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Datasets:
SAIRfoundation
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equational-theories-selected-problems

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id
stringlengths
9
9
index
int64
1
200
difficulty
stringclasses
1 value
eq1_id
int64
54
4.65k
eq2_id
int64
221
4.58k
equation1
stringlengths
19
27
equation2
stringlengths
19
27
answer
bool
2 classes
hard_0001
1
hard
706
4,204
x = y * (y * ((x * y) * x))
x * y = ((z * x) * w) * y
false
hard_0002
2
hard
4,555
4,437
x * (y * z) = (z * w) * w
x * (y * x) = (x * y) * z
false
hard_0003
3
hard
286
1,238
x = ((y * y) * z) * x
x = x * (((y * x) * x) * x)
true
hard_0004
4
hard
4,634
4,340
(x * y) * x = (x * z) * w
x * (y * y) = x * (z * y)
false
hard_0005
5
hard
2,805
3,891
x = ((y * z) * (z * x)) * y
x * x = (y * (y * y)) * y
false
hard_0006
6
hard
363
373
x * x = (x * y) * z
x * x = (y * z) * w
false
hard_0007
7
hard
2,343
3,446
x = (y * (y * (y * z))) * x
x * y = z * (w * (w * y))
true
hard_0008
8
hard
646
1,020
x = x * (y * ((y * z) * y))
x = x * ((x * (x * x)) * x)
false
hard_0009
9
hard
2,656
2,863
x = ((x * x) * (y * z)) * y
x = ((x * (y * x)) * x) * y
false
hard_0010
10
hard
4,609
4,341
(x * x) * y = (y * y) * z
x * (y * y) = x * (z * z)
false
hard_0011
11
hard
1,880
2,031
x = (x * (y * z)) * (w * y)
x = (y * (z * w)) * (u * z)
false
hard_0012
12
hard
4,653
4,270
(x * y) * x = (z * w) * u
x * (x * x) = x * (y * y)
false
hard_0013
13
hard
4,208
3,356
x * y = ((z * y) * x) * x
x * y = y * (y * (y * y))
true
hard_0014
14
hard
1,480
572
x = (y * x) * (x * (x * z))
x = y * (z * (z * (x * y)))
false
hard_0015
15
hard
3,265
4,269
x * x = x * (y * (z * y))
x * (x * x) = x * (y * x)
false
hard_0016
16
hard
1,446
1,448
x = (x * y) * (x * (y * z))
x = (x * y) * (x * (z * y))
false
hard_0017
17
hard
1,286
2,891
x = y * (((x * y) * x) * y)
x = ((x * (y * z)) * z) * y
false
hard_0018
18
hard
2,105
1,229
x = ((y * x) * y) * (z * z)
x = x * (((x * y) * x) * y)
true
hard_0019
19
hard
3,926
3,920
x * y = (x * (y * x)) * z
x * y = (x * (x * z)) * x
false
hard_0020
20
hard
4,508
4,527
x * (y * z) = (x * x) * z
x * (y * z) = (y * y) * x
false
hard_0021
21
hard
907
221
x = y * ((y * x) * (x * y))
x = (y * (x * y)) * x
false
hard_0022
22
hard
3,961
3,722
x * y = (y * (y * x)) * x
x * y = (x * y) * (x * y)
false
hard_0023
23
hard
164
3,470
x = (x * y) * (z * z)
x * x = x * ((y * z) * w)
true
hard_0024
24
hard
706
860
x = y * (y * ((x * y) * x))
x = x * ((y * z) * (z * x))
false
hard_0025
25
hard
3,008
1,131
x = ((y * (z * z)) * x) * y
x = y * ((y * (z * x)) * z)
false
hard_0026
26
hard
2,890
3,261
x = ((x * (y * z)) * z) * x
x * x = x * (y * (y * x))
false
hard_0027
27
hard
54
2,695
x = x * (y * (x * z))
x = ((x * y) * (z * w)) * u
true
hard_0028
28
hard
3,583
4,406
x * y = z * ((x * x) * y)
x * (x * y) = (y * x) * y
false
hard_0029
29
hard
1,738
3,972
x = (y * y) * ((z * x) * x)
x * y = (y * (z * x)) * y
false
hard_0030
30
hard
2,301
4,303
x = (y * (x * (y * x))) * y
x * (x * y) = z * (y * x)
false
hard_0031
31
hard
2,089
900
x = ((y * x) * x) * (x * z)
x = y * ((x * z) * (z * w))
false
hard_0032
32
hard
2,089
404
x = ((y * x) * x) * (x * z)
x * y = (z * z) * z
false
hard_0033
33
hard
2,994
2,623
x = ((y * (z * y)) * y) * x
x = (y * ((z * w) * y)) * x
false
hard_0034
34
hard
386
3,639
x * y = (y * x) * z
x * y = z * ((w * y) * y)
true
hard_0035
35
hard
868
2,243
x = x * ((y * z) * (w * u))
x = (x * (x * (y * x))) * x
true
hard_0036
36
hard
3,290
3,264
x * x = y * (z * (x * z))
x * x = x * (y * (z * x))
false
hard_0037
37
hard
1,336
419
x = y * (((y * z) * y) * x)
x = x * (x * (y * (y * x)))
false
hard_0038
38
hard
3,919
3,330
x * y = (x * (x * y)) * z
x * y = x * (z * (y * y))
false
hard_0039
39
hard
830
1,229
x = x * ((x * y) * (z * z))
x = x * (((x * y) * x) * y)
false
hard_0040
40
hard
1,070
3,950
x = x * ((y * (z * w)) * w)
x * y = (x * (z * w)) * u
true
hard_0041
41
hard
3,118
3,441
x = (((y * x) * y) * z) * x
x * y = z * (w * (z * y))
true
hard_0042
42
hard
168
431
x = (y * x) * (x * z)
x = x * (y * (x * (y * z)))
false
hard_0043
43
hard
3,277
3,473
x * x = y * (x * (z * w))
x * x = y * ((x * x) * z)
false
hard_0044
44
hard
234
3,065
x = (y * (y * z)) * x
x = (((x * y) * x) * x) * x
true
hard_0045
45
hard
1,480
1,882
x = (y * x) * (x * (x * z))
x = (x * (y * z)) * (w * w)
false
hard_0046
46
hard
3,550
3,282
x * y = y * ((x * y) * z)
x * x = y * (y * (y * y))
true
hard_0047
47
hard
1,486
913
x = (y * x) * (x * (z * z))
x = y * ((y * x) * (z * y))
false
hard_0048
48
hard
242
3,197
x = (y * (z * y)) * x
x = (((y * z) * y) * y) * x
false
hard_0049
49
hard
4,541
4,578
x * (y * z) = (z * x) * y
x * (y * z) = (w * u) * y
false
hard_0050
50
hard
1,274
2,647
x = x * (((y * z) * w) * u)
x = ((x * x) * (x * y)) * y
true
hard_0051
51
hard
4,150
4,243
x * y = ((x * z) * w) * y
x * y = ((z * w) * x) * y
false
hard_0052
52
hard
169
2,056
x = (y * x) * (y * x)
x = ((x * y) * x) * (z * x)
true
hard_0053
53
hard
3,738
3,727
x * y = (x * z) * (y * w)
x * y = (x * y) * (z * x)
false
hard_0054
54
hard
4,246
4,469
x * y = ((z * w) * x) * u
x * (y * y) = (x * x) * x
true
hard_0055
55
hard
1,400
1,364
x = y * (((z * w) * x) * x)
x = y * (((z * x) * w) * w)
false
hard_0056
56
hard
1,953
608
x = (y * (y * z)) * (w * x)
x = y * (z * (w * (u * x)))
true
hard_0057
57
hard
964
698
x = y * ((z * y) * (y * x))
x = y * (x * ((z * w) * x))
false
hard_0058
58
hard
4,574
4,438
x * (y * z) = (w * w) * z
x * (y * x) = (x * z) * x
false
hard_0059
59
hard
298
1,045
x = ((y * z) * z) * x
x = x * ((y * (y * x)) * x)
true
hard_0060
60
hard
3,184
1,288
x = (((y * z) * x) * z) * x
x = y * (((x * y) * y) * x)
true
hard_0061
61
hard
447
1,247
x = x * (y * (z * (x * y)))
x = x * (((y * x) * z) * w)
true
hard_0062
62
hard
2,116
2,327
x = ((y * x) * z) * (z * y)
x = (y * (y * (x * x))) * x
false
hard_0063
63
hard
302
1,217
x = ((y * z) * w) * x
x = y * ((z * (w * u)) * x)
true
hard_0064
64
hard
862
2,466
x = x * ((y * z) * (z * z))
x = (x * ((y * y) * x)) * x
true
hard_0065
65
hard
59
2,240
x = x * (y * (z * y))
x = (x * (x * (x * y))) * x
true
hard_0066
66
hard
2,180
3,116
x = ((y * z) * y) * (x * z)
x = (((y * x) * y) * y) * y
false
hard_0067
67
hard
1,927
3,409
x = (y * (y * x)) * (z * x)
x * y = z * (y * (w * y))
true
hard_0068
68
hard
3,478
4,316
x * x = y * ((x * z) * y)
x * (y * x) = x * (z * x)
false
hard_0069
69
hard
286
532
x = ((y * y) * z) * x
x = y * (y * (z * (w * x)))
true
hard_0070
70
hard
2,105
1,229
x = ((y * x) * y) * (z * z)
x = x * (((x * y) * x) * y)
true
hard_0071
71
hard
3,926
3,920
x * y = (x * (y * x)) * z
x * y = (x * (x * z)) * x
false
hard_0072
72
hard
4,508
4,527
x * (y * z) = (x * x) * z
x * (y * z) = (y * y) * x
false
hard_0073
73
hard
907
221
x = y * ((y * x) * (x * y))
x = (y * (x * y)) * x
false
hard_0074
74
hard
3,961
3,722
x * y = (y * (y * x)) * x
x * y = (x * y) * (x * y)
false
hard_0075
75
hard
164
3,470
x = (x * y) * (z * z)
x * x = x * ((y * z) * w)
true
hard_0076
76
hard
706
860
x = y * (y * ((x * y) * x))
x = x * ((y * z) * (z * x))
false
hard_0077
77
hard
3,008
1,131
x = ((y * (z * z)) * x) * y
x = y * ((y * (z * x)) * z)
false
hard_0078
78
hard
2,890
3,261
x = ((x * (y * z)) * z) * x
x * x = x * (y * (y * x))
false
hard_0079
79
hard
54
2,695
x = x * (y * (x * z))
x = ((x * y) * (z * w)) * u
true
hard_0080
80
hard
3,583
4,406
x * y = z * ((x * x) * y)
x * (x * y) = (y * x) * y
false
hard_0081
81
hard
1,738
3,972
x = (y * y) * ((z * x) * x)
x * y = (y * (z * x)) * y
false
hard_0082
82
hard
2,301
4,303
x = (y * (x * (y * x))) * y
x * (x * y) = z * (y * x)
false
hard_0083
83
hard
2,089
900
x = ((y * x) * x) * (x * z)
x = y * ((x * z) * (z * w))
false
hard_0084
84
hard
2,089
404
x = ((y * x) * x) * (x * z)
x * y = (z * z) * z
false
hard_0085
85
hard
2,994
2,623
x = ((y * (z * y)) * y) * x
x = (y * ((z * w) * y)) * x
false
hard_0086
86
hard
386
3,639
x * y = (y * x) * z
x * y = z * ((w * y) * y)
true
hard_0087
87
hard
868
2,243
x = x * ((y * z) * (w * u))
x = (x * (x * (y * x))) * x
true
hard_0088
88
hard
3,290
3,264
x * x = y * (z * (x * z))
x * x = x * (y * (z * x))
false
hard_0089
89
hard
1,336
419
x = y * (((y * z) * y) * x)
x = x * (x * (y * (y * x)))
false
hard_0090
90
hard
3,919
3,330
x * y = (x * (x * y)) * z
x * y = x * (z * (y * y))
false
hard_0091
91
hard
830
1,229
x = x * ((x * y) * (z * z))
x = x * (((x * y) * x) * y)
false
hard_0092
92
hard
1,070
3,950
x = x * ((y * (z * w)) * w)
x * y = (x * (z * w)) * u
true
hard_0093
93
hard
3,118
3,441
x = (((y * x) * y) * z) * x
x * y = z * (w * (z * y))
true
hard_0094
94
hard
168
431
x = (y * x) * (x * z)
x = x * (y * (x * (y * z)))
false
hard_0095
95
hard
3,277
3,473
x * x = y * (x * (z * w))
x * x = y * ((x * x) * z)
false
hard_0096
96
hard
234
3,065
x = (y * (y * z)) * x
x = (((x * y) * x) * x) * x
true
hard_0097
97
hard
1,480
1,882
x = (y * x) * (x * (x * z))
x = (x * (y * z)) * (w * w)
false
hard_0098
98
hard
3,550
3,282
x * y = y * ((x * y) * z)
x * x = y * (y * (y * y))
true
hard_0099
99
hard
1,486
913
x = (y * x) * (x * (z * z))
x = y * ((y * x) * (z * y))
false
hard_0100
100
hard
242
3,197
x = (y * (z * y)) * x
x = (((y * z) * y) * y) * x
false
End of preview. Expand in Data Studio

Equational Theories Selected Problems

Update (March 24, 2026)

This dataset was updated on March 24, 2026.

Main changes:

  • added eq1_id and eq2_id to all released subsets
  • added the hard3 subset
  • updated the README and metadata to reflect the current released subsets

This dataset contains public training problem subsets selected from the Equational Theories Project raw implication dataset.

This dataset is intended for Stage 1 of the Mathematics Distillation Challenge: Equational Theories competition.

Competition page:

The full raw implication dataset contains 4694 laws, which yields 4694 * (4694 - 1) = 22,028,942 ordered implications.

The full raw implications table can be downloaded from the Equational Theories Project implications page by selecting Download raw implications table:

The full list of all 4694 equations is available here:

Because the full raw dataset is very large, the competition organizers selected public subsets from it to make testing and experimentation more practical for participants. These selected problems are drawn directly from the full raw dataset above.

Subsets

  • normal: 1000 selected problems, chosen programmatically, with 500 ground-truth TRUE labels and 500 ground-truth FALSE labels
  • hard: 200 selected problems co-curated by human mathematicians and AI, with 74 ground-truth TRUE labels and 126 ground-truth FALSE labels
  • hard1: a deduplicated version of the 200-problem hard subset, containing 69 unique problems total, with 24 ground-truth TRUE labels and 45 ground-truth FALSE labels
  • hard2: 200 selected problems co-curated by human mathematicians and AI, with 100 ground-truth TRUE labels and 100 ground-truth FALSE labels
  • hard3: 400 selected problems with 195 ground-truth TRUE labels and 205 ground-truth FALSE labels

All subsets are public training problems and are exposed as the train split in this repository.

Data Schema

Each record has the following fields:

  • id: stable identifier such as normal_0001, hard_0001, hard1_0001, hard2_0001, or hard3_0001
  • index: 1-based index within the subset
  • difficulty: normal or hard
  • eq1_id: 1-based equation identifier for equation1 in the full 4694-law equation list
  • eq2_id: 1-based equation identifier for equation2 in the full 4694-law equation list
  • equation1: Equation 1
  • equation2: Equation 2
  • answer: whether Equation 1 implies Equation 2 over all magmas

Subset Metadata

Subset-level metadata for the official released subsets is stored under metadata/.

Each metadata file contains:

  • subset_name: subset identifier
  • source: release source, such as official
  • curators: list of named curators, if provided
  • difficulty: subset difficulty label
  • problem_count: total number of problems in the subset
  • true_count: number of ground-truth TRUE labels
  • false_count: number of ground-truth FALSE labels
  • selection_method: high-level selection method such as programmatic, human, or human_ai
  • derived_from: parent subset name if the subset is derived from another subset
  • notes: optional explanatory notes

Records in hard1, hard2, and hard3 use subset-specific IDs and keep difficulty: hard.

Usage 

from datasets import load_dataset

normal = load_dataset(
    "SAIRfoundation/equational-theories-selected-problems",
    "normal",
    split="train",
)

hard = load_dataset(
    "SAIRfoundation/equational-theories-selected-problems",
    "hard",
    split="train",
)

hard1 = load_dataset(
    "SAIRfoundation/equational-theories-selected-problems",
    "hard1",
    split="train",
)

hard2 = load_dataset(
    "SAIRfoundation/equational-theories-selected-problems",
    "hard2",
    split="train",
)

hard3 = load_dataset(
    "SAIRfoundation/equational-theories-selected-problems",
    "hard3",
    split="train",
)

Files

  • data/normal.jsonl
  • data/hard.jsonl
  • data/hard1.jsonl
  • data/hard2.jsonl
  • data/hard3.jsonl
  • metadata/normal.json
  • metadata/hard.json
  • metadata/hard1.json
  • metadata/hard2.json
  • metadata/hard3.json
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