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 |   L |   M |   N |   O |   P |
|:---------------|----:|----:|----:|----:|----:|----:|---------:|----:|----:|----:|----:|----:|----:|----:|----:|
| ALCONA         |   2 |   2 |   0 |   0 |   0 |   0 | 100      |   0 |  47 |   0 |  11 |   0 |  31 |   0 |   0 |
| ALGER          |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |   0 |   0 |   0 |   0 |   3 |   0 |   0 |
| ANTRIM         |   1 |   1 |   0 |   0 |   1 |   1 | 100      |   3 |  46 |   0 |   8 |   2 |  78 |   0 |   0 |
| ARENAC         |   2 |   2 |   0 |   0 |   4 |   4 | 100      |   0 |  38 |   0 |  10 |   2 |  44 |   0 |   0 |
| BARAGA         |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |  50 |   0 |  18 |   0 |  31 |   0 |   0 |
| BARRY          |   2 |   2 |   1 |   1 |   4 |   4 | 100      |   0 |  46 |   0 |   3 |   3 | 110 |   0 |   0 |
| BENZIE         |   2 |   2 |   0 |   0 |   1 |   1 | 100      |   0 |  34 |   0 |   9 |   0 |  36 |   0 |   0 |
| BERRIEN        |   8 |   8 |   2 |   2 |  24 |  20 |  88.2353 |   3 | 604 |   0 | 148 |  11 | 452 |   0 |   0 |
| BRANCH         |   2 |   2 |   1 |   1 |   4 |   4 | 100      |   0 | 192 |   0 |  40 |   1 | 136 |   0 |   0 |
| CASS           |   2 |   2 |   0 |   0 |   1 |   1 | 100      |   0 | 126 |   0 |  27 |   2 | 123 |   0 |   0 |
| CHARLEVOIX     |   2 |   2 |   0 |   0 |   0 |   0 | 100      |   0 |   1 |   0 |   0 |   5 |  40 |   0 |   0 |
| CHEBOYGAN      |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |  77 |   0 |  16 |   0 |  34 |   0 |   0 |
| CHIPPEWA       |   2 |   2 |   1 |   1 |   1 |   1 | 100      |   0 |   9 |   0 |   1 |   0 |  19 |   0 |   0 |
| CRAWFORD       |   2 |   2 |   0 |   0 |   1 |   0 |  66.6667 |   0 |  36 |   0 |   4 |   1 |  53 |   0 |   0 |
| DELTA          |   3 |   3 |   2 |   2 |   6 |   6 | 100      |   0 | 180 |   0 |  41 |   3 | 148 |   0 |   0 |
| EATON          |   4 |   4 |   6 |   6 |   6 |   6 | 100      |   1 | 282 |   0 |  49 |   6 | 303 |   0 |   0 |
| EMMET          |   2 |   2 |   1 |   1 |   2 |   2 | 100      |   0 |  98 |   0 |  21 |   1 |  85 |   0 |   0 |
| GLADWIN        |   2 |   2 |   0 |   0 |   3 |   3 | 100      |   0 |  63 |   0 |  10 |   1 |  57 |   0 |   0 |
| GOGEBIC        |   2 |   2 |   1 |   1 |   1 |   1 | 100      |   0 | 135 |   0 |  20 |   0 |  89 |   0 |   0 |
| GRAND TRAVERSE |   0 |   0 |   1 |   1 |  10 |   6 |  63.6364 |   1 |  17 |   0 |   7 |   3 |  28 |   0 |   0 |
| GRATIOT        |   5 |   5 |   2 |   2 |   4 |   4 | 100      |   0 | 234 |   0 |  60 |   2 | 239 |   0 |   0 |
| HILLSDALE      |   2 |   2 |   0 |   0 |   7 |   6 |  88.8889 |   0 | 108 |   0 |  31 |   3 | 140 |   0 |   0 |
| HOUGHTON       |   4 |   4 |   1 |   1 |   0 |   0 | 100      |   0 |  86 |   0 |  13 |   0 | 109 |   0 |   0 |
| IONIA          |   2 |   2 |   0 |   0 |   4 |   4 | 100      |   0 | 138 |   0 |  36 |   1 | 113 |   0 |   0 |
| IOSCO          |   3 |   3 |   1 |   1 |   1 |   1 | 100      |   0 |  99 |   0 |  48 |   4 | 101 |   0 |   0 |
| KALKASKA       |   1 |   1 |   1 |   1 |   1 |   1 | 100      |   0 |   3 |   0 |   0 |   1 |  30 |   0 |   0 |
| LAKE           |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |  23 |   0 |   7 |   2 |  22 |   0 |   0 |
| LAPEER         |   4 |   4 |   1 |   1 |   9 |   6 |  78.5714 |   2 | 132 |   0 |  45 |  19 | 170 |   0 |   0 |
| LEELANAU       |   3 |   3 |   1 |   1 |   1 |   1 | 100      |   0 |  24 |   0 |   1 |   0 |  49 |   0 |   0 |
| LENAWEE        |   4 |   4 |   5 |   5 |   9 |   9 | 100      |   0 | 214 |   0 |  29 |   0 | 238 |   0 |   0 |
| LIVINGSTON     |   6 |   6 |   4 |   4 |  15 |  15 | 100      |   2 | 404 |   0 |  55 |  10 | 342 |   0 |   0 |
| LUCE           |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |   0 |   0 |   0 |   0 |   7 |   0 |   0 |
| MACKINAC       |   1 |   1 |   1 |   1 |   1 |   0 |  66.6667 |   0 |   1 |   0 |   0 |   0 |   9 |   0 |   0 |
| MANISTEE       |   1 |   1 |   0 |   0 |   1 |   1 | 100      |   0 |   1 |   0 |   0 |   0 |  11 |   0 |   0 |
| MASON          |   2 |   2 |   0 |   0 |   3 |   3 | 100      |   0 |  43 |   0 |   8 |   1 |  45 |   0 |   0 |
| MECOSTA        |   2 |   2 |   0 |   0 |   2 |   1 |  75      |   0 |  63 |   0 |   6 |   0 |  61 |   0 |   0 |
| MENOMINEE      |   3 |   3 |   0 |   0 |   0 |   0 | 100      |   0 |  90 |   0 |  18 |   0 |  76 |   0 |   0 |
| MISSAUKEE      |   1 |   1 |   0 |   0 |   3 |   3 | 100      |   0 |  48 |   0 |   7 |   0 |  57 |   0 |   0 |
| MONTCALM       |   4 |   4 |   2 |   2 |   3 |   3 | 100      |   1 | 154 |   0 |  36 |   3 | 153 |   0 |   0 |
| NEWAYGO        |   2 |   2 |   1 |   1 |   3 |   3 | 100      |   2 |  93 |   0 |   8 |   1 | 118 |   0 |   0 |
| OGEMAW         |   2 |   2 |   0 |   0 |   1 |   1 | 100      |   0 |  73 |   0 |   8 |   0 |  43 |   0 |   0 |
| ONTONAGON      |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |  87 |   0 |   8 |   1 |  29 |   0 |   0 |
| OSCEOLA        |   1 |   1 |   0 |   0 |   2 |   2 | 100      |   1 |  24 |   0 |   2 |   0 |  31 |   0 |   0 |
| OSCODA         |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |  34 |   0 |  10 |   0 |  22 |   0 |   0 |
| OTSEGO         |   2 |   2 |   0 |   0 |   5 |   4 |  85.7143 |   0 |  65 |   0 |  11 |   1 |  50 |   0 |   0 |
| PRESQUE ISLE   |   1 |   1 |   0 |   0 |   1 |   1 | 100      |   0 |  34 |   0 |  11 |   1 |  24 |   0 |   0 |
| ROSCOMMON      |   2 |   2 |   0 |   0 |   2 |   2 | 100      |   1 |  59 |   0 |  13 |   0 |  42 |   0 |   0 |
| SAINT CLAIR    |   5 |   5 |   2 |   2 |   8 |   8 | 100      |   2 | 470 |   0 |  97 |   7 | 370 |   0 |   0 |
| SAINT JOSEPH   |   4 |   4 |   2 |   2 |   1 |   1 | 100      |   0 | 204 |   0 |  43 |   2 | 143 |   0 |   0 |
| SANILAC        |   3 |   3 |   1 |   1 |   1 |   0 |  80      |   0 | 116 |   0 |  38 |   2 | 214 |   0 |   0 |
| SCHOOLCRAFT    |   1 |   1 |   0 |   0 |   0 |   0 | 100      |   0 |   0 |   0 |   1 |   0 |   2 |   0 |   0 |
| SHIAWASSEE     |   3 |   3 |   1 |   1 |   8 |   8 | 100      |   5 | 160 |   0 |  27 |  18 | 173 |   0 |   0 |
| WEXFORD        |   1 |   1 |   1 |   1 |   7 |   7 | 100      |   1 |  43 |   0 |  10 |   0 |  40 |   0 |   0 |



labeled_formulas: K=B-C
O=B*K
O=C*K
O=D-E
O=D*K
O=D*P
O=E*K
O=E*P
O=F*K
O=G*K
O=H*K
O=I*P
O=J*K
O=K*L
O=K*M
O=K*N
O=M*P
E=(B-C)+D
K=(B-C)*D
C=(D*P)+B
K=(B-C)*E
C=(E*P)+B
K=(B-C)*F
K=(B-C)*G
K=(B-C)*H
C=(I*P)+B
K=(B-C)*J
K=(B-C)*L
K=(B-C)*M
K=(B-C)*N
E=(B*O)+D
O=(B*D)*K
O=(B-D)*P
O=(B*E)*K
O=(B-E)*P
O=(B*F)*K
O=(B-F)*P
O=(B*G)*K
O=(B-G)*P
O=(B+H)*K
O=(B*I)*P
O=(B+J)*K
O=(B*K)*L
O=(B*K)*N
O=(B-M)*P
E=(C*O)+D
O=(C*D)*K
O=(C-D)*P
O=(C*E)*K
O=(C-E)*P
O=(C*F)*K
O=(C-F)*P
O=(C*G)*K
O=(C-G)*P
O=(C+H)*K
O=(C*I)*P
O=(C+J)*K
O=(C*K)*L
O=(C-M)*K
O=(C*K)*N
O=(C-M)*P
O=(D-E)*F
O=(D-E)*G
O=(D-E)*H
K=(D-E)*I
O=(D-E)*I
O=(D-E)*J
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_81811___xlsx_1959f5b272bdffe744f4b56bf9157864babe6e8d.xlsx___line_76