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 |
|----:|:-------------------|--------:|--------:|--------:|-------------:|---------:|
|   1 | 01-AGRA            |  122699 |  107363 |  -15336 | -0.124989    | 0.875011 |
|   2 | 02-ALIGARH         |   96120 |   94028 |   -2092 | -0.0217645   | 0.978236 |
|   3 | 03-ALLAHABAD       |  180115 |  178299 |   -1816 | -0.0100824   | 0.989918 |
|   4 | 04-AMBEDKAR NAGAR  |   79143 |   77774 |   -1369 | -0.0172978   | 0.982702 |
|   5 | 05-AURAIYA         |   59213 |   55388 |   -3825 | -0.0645973   | 0.935403 |
|   6 | 06-AZAMGARH        |  172632 |  155406 |  -17226 | -0.0997845   | 0.900215 |
|   7 | 07-BADAUN          |  144817 |  134311 |  -10506 | -0.0725467   | 0.927453 |
|   8 | 08-BAGHPAT         |   35912 |   35634 |    -278 | -0.00774115  | 0.992259 |
|   9 | 09-BAHRAICH        |  210329 |  187903 |  -22426 | -0.106623    | 0.893377 |
|  10 | 10-BALLIA          |  133357 |  138196 |    4839 |  0.0362861   | 1.03629  |
|  11 | 11-BALRAMPUR       |  131683 |  126391 |   -5292 | -0.0401874   | 0.959813 |
|  12 | 12-BANDA           |   99455 |   85481 |  -13974 | -0.140506    | 0.859494 |
|  13 | 13-BARABANKI       |  130818 |  150787 |   19969 |  0.152647    | 1.15265  |
|  14 | 14-BAREILY         |  142853 |  141539 |   -1314 | -0.00919827  | 0.990802 |
|  15 | 15-BASTI           |  102307 |   89427 |  -12880 | -0.125896    | 0.874104 |
|  16 | 16-BHADOHI         |   62535 |   61216 |   -1319 | -0.0210922   | 0.978908 |
|  17 | 17-BIJNOUR         |   93110 |   82494 |  -10616 | -0.114016    | 0.885984 |
|  18 | 18-BULANDSHAHAR    |  112789 |  101487 |  -11302 | -0.100205    | 0.899795 |
|  19 | 19-CHANDAULI       |   92750 |   92472 |    -278 | -0.0029973   | 0.997003 |
|  20 | 20-CHITRAKOOT      |   69581 |   60664 |   -8917 | -0.128153    | 0.871847 |
|  21 | 21-AMETHI          |   73294 |   65740 |   -7554 | -0.103064    | 0.896936 |
|  22 | 22-DEORIA          |  123241 |  112480 |  -10761 | -0.0873167   | 0.912683 |
|  23 | 23-ETAH            |   77730 |   66450 |  -11280 | -0.145118    | 0.854882 |
|  24 | 24-FAIZABAD        |   95021 |   85575 |   -9446 | -0.0994096   | 0.90059  |
|  25 | 25-FARRUKHABAD     |   86235 |   78889 |   -7346 | -0.0851858   | 0.914814 |
|  26 | 26-FATEHPUR        |  124554 |  117714 |   -6840 | -0.0549159   | 0.945084 |
|  27 | 27-FIROZABAD       |   78478 |   75823 |   -2655 | -0.0338311   | 0.966169 |
|  28 | 28-G.B. NAGAR      |   42142 |   37808 |   -4334 | -0.102843    | 0.897157 |
|  29 | 29-GHAZIPUR        |  143817 |  134203 |   -9614 | -0.0668488   | 0.933151 |
|  30 | 30-GHAZIYABAD      |   45440 |   42576 |   -2864 | -0.0630282   | 0.936972 |
|  31 | 31-GONDA           |  169519 |  152313 |  -17206 | -0.101499    | 0.898501 |
|  32 | 32-GORAKHPUR       |  147157 |  147187 |      30 |  0.000203864 | 1.0002   |
|  33 | 33-HAMEERPUR       |   50999 |   44519 |   -6480 | -0.127061    | 0.872939 |
|  34 | 34-HARDOI          |  217727 |  196228 |  -21499 | -0.0987429   | 0.901257 |
|  35 | 35-HATHRAS         |   56394 |   52768 |   -3626 | -0.0642976   | 0.935702 |
|  36 | 36-ITAWAH          |   59062 |   54129 |   -4933 | -0.0835224   | 0.916478 |
|  37 | 37-J.P. NAGAR      |   55842 |   55760 |     -82 | -0.00146843  | 0.998532 |
|  38 | 38-JALAUN          |   60878 |   55068 |   -5810 | -0.0954368   | 0.904563 |
|  39 | 39-JAUNPUR         |  184155 |  158306 |  -25849 | -0.140365    | 0.859635 |
|  40 | 40-JHANSI          |   58904 |   58794 |    -110 | -0.00186745  | 0.998133 |
|  41 | 41-KANNAUJ         |   74223 |   73216 |   -1007 | -0.0135672   | 0.986433 |
|  42 | 42-KANPUR DEHAT    |   63191 |   64951 |    1760 |  0.0278521   | 1.02785  |
|  43 | 43-KANPUR NAGAR    |   78270 |   70303 |   -7967 | -0.101789    | 0.898211 |
|  44 | 44-KAAS GANJ       |   67630 |   64782 |   -2848 | -0.0421115   | 0.957889 |
|  45 | 45-KAUSHAMBI       |   72644 |   75250 |    2606 |  0.0358736   | 1.03587  |
|  46 | 46-KUSHINAGAR      |  159481 |  147535 |  -11946 | -0.0749055   | 0.925095 |
|  47 | 47-LAKHIMPUR KHERI |  211297 |  214788 |    3491 |  0.0165218   | 1.01652  |
|  48 | 48-LALITPUR        |   64136 |   61214 |   -2922 | -0.0455594   | 0.954441 |
|  49 | 49-LUCKNOW         |   92076 |   94521 |    2445 |  0.0265542   | 1.02655  |
|  50 | 50-MAHOBA          |   49784 |   40725 |   -9059 | -0.181966    | 0.818034 |
|  51 | 51-MAHRAJGANJ      |  120855 |  110814 |  -10041 | -0.083083    | 0.916917 |
|  52 | 52-MAINPURI        |   76727 |   62743 |  -13984 | -0.182257    | 0.817743 |
|  53 | 53-MATHURA         |   77453 |   74005 |   -3448 | -0.0445173   | 0.955483 |
|  54 | 54-MAU             |   83519 |   77058 |   -6461 | -0.0773596   | 0.92264  |
|  55 | 55-MEERUT          |   61097 |   62737 |    1640 |  0.0268426   | 1.02684  |
|  56 | 56-MIRZAPUR        |  132781 |  116405 |  -16376 | -0.123331    | 0.876669 |
|  57 | 57-MORADABAD       |   78358 |   75632 |   -2726 | -0.034789    | 0.965211 |
|  58 | 58-MUZAFFARNAGAR   |   69079 |   69832 |     753 |  0.0109006   | 1.0109   |
|  59 | 59-PILIBHIT        |   77734 |   73167 |   -4567 | -0.0587516   | 0.941248 |
|  60 | 60-PRATAPGARH      |  120941 |  113072 |   -7869 | -0.0650648   | 0.934935 |
|  61 | 61-RAI BAREILY     |  108810 |  101275 |   -7535 | -0.0692491   | 0.930751 |
|  62 | 62-RAMPUR          |   70505 |   70346 |    -159 | -0.00225516  | 0.997745 |
|  63 | 63-SAHARANPUR      |   93112 |   87993 |   -5119 | -0.0549768   | 0.945023 |
|  64 | 64-SANTKABIR NAGAR |   69194 |   62141 |   -7053 | -0.101931    | 0.898069 |
|  65 | 65-SHAHJAHANPUR    |  155444 |  143296 |  -12148 | -0.0781503   | 0.92185  |
|  66 | 66-SHRAWASTI       |   59689 |   57008 |   -2681 | -0.0449161   | 0.955084 |
|  67 | 67-SIDDHARTHNAGAR  |  158896 |  123420 |  -35476 | -0.223266    | 0.776734 |
|  68 | 68-SITAPUR         |  212810 |  212884 |      74 |  0.000347728 | 1.00035  |
|  69 | 69-SONBHADRA       |  115679 |  104384 |  -11295 | -0.0976409   | 0.902359 |
|  70 | 70-SULTANPUR       |  120336 |  103726 |  -16610 | -0.13803     | 0.86197  |
|  71 | 71-UNNAO           |  121656 |  110790 |  -10866 | -0.0893174   | 0.910683 |
|  72 | 72-VARANASI        |  108100 |  110554 |    2454 |  0.0227012   | 1.0227   |
|  73 | 73-SAMBHAL         |   98810 |   90110 |   -8700 | -0.0880478   | 0.911952 |
|  74 | 74-HAPUR           |   30867 |   29917 |    -950 | -0.0307772   | 0.969223 |
|  75 | 75-SHAMLI          |   41354 |   39364 |   -1990 | -0.0481211   | 0.951879 |
|   0 | TOTAL              | 7619345 | 7117560 | -501785 | -0.0658567   | 0.934143 |



label: E=D-C
F=E/C
G=D/C
labeled_formulas: E=D-C
F=E/C
G=D/C
output: {"Arithmetic-Relationship": ["E=D-C", "F=E/C", "G=D/C"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_377335___xlsx_5e696bd67948a6dabbd8fc6de2fbfac425952fd0.xlsx___line_198