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 |
|-------:|:---------------------------------|:-------------------------------------------------------|-------:|----:|-------:|
| 100032 | BALLANTINE BAL BURT              | 1/6bbl Keg                                             |  70.08 |   0 |  70.08 |
| 102753 | BALLANTINE BAL ALE               | 12/40oz Btl LS                                         |  21.15 |   0 |  21.15 |
| 102742 | BALLANTINE BAL ALE               | 24/12oz 4/6 Btl Bskt                                   |  15.84 |   0 |  15.84 |
| 100601 | BALLANTINE BAL BURT              | 24/12oz 4/6 Btl Bskt                                   |  26.67 |   0 |  26.67 |
| 102764 | BALLANTINE BAL ALE               | 24/16oz 4/6 Can HiC                                    |  16    |   0 |  16    |
| 102668 | BLACK LABEL BLK LABL             | 24/12oz 2/12 Can Fibr                                  |   9.21 |   0 |   9.21 |
| 102570 | BLACK LABEL BLK LABL             | 24/12oz Can Fibr                                       |   9.21 |   0 |   9.21 |
| 102581 | BULL SCHLZ BULL ICE              | 12/40oz Btl LS                                         |  17.24 |   0 |  17.24 |
| 102729 | COLT 45 COLT 45 MALT             | 12/24oz Can LS                                         |  10.01 |   0 |  10.01 |
| 102657 | COLT 45 COLT HiGRV               | 12/24oz Can LS                                         |  10.01 |   0 |  10.01 |
| 102726 | COLT 45 COLT 45 MALT             | 12/40oz Btl LS                                         |  17.67 |   0 |  17.67 |
| 102691 | COLT 45 COLT 45 MALT             | 15/22oz Btl LS                                         |  13.71 |   0 |  13.71 |
| 102739 | COLT 45 COLT 45 MALT             | 24/16oz 4/6 Can HiC                                    |  14.09 |   0 |  14.09 |
| 102733 | COLT 45 COLT 45 MALT             | 30/12oz Can StkPk                                      |  11.92 |   0 |  11.92 |
| 102602 | COUNTRY CLUB CTRY CLB MALT       | 12/40oz Btl LS                                         |  15.6  |   0 |  15.6  |
| 104570 | Jack Daniel's JDCC MWP           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104571 | Jack Daniel's JDCC MBJC          | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104572 | Jack Daniel's JDCC MDP           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104573 | Jack Daniel's JDCC MBP           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104574 | Jack Daniel's JDCC MCL           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104575 | Jack Daniel's JDCC MSP           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104576 | Jack Daniel's JDCC MSC           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104579 | Jack Daniel's JDCC MLL           | 24/10oz 4/6 PrpLN Btl Bskt                             |  18    |   0 |  18    |
| 104563 | Jack Daniel's JDCC MDP           | 24/16oz LS Can                                         |  24.4  |   0 |  24.4  |
| 104564 | Jack Daniel's JDCC MLL           | 24/16oz LS Can                                         |  24.4  |   0 |  24.4  |
| 104565 | Jack Daniel's JDCC MWP           | 24/16oz LS Can                                         |  24.4  |   0 |  24.4  |
| 104566 | Jack Daniel's JDCC MSP           | 24/16oz LS Can                                         |  24.4  |   0 |  24.4  |
| 104567 | Jack Daniel's JDCC MSC           | 24/16oz LS Can                                         |  24.4  |   0 |  24.4  |
| 104568 | Jack Daniel's JDCC MDP           | 12/23.5oz Can LS                                       |  24.4  |   0 |  24.4  |
| 104569 | Jack Daniel's JDCC MSP           | 12/23.5oz Can LS                                       |  24.4  |   0 |  24.4  |
| 102743 | LONE STAR LONE STAR              | 24/12oz 4/6 Btl Bskt                                   |  15.28 |   0 |  15.28 |
| 100960 | NEW HOLLAND NH CABN FEVR         | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 100962 | NEW HOLLAND NH CABN FEVR         | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 100959 | NEW HOLLAND NH CABN FEVR         | 24/12oz 4/6 Btl Bskt                                   |  22.97 |   0 |  22.97 |
| 104401 | NEW HOLLAND NH DM 2021 RES1      | 1/6bbl Keg                                             |  84.33 |   0 |  84.33 |
| 104389 | NEW HOLLAND NH DM 2021 RES1      | 24/12oz 6/4 Btl Bskt                                   |  55.47 |   0 |  55.47 |
| 104402 | NEW HOLLAND NH DM 2021 RES2      | 1/6bbl Keg                                             |  84.33 |   0 |  84.33 |
| 104411 | NEW HOLLAND NH DM 2021 RES2      | 24/12oz 6/4 Btl Bskt                                   |  55.47 |   0 |  55.47 |
| 102104 | NEW HOLLAND NH DM WHT STOUT      | 1/2bbl Keg                                             | 130.32 |   0 | 130.32 |
| 102105 | NEW HOLLAND NH DM WHT STOUT      | 1/6bbl Keg                                             |  54.3  |   0 |  54.3  |
| 102103 | NEW HOLLAND NH DM WHT STOUT      | 24/12oz 4/6 Can Wrap                                   |  26.18 |   0 |  26.18 |
| 100965 | NEW HOLLAND NH DRGN MLK          | 1/2bbl Keg                                             | 214.39 |   0 | 214.39 |
| 100964 | NEW HOLLAND NH DRGN MLK          | 1/6bbl Keg                                             |  80.33 |   0 |  80.33 |
| 100963 | NEW HOLLAND NH DRGN MLK          | 12/22oz Btl LS                                         |  50.15 |   0 |  50.15 |
| 100961 | NEW HOLLAND NH DRGN MLK          | 24/12oz 6/4 Btl Bskt                                   |  51.47 |   0 |  51.47 |
| 100974 | NEW HOLLAND NH ICHABOD           | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 100973 | NEW HOLLAND NH ICHABOD           | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 100972 | NEW HOLLAND NH ICHABOD           | 24/12oz 4/6 Btl Bskt                                   |  22.97 |   0 |  22.97 |
| 101886 | NEW HOLLAND NH ICHABOD           | 24/12oz 4/6 Can Wrap                                   |  22.97 |   0 |  22.97 |
| 102196 | NEW HOLLAND NH LAKE & TRAIL      | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 102197 | NEW HOLLAND NH LAKE & TRAIL      | 1/6bbl Keg                                             |  50.34 |   0 |  50.34 |
| 102195 | NEW HOLLAND NH LAKE & TRAIL      | 24/12oz 4/6 Can Wrap                                   |  22.97 |   0 |  22.97 |
| 102330 | NEW HOLLAND NH LIGHTPOINT        | 1/2bbl Keg                                             | 102.51 |   0 | 102.51 |
| 102331 | NEW HOLLAND NH LIGHTPOINT        | 1/6bbl Keg                                             |  47.88 |   0 |  47.88 |
| 102328 | NEW HOLLAND NH LIGHTPOINT        | 24/12oz 2/12 Can Fibr                                  |  16.78 |   0 |  16.78 |
| 102329 | NEW HOLLAND NH LIGHTPOINT        | 24/12oz 4/6 Can Wrap                                   |  17.82 |   0 |  17.82 |
| 100987 | NEW HOLLAND NH MAD HAT           | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 100986 | NEW HOLLAND NH MAD HAT           | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 101547 | NEW HOLLAND NH MAD HAT           | 24/12oz 2/12 Can Fibr                                  |  21.97 |   0 |  21.97 |
| 100985 | NEW HOLLAND NH MAD HAT           | 24/12oz 4/6 Btl Bskt                                   |  22.97 |   0 |  22.97 |
| 101546 | NEW HOLLAND NH MAD HAT           | 24/12oz 4/6 Can Wrap                                   |  22.97 |   0 |  22.97 |
| 100999 | NEW HOLLAND NH POET              | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 100998 | NEW HOLLAND NH POET              | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 100997 | NEW HOLLAND NH POET              | 24/12oz 4/6 Btl Bskt                                   |  22.97 |   0 |  22.97 |
| 101538 | NEW HOLLAND NH SB SPCL           | 1/2bbl Keg                                             | 120    |   0 | 120    |
| 101539 | NEW HOLLAND NH SB SPCL           | 1/6bbl Keg                                             |  50    |   0 |  50    |
| 101549 | NEW HOLLAND NH SB SPCL GR1       | 1/2bbl Keg                                             | 145    |   0 | 145    |
| 101552 | NEW HOLLAND NH SB SPCL GR1       | 1/6bbl Keg                                             |  72    |   0 |  72    |
| 101551 | NEW HOLLAND NH SB SPCL GR2       | 1/2bbl Keg                                             | 230    |   0 | 230    |
| 101553 | NEW HOLLAND NH SB SPCL GR2       | 1/6bbl Keg                                             |  89    |   0 |  89    |
| 104498 | NEW HOLLAND NH TAN SUMMER ALE    | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 104497 | NEW HOLLAND NH TAN SUMMER ALE    | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 104499 | NEW HOLLAND NH TAN SUMMER ALE    | 24/12oz 4/6 | Aluminum Can Bulk | End Load Wrap | Tray |  22.97 |   0 |  22.97 |
| 101666 | NEW HOLLAND NH TAN SP MACH       | 1/2bbl Keg                                             | 158.38 |   0 | 158.38 |
| 101667 | NEW HOLLAND NH TAN SP MACH       | 1/6bbl Keg                                             |  73.32 |   0 |  73.32 |
| 101668 | NEW HOLLAND NH TAN SP MACH       | 24/12oz 4/6 | Aluminum Can Bulk | End Load Wrap | Tray |  31.97 |   0 |  31.97 |
| 104406 | NEW HOLLAND NH WATERMELON SPRITZ | 1/2bbl Keg                                             | 117.39 |   0 | 117.39 |
| 104405 | NEW HOLLAND NH WATERMELON SPRITZ | 1/6bbl Keg                                             |  50.33 |   0 |  50.33 |
| 104408 | NEW HOLLAND NH WATERMELON SPRITZ | 24/12oz 4/6 | Aluminum Can Bulk | End Load Wrap | Tray |  22.97 |   0 |  22.97 |
| 102791 | OLD MILWAUKEE OLD MIL            | 24/12oz 2/12 Can Fibr                                  |   8.89 |   0 |   8.89 |
| 102579 | OLD MILWAUKEE OLD MIL NA         | 24/12oz 2/12 Can Fibr                                  |   8.89 |   0 |   8.89 |
| 102792 | OLD MILWAUKEE OLD MIL            | 30/12oz Can StkPk                                      |  11.48 |   0 |  11.48 |
| 102573 | OLD MILWAUKEE OLD MIL LT         | 30/12oz Can StkPk                                      |  11.48 |   0 |  11.48 |
| 102612 | PABST PBR                        | 1/2bbl Keg                                             |  41.39 |   0 |  41.39 |
| 102599 | PABST PBR                        | 12/24oz Can LS                                         |  11.95 |   0 |  11.95 |
| 102716 | PABST PBR EASY                   | 12/24oz Can LS                                         |  11.95 |   0 |  11.95 |
| 102715 | PABST PBR EXTRA                  | 12/24oz Can LS                                         |  11.95 |   0 |  11.95 |
| 102402 | PABST PBR SELTZER BRY            | 12/24oz Can LS                                         |  17    |   0 |  17    |
| 102401 | PABST PBR SELTZERLIME            | 12/24oz Can LS                                         |  17    |   0 |  17    |
| 102590 | PABST PBR                        | 12/40oz Btl LS                                         |  20.13 |   0 |  20.13 |
| 102585 | PABST PBR                        | 24/12oz 2/12 Btl Fibr                                  |  10.36 |   0 |  10.36 |
| 102600 | PABST PBR                        | 24/12oz 2/12 Can Fibr                                  |  10.2  |   0 |  10.2  |
| 102614 | PABST PBR EASY                   | 24/12oz 2/12 Can Fibr                                  |  10.2  |   0 |  10.2  |
| 102703 | PABST PBR EXTRA                  | 24/12oz 2/12 Can Fibr                                  |  10.2  |   0 |  10.2  |
| 102806 | PABST PBR NON-ALC                | 24/12oz 2/12 Can Fibr                                  |  10.2  |   0 |  10.2  |
| 102589 | PABST PBR                        | 24/12oz 4/6 Btl Bskt                                   |  14.57 |   0 |  14.57 |
| 102428 | PABST PBR HARD T PCH             | 24/12oz 4/6 Can Wrap                                   |  20.24 |   0 |  20.24 |
| 104263 | PABST PBR HARD TEA VARIETY       | 24/12oz 2/12 Can Fibr                                  |  17.44 |   0 |  17.44 |
| 102586 | PABST PBR                        | 24/16oz 2/12 Can Fibr                                  |  12.65 |   0 |  12.65 |
| 102608 | PABST PBR                        | 24/16oz 4/6 Can HiC                                    |  13.19 |   0 |  13.19 |
| 102615 | PABST PBR EASY                   | 24/16oz 4/6 Can HiC                                    |  13.19 |   0 |  13.19 |
| 102704 | PABST PBR EXTRA                  | 24/16oz 4/6 Can HiC                                    |  13.19 |   0 |  13.19 |
| 102387 | PABST PBR SELTZER BRY            | 24/16oz 6/4 Can HiC                                    |  27.72 |   0 |  27.72 |
| 102386 | PABST PBR SELTZERLIME            | 24/16oz 6/4 Can HiC                                    |  27.72 |   0 |  27.72 |
| 104226 | PABST PBR SELTZER VARIETY        | 24/12oz 2/12 Can Fibr                                  |  17.44 |   0 |  17.44 |
| 102640 | SCHAEFER SCHFR                   | 24/12oz 2/12 Can Fibr                                  |   9.13 |   0 |   9.13 |
| 102666 | SCHLITZ SCHLZ PREM               | 1/2bbl Keg                                             |  58.97 |   0 |  58.97 |
| 102662 | SCHLITZ SCHLZ PREM               | 24/12oz 2/12 Btl Fibr                                  |  13.13 |   0 |  13.13 |
| 102664 | SCHLITZ SCHLZ PREM               | 24/12oz 2/12 Can Fibr                                  |  13.13 |   0 |  13.13 |
| 102663 | SCHLITZ SCHLZ PREM               | 24/12oz 4/6 Btl Bskt                                   |  15.54 |   0 |  15.54 |
| 102665 | SCHLITZ SCHLZ PREM               | 24/16oz 4/6 Can HiC                                    |  19.02 |   0 |  19.02 |
| 102655 | SCHLITZ ML SCHLZ MALT            | 12/24oz Can LS                                         |  10.01 |   0 |  10.01 |
| 102654 | SCHLITZ ML SCHLZ MALT            | 12/40oz Btl LS                                         |  15.15 |   0 |  15.15 |
| 102675 | SCHMIDT SCHMIDT                  | 24/12oz Can Fibr                                       |   8.93 |   0 |   8.93 |
| 102084 | SMALL TOWN NYF LEMONADE          | 24/12oz 4/6 Btl Bskt                                   |  22.75 |   0 |  22.75 |
| 100013 | SMALL TOWN NYF ROOT BR           | 24/12oz 4/6 Btl Bskt                                   |  22.75 |   0 |  22.75 |
| 102670 | ST. IDES ML ST IDES MALT         | 12/40oz Btl LS                                         |  17.37 |   0 |  17.37 |
| 102667 | ST. IDES ML ST IDES MALT         | 15/22oz Btl LS                                         |  13.66 |   0 |  13.66 |
| 102672 | ST. IDES ML ST IDES MALT         | 24/16oz 4/6 Can HiC                                    |  13.88 |   0 |  13.88 |
| 102696 | STROH STROH                      | 30/12oz Can StkPk                                      |  11.24 |   0 |  11.24 |



label: F=D-E
labeled_formulas: F=D-E
F=(A*E)+D
output: {"Arithmetic-Relationship": ["F=D-E"]}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_277121___xlsx_da25293d4cded3e9982f2352fe2a970b2cf86e70.xlsx___line_638