07 | 541990           |  4006219 |   0 |  5755339 |  4171193 |   0 |  5590365 | 11345704 |
| Burke 26-2                | 26002 |      1.66327e+06 |  35853.4         |   223191 |   0 |   590519 |   227363 |   0 |   586347 |  1176866 |
| Canistota 43-1            | 43001 |      1.62077e+06 |  40459.6         |   254143 |   0 |   536014 |   257095 |   0 |   533062 |  1069076 |
| Canton 41-1               | 41001 |      5.22518e+06 | 124979           |  1055995 |   0 |  1494107 |  1123931 |   0 |  1426171 |  2920278 |
| Castlewood 28-1           | 28001 |      2.20134e+06 |  55499.9         |   327135 |   0 |   745787 |   324385 |   0 |   748537 |  1494324 |
| Centerville 60-1          | 60001 |      2.00714e+06 |  50559.8         |   303118 |   0 |   675170 |   292081 |   0 |   686207 |  1361377 |
| Chamberlain 07-1          |  7001 |      5.22512e+06 | 213446           |   947614 |   0 |  1558225 |   918260 |   0 |  1587579 |  3145804 |
| Chester 39-1              | 39001 |      3.37127e+06 |  97862.5         |   594278 |   0 |  1042428 |   616833 |   0 |  1019873 |  2062301 |
| Clark 12-2                | 12002 |      2.96088e+06 | 162390           |   891505 |   0 |   507739 |   986053 |   0 |   413191 |   920930 |
| Colman-Egan 50-5          | 50005 |      1.98689e+06 |  51788.5         |   297745 |   0 |   669806 |   307020 |   0 |   660531 |  1330337 |
| Colome Consolidated 59-3  | 59003 |      1.50319e+06 |  26814.8         |   271881 |   0 |   466307 |   274164 |   0 |   464024 |   930331 |
| Corsica-Stickney 21-3     | 21003 |      1.85867e+06 |  86478.1         |   471046 |   0 |   415047 |   458229 |   0 |   427864 |   842911 |
| Custer 16-1               | 16001 |      5.31042e+06 | 216977           |  2335502 |   0 |   211220 |  2489680 |   0 |    57042 |   268262 |
| Dakota Valley 61-8        | 61008 |      8.25285e+06 | 237738           |  2104568 |   0 |  1902986 |  2101954 |   0 |  1905600 |  3808586 |
| De Smet 38-2              | 38002 |      2.01049e+06 |  58511.7         |   556494 |   0 |   419494 |   561034 |   0 |   414954 |   834448 |
| Dell Rapids 49-3          | 49003 |      5.85731e+06 | 197796           |  1032182 |   0 |  1797576 |  1106109 |   0 |  1723649 |  3521225 |
| Deubrook 05-6             |  5006 |      2.61322e+06 | 319548           |   540674 |   0 |   606160 |   605286 |   0 |   541548 |  1147708 |
| Deuel 19-4                | 19004 |      3.23151e+06 | 168113           |   871135 |   0 |   660564 |   891117 |   0 |   640582 |  1301146 |
| Doland 56-2               | 56002 |      1.07106e+06 |  30297.3         |   456569 |   0 |    63811 |   456085 |   0 |    64295 |   128106 |
| Douglas 51-1              | 51001 |      1.64455e+07 | 211245           |  1435843 |   0 |  6681279 |  1471797 |   0 |  6645325 | 13326604 |
| Dupree 64-2               | 64002 |      2.63371e+06 |      0           |   173590 |   0 |  1143264 |   172408 |   0 |  1144446 |  2287710 |
| Eagle Butte 20-1          | 20001 |      2.21138e+06 |  40102.9         |   214329 |   0 |   871307 |   211927 |   0 |   873709 |  1745016 |
| Edgemont 23-1             | 23001 |      1.17723e+06 |  37820           |   377908 |   0 |   191798 |   402503 |   0 |   167203 |   359001 |
| Elk Point-Jefferson 61-7  | 61007 |      4.15459e+06 | 104576           |   754124 |   0 |  1270881 |   747051 |   0 |  1277954 |  2548835 |
| Elkton 05-3               |  5003 |      2.32494e+06 | 161272           |   855770 |   0 |   226063 |   856343 |   0 |   225490 |   451553 |
| Estelline 28-2            | 28002 |      1.95423e+06 |  90176.5         |   524574 |   0 |   407453 |   526391 |   0 |   405636 |   813089 |
| Ethan 17-1                | 17001 |      1.93396e+06 |  25660.7         |   166766 |   0 |   787385 |   167306 |   0 |   786845 |  1574230 |
| Eureka 44-1               | 44001 |      1.10421e+06 |  40021           |   501165 |   0 |    30932 |   496995 |   0 |    35102 |    66034 |
| Faith 46-2                | 46002 |      1.37095e+06 |  15162.2         |   137951 |   0 |   539945 |   138670 |   0 |   539226 |  1079171 |
| Faulkton Area 24-4        | 24004 |      2.44401e+06 |  91582           |   841150 |   0 |   335064 |   813447 |   0 |   362767 |   697831 |
| Flandreau 50-3            | 50003 |      4.29037e+06 | 106861           |   578751 |   0 |  1513004 |   571586 |   0 |  1520169 |  3033173 |
| Florence 14-1             | 14001 |      2.03403e+06 |  44786           |   145031 |   0 |   849592 |   142302 |   0 |   852321 |  1701913 |
| Frederick Area 06-2       |  6002 |      1.20331e+06 |  59109           |   355811 |   0 |   216290 |   342940 |   0 |   229161 |   445451 |
| Freeman 33-1              | 33001 |      2.3737e+06  | 111593           |   515576 |   0 |   615478 |   510988 |   0 |   620066 |  1235544 |
| Garretson 49-4            | 49004 |      2.98196e+06 | 148826           |   461255 |   0 |   955315 |   484308 |   0 |   932262 |  1887577 |
| Gayville-Volin 63-1       | 63001 |      1.98275e+06 |  36308.7         |   153494 |   0 |   819726 |   151113 |   0 |   822107 |  1641833 |
| Gettysburg 53-1           | 53001 |      1.64938e+06 |  61820.2         |   394661 |   0 |   399121 |   352569 |   0 |   441213 |   840334 |
| Gregory 26-4              | 26004 |      2.5677e+06  |  93608.6         |   395187 |   0 |   841858 |   416050 |   0 |   820995 |  1662853 |
| Haakon 27-1               | 27001 |      2.19137e+06 |  97645.3         |   505792 |   0 |   541071 |   524820 |   0 |   522043 |  1063114 |
| Hamlin 28-3               | 28003 |      5.06955e+06 | 158601           |   901626 |   0 |  1553848 |   902220 |   0 |  1553254 |  3107102 |
| Hanson 30-1               | 30001 |      2.63255e+06 |  76048           |   435390 |   0 |   842859 |   430980 |   0 |   847269 |  1690128 |
| Harding County 31-1       | 31001 |      1.44733e+06 |      0           |   437000 |   0 |   286663 |   430228 |   0 |   293435 |   580098 |
| Harrisburg 41-2           | 41002 |      3.24322e+07 | 642397           |  6236400 |   0 |  9658495 |  6565914 |   0 |  9328981 | 18987476 |
| Henry 14-2                | 14002 |      1.25174e+06 |  32407.4         |   125525 |   0 |   484142 |   125930 |   0 |   483737 |   967879 |
| Herreid 10-1              | 10001 | 887991           |  53135.1         |   330450 |   0 |    86978 |   341287 |   0 |    76141 |   163119 |
| Highmore-Harrold 34-2     | 34002 |      1.59345e+06 |  94794.9         |   737876 |   0 |    11450 |   715291 |   0 |    34035 |    45485 |
| Hitchcock-Tulare 56-6     | 56006 |      1.65492e+06 |  75632.2         |   689890 |   0 |    99756 |   713554 |   0 |    76092 |   175848 |
| Hot Springs 23-2          | 23002 |      4.36035e+06 | 163415           |  1029411 |   0 |  1069056 |  1052693 |   0 |  1045774 |  2114830 |
| Howard 48-3               | 48003 |      2.34279e+06 | 220662           |   790806 |   0 |   270260 |   795444 |   0 |   265622 |   535882 |
| Huron 02-2                |  2002 |      1.76741e+07 | 344005           |  2295677 |   0 |  6369348 |  2320658 |   0 |  6344367 | 12713715 |
| Ipswich Public 22-6       | 22006 |      2.76224e+06 | 186000           |   916227 |   0 |   371891 |   900891 |   0 |   387227 |   759118 |
| Irene-Wakonda 13-3        | 13003 |      2.06824e+06 |  64740.9         |   492843 |   0 |   508905 |   485892 |   0 |   515856 |  1024761 |
| Iroquois 02-3             |  2003 |      1.5199e+06  |  48001.3         |   611616 |   0 |   124332 |   612588 |   0 |   123360 |   247692 |
| Jones County 37-3         | 37003 |      1.32625e+06 |  73657.1         |   306235 |   0 |   320061 |   310738 |   0 |   315558 |   635619 |
| Kadoka Area 35-2          | 35002 |      2.18634e+06 |  89153.2         |   376018 |   0 |   672576 |   375312 |   0 |   673282 |  1345858 |
| Kimball 07-2              |  7002 |      2.29164e+06 | 103138           |   488059 |   0 |   606191 |   477274 |   0 |   616976 |  1223167 |
| Lake Preston 38-3         | 38003 |      1.26478e+06 |  35808           |   352298 |   0 |   262188 |   357139 |   0 |   257347 |   519535 |
| Langford Area 45-5        | 45005 |      1.58348e+06 |  64291.4         |   501618 |   0 |   257974 |   489680 |   0 |   269912 |   527886 |
| Lemmon 52-4               | 52004 |      1.82413e+06 | 103127           |   485787 |   0 |   374716 |   479213 |   0 |   381290 |   756006 |
| Lennox 41-4               | 41004 |      6.77001e+06 | 244048           |  1243409 |   0 |  2019572 |  1189556 |   0 |  2073425 |  4092997 |
| Leola 44-2                | 44002 |      1.62513e+06 | 142804           |   356248 |   0 |   384917 |   347717 |   0 |   393448 |   778365 |
| Lyman 42-1                | 42001 |      2.40059e+06 |  77798           |   559976 |   0 |   601420 |   557041 |   0 |   604355 |  1205775 |
| Madison Central 39-2      | 39002 |      6.76757e+06 | 188923           |  1655892 |   0 |  1633430 |  1826440 |   0 |  1462882 |  3096312 |
| Marion 60-3               | 60003 |      1.33348e+06 | 139196           |   417487 |   0 |   179653 |   411607 |   0 |   185533 |   365186 |
| McCook Central 43-7       | 43007 |      2.64901e+06 |  80038.6         |   451443 |   0 |   833043 |   456203 |   0 |   828283 |  1661326 |
| McIntosh 15-1             | 15001 |      1.02822e+06 |  23101.6         |   147980 |   0 |   354577 |   161069 |   0 |   341488 |   696065 |
| McLaughlin 15-2           | 15002 |      2.87277e+06 |  43259.2         |   191862 |   0 |  1222892 |   216863 |   0 |  1197891 |  2420783 |
| Menno 33-2                | 33002 |      2.04887e+06 | 170802           |   360034 |   0 |   579000 |   349404 |   0 |   589630 |  1168630 |
| Milbank 25-4              | 25004 |      5.90392e+06 | 200772           |  1497385 |   0 |  1354191 |  1506973 |   0 |  1344603 |  2698794 |
| Miller 29-4               | 29004 |      2.77951e+06 | 136169           |  1146110 |   0 |   175560 |  1154386 |   0 |   167284 |   342844 |
| Mitchell 17-2             | 17002 |      1.68208e+07 | 287100           |  3062107 |   0 |  5204760 |  3163848 |   0 |  5103019 | 10307779 |
| Mobridge-Pollock 62-6     | 62006 |      3.70241e+06 | 229278           |   561922 |   0 |  1174644 |   577195 |   0 |  1159371 |  2334015 |
| Montrose 43-2             | 43002 |      1.73974e+06 |  55046.4         |   232517 |   0 |   609831 |   233388 |   0 |   608960 |  1218791 |
| Mount Vernon 17-3         | 17003 |      1.62022e+06 |  45067.5         |   244433 |   0 |   543143 |   237953 |   0 |   549623 |  1092766 |
| New Underwood 51-3        | 51003 |      1.91721e+06 |  31250.3         |   165780 |   0 |   777198 |   177164 |   0 |   765814 |  1543012 |
| Newell 09-2               |  9002 |      1.83981e+06 |  83092.7         |   336648 |   0 |   541710 |   357699 |   0 |   520659 |  1062369 |
| Northwestern Area 56-7    | 56007 |      2.15547e+06 |  68109.2         |   770069 |   0 |   273610 |   766883 |   0 |   276796 |   550406 |
| Oelrichs 23-3             | 23003 | 953707           |  20914.8         |    78781 |   0 |   387615 |    80182 |   0 |   386214 |   773829 |
| Oglala Lakota County 65-1 | 65001 |      1.04819e+07 | 248921           |    66642 |   0 |  5049847 |    63875 |   0 |  5052614 | 10102461 |
| Oldham-Ramona 39-5        | 39005 |      1.17537e+06 |  21125.7         |   269986 |   0 |   307136 |   268693 |   0 |   308429 |   615565 |
| Parker 60-4               | 60004 |      3.05496e+06 |  61973.9         |   443218 |   0 |  1053276 |   456820 |   0 |  1039674 |  2092950 |
| Parkston 33-3             | 33003 |      3.31929e+06 | 113538           |   555411 |   0 |  1047463 |   554845 |   0 |  1048029 |  2095492 |
| Pierre 32-2               | 32002 |      1.65797e+07 | 449341           |  2817637 |   0 |  5247542 |  2850645 |   0 |  5214534 | 10462076 |
| Platte-Geddes 11-5        | 11005 |      3.18474e+06 | 131436           |   837736 |   0 |   688917 |   844113 |   0 |   682540 |  1371457 |
| Rapid City 51-4           | 51004 |      7.56775e+07 |      1.22052e+06 | 18956954 |   0 | 18271551 | 19373232 |   0 | 17855273 | 36126824 |
| Redfield 56-4             | 56004 |      3.40827e+06 |  92421.8         |   786291 |   0 |   871633 |   825665 |   0 |   832259 |  1703892 |
| Rosholt 54-4              | 54004 |      1.77077e+06 |  24781.7         |   224035 |   0 |   648957 |   228575 |   0 |   644417 |  1293374 |
| Rutland 39-4              | 39004 |      1.34115e+06 |  32311.3         |   186894 |   0 |   467526 |   184431 |   0 |   469989 |   937515 |
| Sanborn Central 55-5      | 55005 |      1.40448e+06 |  40613.9         |   379787 |   0 |   302148 |   367491 |   0 |   314444 |   616592 |
| Scotland 04-3             |  4003 |      1.82462e+06 |  70830.5         |   396644 |   0 |   480251 |   381806 |   0 |   495089 |   975340 |
| Sioux Falls 49-5          | 49005 |      1.46032e+08 |      3.6959e+06  | 30481363 |   0 | 40686462 | 32085749 |   0 | 39082076 | 79768538 |
| Sioux Valley 05-5         |  5005 |      4.01457e+06 | 101888           |   635972 |   0 |  1320370 |   633966 |   0 |  1322376 |  2642746 |
| Sisseton 54-2             | 54002 |      5.48864e+06 | 429725           |   937546 |   0 |  1591914 |   980951 |   0 |  1548509 |  3140423 |
| Smee 15-3                 | 15003 |      1.47713e+06 |  15853.5         |    12206 |   0 |   718432 |    13136 |   0 |   717502 |  1435934 |
| South Central 26-5        | 26005 | 424698           |  27471.8         |   155650 |   0 |    42963 |   161839 |   0 |    36774 |    79737 |
| Stanley County 57-1       | 57001 |      2.72582e+06 |  31189           |   857772 |   0 |   489542 |   832714 |   0 |   514600 |  1004142 |
| Summit 54-6               | 54006 |      1.32625e+06 |  38935           |   153078 |   0 |   490579 |   158609 |   0 |   485048 |   975627 |
| Tea Area 41-5             | 41005 |      1.23296e+07 | 234238           |  1331529 |   0 |  4716175 |  1465434 |   0 |  4582270 |  9298445 |
| Timber Lake 20-3          | 20003 |      2.29799e+06 |  23298.9         |   203626 |   0 |   933718 |   199568 |   0 |   937776 |  1871494 |
| Todd County 66-1          | 66001 |      1.30269e+07 |  90922.5         |   193443 |   0 |  6274553 |   203932 |   0 |  6264064 | 12538617 |
| Tripp-Delmont 33-5        | 33005 |      1.07665e+06 | 182158           |   364272 |   0 |    82972 |   355644 |   0 |    91600 |   174572 |
| Tri-Valley 49-6           | 49006 |      5.93236e+06 | 230581           |  1190425 |   0 |  1660463 |  1296601 |   0 |  1554287 |  3214750 |
| Vermillion 13-1           | 13001 |      7.50961e+06 | 246002           |  1400816 |   0 |  2230988 |  1451009 |   0 |  2180795 |  4411783 |
| Viborg -Hurley 60-6       | 60006 |      2.39112e+06 |  71499.4         |   490013 |   0 |   669800 |   492368 |   0 |   667445 |  1337245 |
| Wagner 11-4               | 11004 |      4.88775e+06 | 153099           |   433326 |   0 |  1933999 |   440121 |   0 |  1927204 |  3861203 |
| Wall 51-5                 | 51005 |      1.96318e+06 |  68876.7         |   368373 |   0 |   578779 |   367615 |   0 |   579537 |  1158316 |
| Warner 06-5               |  6005 |      2.19417e+06 |  47564.5         |   282124 |   0 |   791177 |   287855 |   0 |   785446 |  1576623 |
| Watertown 14-4            | 14004 |      2.31194e+07 | 665097           |  4819191 |   0 |  6407938 |  4934144 |   0 |  6292985 | 12700923 |
| Waubay 18-3               | 18003 |      1.37095e+06 |  45213.7         |   285415 |   0 |   377455 |   268126 |   0 |   394744 |   772199 |
| Waverly 14-5              | 14005 |      1.82462e+06 |  49368.6         |   234157 |   0 |   653469 |   231916 |   0 |   655710 |  1309179 |
| Webster Area 18-5         | 18005 |      3.15878e+06 | 179823           |   987750 |   0 |   501728 |   961666 |   0 |   527812 |  1029540 |
| Wessington Springs 36-2   | 36002 |      2.55683e+06 | 178554           |   591805 |   0 |   597331 |   581697 |   0 |   607439 |  1204770 |
| West Central 49-7         | 49007 |      8.25553e+06 | 206280           |  1274363 |   0 |  2750261 |  1349004 |   0 |  2675620 |  5425881 |
| White Lake 01-3           |  1003 | 953075           | 215153           |   222767 |   0 |   146194 |   223946 |   0 |   145015 |   291209 |
| White River 47-1          | 47001 |      2.56245e+06 |  25754.4         |   152197 |   0 |  1116152 |   147553 |   0 |  1120796 |  2236948 |
| Willow Lake 12-3          | 12003 |      2.03423e+06 | 209773           |   428842 |   0 |   483384 |   466438 |   0 |   445788 |   929172 |
| Wilmot 54-7               | 54007 |      1.60621e+06 |  68539.4         |   262637 |   0 |   506200 |   284456 |   0 |   484381 |   990581 |
| Winner 59-2               | 59002 |      4.24102e+06 | 105983           |   852829 |   0 |  1214688 |   871228 |   0 |  1196289 |  2410977 |
| Wolsey-Wessington 02-6    |  2006 |      2.40569e+06 |  66699.8         |   513111 |   0 |   656384 |   502764 |   0 |   666731 |  1323115 |
| Woonsocket 55-4           | 55004 |      1.8142e+06  |  42330.1         |   257868 |   0 |   628065 |   241661 |   0 |   644272 |  1272337 |
| Yankton 63-3              | 63003 |      1.68016e+07 | 586417           |  3189178 |   0 |  4918403 |  3243903 |   0 |  4863678 |  9782081 |



label: K=J+G
labeled_formulas: K=J+G
I=B*F
I=C*F
I=D*F
I=E*F
I=F*H
I=(B+C)*F
I=(B+D)*F
I=(B+E)*F
I=(B*F)*H
I=(C+D)*F
I=(C+E)*F
I=(C*F)*H
I=(D+E)*F
I=(D*F)*H
I=(E*F)*H
J=(E+G)-H
output: {"Arithmetic-Relationship": ["K=J+G"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_57858___xlsx_f1b04249f551159bb7e57bd79139a6ec16b6d448.xlsx___line_835