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Datasets:
HyperCactus0
/
LeanTransitionCorpus

Tasks:
Text Generation
Tags:
lean
lean4
automated-theorem-proving
theorem-proving
formal-mathematics
mathematics
License:
Dataset card Data Studio Files
xet
 Community
1
HyperCactus0 commited on
Commit
a61a84f
·
verified ·
1 Parent(s): 245cb32

Record 19 verified HPC batch manifests

Browse files
Files changed (19) hide show
  1. manifests/batches/deepseek/000147.json +148 -0
  2. manifests/batches/deepseek/000149.json +166 -0
  3. manifests/batches/deepseek/000151.json +166 -0
  4. manifests/batches/deepseek/000152.json +166 -0
  5. manifests/batches/mathlib/000076.json +1280 -0
  6. manifests/batches/mathlib/000077.json +80 -0
  7. manifests/batches/mathlib/000078.json +372 -0
  8. manifests/batches/mathlib/000079.json +854 -0
  9. manifests/batches/mathlib/000080.json +0 -0
  10. manifests/batches/mathlib/000081.json +365 -0
  11. manifests/batches/mathlib/000082.json +994 -0
  12. manifests/batches/mathlib/000083.json +295 -0
  13. manifests/batches/numina/000836.json +44 -0
  14. manifests/batches/numina/000837.json +51 -0
  15. manifests/batches/numina/000838.json +44 -0
  16. manifests/batches/numina/000839.json +51 -0
  17. manifests/batches/numina/000840.json +51 -0
  18. manifests/batches/numina/000842.json +51 -0
  19. manifests/batches/numina/000843.json +51 -0
manifests/batches/deepseek/000147.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000147",
4
+ "batch_fingerprint": "bfe1ceb56087f2283913f9c7a76e08b37c09c250794ca7061337c2c0ca913f9e",
5
+ "evidence": {
6
+ "batch_fingerprint": "bfe1ceb56087f2283913f9c7a76e08b37c09c250794ca7061337c2c0ca913f9e",
7
+ "contamination_matches": [],
8
+ "failures": [],
9
+ "input_proofs": 50,
10
+ "input_units": 50,
11
+ "proofs": 50,
12
+ "published_proofs": 50,
13
+ "published_transitions": 122,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 122,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
20
+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000147-0000-2e7dc364d2c18a6c.parquet",
24
+ "sha256": "2e7dc364d2c18a6ccdbef2a06aabe24e15200190787762d8580c9b254bc0703d",
25
+ "bytes": 313029,
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+ "rows": 101,
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+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
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+ },
29
+ {
30
+ "path": "data/dev/deepseek-000147-0000-fca5ee570bc2e270.parquet",
31
+ "sha256": "fca5ee570bc2e2700c8be5126edfe4b388dd8b302e062b03283c463fdf4b25f4",
32
+ "bytes": 72255,
33
+ "rows": 7,
34
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
35
+ },
36
+ {
37
+ "path": "data/internal_test/deepseek-000147-0000-0c0d469a3edc6ff1.parquet",
38
+ "sha256": "0c0d469a3edc6ff17245fefa9aaea48db587bf9c3bd64253966a77619d3109da",
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+ "bytes": 55078,
40
+ "rows": 14,
41
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
42
+ }
43
+ ],
44
+ "proof_ids": [
45
+ "deepseek:thm_7382",
46
+ "deepseek:thm_7383",
47
+ "deepseek:thm_7384",
48
+ "deepseek:thm_7385",
49
+ "deepseek:thm_7386",
50
+ "deepseek:thm_7387",
51
+ "deepseek:thm_7388",
52
+ "deepseek:thm_7389",
53
+ "deepseek:thm_7390",
54
+ "deepseek:thm_7391",
55
+ "deepseek:thm_7392",
56
+ "deepseek:thm_7393",
57
+ "deepseek:thm_7394",
58
+ "deepseek:thm_7395",
59
+ "deepseek:thm_7396",
60
+ "deepseek:thm_7397",
61
+ "deepseek:thm_7398",
62
+ "deepseek:thm_7399",
63
+ "deepseek:thm_7400",
64
+ "deepseek:thm_7401",
65
+ "deepseek:thm_7402",
66
+ "deepseek:thm_7403",
67
+ "deepseek:thm_7404",
68
+ "deepseek:thm_7405",
69
+ "deepseek:thm_7406",
70
+ "deepseek:thm_7407",
71
+ "deepseek:thm_7408",
72
+ "deepseek:thm_7409",
73
+ "deepseek:thm_7410",
74
+ "deepseek:thm_7411",
75
+ "deepseek:thm_7412",
76
+ "deepseek:thm_7413",
77
+ "deepseek:thm_7414",
78
+ "deepseek:thm_7415",
79
+ "deepseek:thm_7416",
80
+ "deepseek:thm_7417",
81
+ "deepseek:thm_7418",
82
+ "deepseek:thm_7419",
83
+ "deepseek:thm_7420",
84
+ "deepseek:thm_7421",
85
+ "deepseek:thm_7422",
86
+ "deepseek:thm_7423",
87
+ "deepseek:thm_7424",
88
+ "deepseek:thm_7425",
89
+ "deepseek:thm_7426",
90
+ "deepseek:thm_7427",
91
+ "deepseek:thm_7428",
92
+ "deepseek:thm_7429",
93
+ "deepseek:thm_7430",
94
+ "deepseek:thm_7431"
95
+ ],
96
+ "proof_keys": [
97
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007350.lean\",\"565262dbbe99bf9d526955951ee032a509a959096f28dbbc351acd9511a597a5\",0,\"thm_7382\"]",
98
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007351.lean\",\"94cf987010d4b2f22ea365d7900057af41bf58cdb597530b87b6739be2b01779\",0,\"thm_7383\"]",
99
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007352.lean\",\"6cbe7dcce6373b75ab7bbbf7982abbb30f475556c3f8cd9ab49efa0a896233b8\",0,\"thm_7384\"]",
100
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007353.lean\",\"8a15a5e33ad5a5926695ad5b395e779a01337e81f7f3688441400647442f4c4e\",0,\"thm_7385\"]",
101
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007354.lean\",\"bf2928c712845ab5e695bf7cc95799089502a814a0a22e9ada8fb8428d5e3de7\",0,\"thm_7386\"]",
102
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007355.lean\",\"8903bc9653add81b278fcd16e7d29cd48cfa05a056904f19fc956cd9c1828d9f\",0,\"thm_7387\"]",
103
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007356.lean\",\"245a9b3c5cbe26d385f0748eafa7e4f05dc94cf5c32face7487a938a25adb55b\",0,\"thm_7388\"]",
104
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007357.lean\",\"06f3170e5329d0002049edcff91c05a4442f9e9d55b1dc52ebbad98315f5e9d8\",0,\"thm_7389\"]",
105
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007358.lean\",\"9a75266d9ef5dd45e6bab91cb2b29dad75d7e85f2f496844d44add802746151f\",0,\"thm_7390\"]",
106
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007359.lean\",\"b71e494232fed7ade58791f7e17153192e7eb5c90707397e85caa034f3216c34\",0,\"thm_7391\"]",
107
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007360.lean\",\"460d1466a7f639cb9354da86d90293c2f244043b079569ca6235149ad960075f\",0,\"thm_7392\"]",
108
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007361.lean\",\"0be54a4e46da2b536bb1b5d6f8e0683869367184969cd989f169d70c164dc385\",0,\"thm_7393\"]",
109
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007362.lean\",\"18e85295c35fb88a0783a047dfcba662614af8ea723fc3a4cc4af6c4b56d4044\",0,\"thm_7394\"]",
110
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007363.lean\",\"312e8a7308d8c24c701bedfa080e740cc49528e917b60c833dda10a142640eaf\",0,\"thm_7395\"]",
111
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007364.lean\",\"c6ec2da5d093aa520a0d0dddb8a4c477f8d982d3d1500115b6275c674078ba66\",0,\"thm_7396\"]",
112
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007365.lean\",\"fe7a84b8fed18b303fb86602d2bafc25ec46315ae6373038e3d035025f87fefc\",0,\"thm_7397\"]",
113
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007366.lean\",\"639df2278130199f60b809bc0bce0a9452295c721eb75a6e5e9af4b568b26891\",0,\"thm_7398\"]",
114
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007367.lean\",\"e81f54a431f908677e600b4dcc3962b94f68c6084da1d1d06ba1b557b73b4b41\",0,\"thm_7399\"]",
115
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007368.lean\",\"c87cd796b1248b924867a5c9c44841b70c5d4b039b1a61e2928df94684075428\",0,\"thm_7400\"]",
116
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007369.lean\",\"43dc4fbba711d19d7d3fa114a014f0cd3252176cfe29d1ad31d844112d9531f0\",0,\"thm_7401\"]",
117
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007370.lean\",\"db682efeee38990d5c36b65eb74c1aa27058fbe1564e5ef3b49c1508ac8bd93a\",0,\"thm_7402\"]",
118
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007371.lean\",\"c5ba22b60b58fd2ead38bd334d34e496db13c45158e767db5895ae72a64289a4\",0,\"thm_7403\"]",
119
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007372.lean\",\"f27a7d3c7436405782f1883890ef7bc26622157c18a5680caebfa8a33b5f60d3\",0,\"thm_7404\"]",
120
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007373.lean\",\"9dbaee1a704fabcc84acae9e9df903bccf6acbf92f74bc08dd3061d5a25f2b10\",0,\"thm_7405\"]",
121
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007374.lean\",\"c8ea82ec27a48dfc59160ae16c5b560b4d5a86897535b3d1ea25da2a0a2c479e\",0,\"thm_7406\"]",
122
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007375.lean\",\"46ab13341bfed6acdad8f0adb47394cf5338716200b588e2c2387b27c9a4e3d3\",0,\"thm_7407\"]",
123
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007376.lean\",\"2db2e2e9ce857f3d9ae2caf71441013a4c153ca9d40cde85e865534fd7b0d479\",0,\"thm_7408\"]",
124
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007377.lean\",\"d7cd0fff707e103ca07f34b89876ed2d6e992d477f587d02c20c3152e61733e9\",0,\"thm_7409\"]",
125
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007378.lean\",\"078b28de5185eae5a275a8450f8d1eef7c95dd2034404432ab497a3ef3fb862a\",0,\"thm_7410\"]",
126
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007379.lean\",\"88103d4040e0fd15292a003004140548d17ca2e60b065c7072a6791d5cc95dd4\",0,\"thm_7411\"]",
127
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007380.lean\",\"0d1b079e381f720a1d6450a29df57fc78e8662b7b8de71dd1f08c7f002ce51da\",0,\"thm_7412\"]",
128
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007381.lean\",\"140133a4d0d87179aa36a74fe917f7b69324521d0b324c2521f10676ef825372\",0,\"thm_7413\"]",
129
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007382.lean\",\"2dfbb1256c8115c764e87c61dd287bb98559a5322cfc5578a4ac94f81068b259\",0,\"thm_7414\"]",
130
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007383.lean\",\"d8e74c92c36ab7ab9aebe03135b4256a6a16baa52a3d1df4efc6997f52d77731\",0,\"thm_7415\"]",
131
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007384.lean\",\"f1195ca9ff7abd404d7d4ddab61c44fab2d5539eeb5336e2eebbda98794d63b3\",0,\"thm_7416\"]",
132
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007385.lean\",\"ca273b50edba7987d06d9f6885c59605ff1d9185c9e0b6825427f976d6bba852\",0,\"thm_7417\"]",
133
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007386.lean\",\"d4cd030e6a8f25633933bd662bafe6aeae8c08b0adc9bbbab38b1edbc828e3ca\",0,\"thm_7418\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007387.lean\",\"fcd2ed395b14abbd6f86674725f378a7c2049f9f61646e704c2785afd666d61c\",0,\"thm_7419\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007388.lean\",\"a513308c78031fa21850acabd86b1278363dd6e5c2e14721a3cf29b934c314e7\",0,\"thm_7420\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007389.lean\",\"80666110173a60ce93b4f25025c52df57ce19c207488f8d92cb27e5e12811b4b\",0,\"thm_7421\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007390.lean\",\"4b20b837a210550d1b915be57b5a98e99794234873b08eb1abad1d9f0f7379c2\",0,\"thm_7422\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007391.lean\",\"23b43a69cb1854c452552659e054f1075fdb0f7ab357ecaf95dcc5e48a6663a9\",0,\"thm_7423\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007392.lean\",\"c786797bd6841106627cc994ab954fd53f5dcdab02e7a0d987b2a5cc1e0e674f\",0,\"thm_7424\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007393.lean\",\"6bc14723e094180177493baad2b4bcb84be81d384a3e4b151ac08ab0f6d58e49\",0,\"thm_7425\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007394.lean\",\"00c5d9218caf10deaa2b4bb14209cc2a41a8d712810ea373b237ce57f6fb3da5\",0,\"thm_7426\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007395.lean\",\"27a18676e385985a32a66b3f9cc5564608722edcf97f80e75ff47f2741ecdde3\",0,\"thm_7427\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007396.lean\",\"e52ef3c0f70a9ffbfd2cb004736097ee29fb5af9a797acd81af3f0b489eb2935\",0,\"thm_7428\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007397.lean\",\"45387a654f6bb7bc854da06afe3cdccd3bb6bda01c8538f542e5bd3b8f43a7fb\",0,\"thm_7429\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007398.lean\",\"bbdd8f7f87311fd87334d8072407c8c3dfe86ffee3e32c6af38d641ed24f3ca4\",0,\"thm_7430\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_007399.lean\",\"177ebe73d5af332a506ddff7134424aec1bdf06571d1b5d8def3b815e0873ce1\",0,\"thm_7431\"]"
147
+ ]
148
+ }
manifests/batches/deepseek/000149.json ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000149",
4
+ "batch_fingerprint": "c345e3d452f025ea1c6e7c99cb9798a899a35199b27d058306d809988bace58b",
5
+ "evidence": {
6
+ "batch_fingerprint": "c345e3d452f025ea1c6e7c99cb9798a899a35199b27d058306d809988bace58b",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "final replayable tactic does not close all goals",
11
+ "id": "thm_7527",
12
+ "index": 7495,
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "file_path": "Full/Proof_007495.lean",
16
+ "id": "thm_7527",
17
+ "index": 7495,
18
+ "metadata": {
19
+ "declaration_kind": "theorem",
20
+ "source_dataset": "deepseek",
21
+ "theorem_name": "thm_7527",
22
+ "theorem_ordinal": 0
23
+ },
24
+ "source_sha256": "1b0c0363cb5474dcb33f682ba033298f53808609f9ab2df961f27c0f50fad8ed"
25
+ },
26
+ "source_file": "Full/Proof_007495.lean"
27
+ }
28
+ ],
29
+ "input_proofs": 50,
30
+ "input_units": 50,
31
+ "proofs": 49,
32
+ "published_proofs": 49,
33
+ "published_transitions": 103,
34
+ "replay_failures": [],
35
+ "trace_mode": "individual",
36
+ "transitions": 103,
37
+ "validation_failures": [],
38
+ "validation_rejected_proofs": 0,
39
+ "zero_transition_proofs": 0
40
+ },
41
+ "shards": [
42
+ {
43
+ "path": "data/train/deepseek-000149-0000-8d724ab0654c1e68.parquet",
44
+ "sha256": "8d724ab0654c1e68defba1c59548643360a15aa4a63cd2d5dc6d4e842b349981",
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+ "bytes": 290760,
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+ "rows": 87,
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+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
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+ },
49
+ {
50
+ "path": "data/dev/deepseek-000149-0000-e84322fd609c91f9.parquet",
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manifests/batches/deepseek/000151.json ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ ]
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+ }
manifests/batches/deepseek/000152.json ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "contamination_matches": [],
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+ "failures": [
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+ "error": "final replayable tactic does not close all goals",
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+ "id": "thm_7651",
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+ "index": 7619,
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+ "reason": "import_failure",
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+ "selection": {
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+ "id": "thm_7651",
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+ "metadata": {
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+ "theorem_name": "thm_7651",
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+ "mathlib:Algebra.ringChar_eq",
362
+ "mathlib:RingHom.charP",
363
+ "mathlib:charP_of_injective_ringHom",
364
+ "mathlib:charZero_of_injective_ringHom"
365
+ ],
366
+ "proof_keys": [
367
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Algebra.lean\",null,null,\"Algebra.ringChar_eq\"]",
368
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Algebra.lean\",null,null,\"RingHom.charP\"]",
369
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Algebra.lean\",null,null,\"charP_of_injective_ringHom\"]",
370
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Algebra.lean\",null,null,\"charZero_of_injective_ringHom\"]"
371
+ ]
372
+ }
manifests/batches/mathlib/000079.json ADDED
@@ -0,0 +1,854 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000079",
4
+ "batch_fingerprint": "a8033b03e7885818d0df3a8cdc35dc2295666027681b0a44274b0b0c484ac9c5",
5
+ "evidence": {
6
+ "batch_fingerprint": "a8033b03e7885818d0df3a8cdc35dc2295666027681b0a44274b0b0c484ac9c5",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "CharP.char_is_prime",
12
+ "id": "CharP.char_is_prime",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
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19
+ ],
20
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
21
+ "full_name": "CharP.char_is_prime",
22
+ "start": [
23
+ 179,
24
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25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "CharP.ringChar_ne_zero_of_finite",
34
+ "id": "CharP.ringChar_ne_zero_of_finite",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
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41
+ ],
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+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
43
+ "full_name": "CharP.ringChar_ne_zero_of_finite",
44
+ "start": [
45
+ 169,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "Commute.exists_add_pow_prime_eq",
56
+ "id": "Commute.exists_add_pow_prime_eq",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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63
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64
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
65
+ "full_name": "Commute.exists_add_pow_prime_eq",
66
+ "start": [
67
+ 54,
68
+ 11
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "Commute.exists_add_pow_prime_pow_eq",
78
+ "id": "Commute.exists_add_pow_prime_pow_eq",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
83
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85
+ ],
86
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
87
+ "full_name": "Commute.exists_add_pow_prime_pow_eq",
88
+ "start": [
89
+ 49,
90
+ 11
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
96
+ },
97
+ {
98
+ "error": "theorem has no selected tactic transitions",
99
+ "full_name": "add_pow_char",
100
+ "id": "add_pow_char",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 121,
106
+ 50
107
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108
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
109
+ "full_name": "add_pow_char",
110
+ "start": [
111
+ 119,
112
+ 1
113
+ ],
114
+ "traced_tactics": [],
115
+ "url": "https://github.com/leanprover-community/mathlib4"
116
+ },
117
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
118
+ },
119
+ {
120
+ "error": "theorem has no selected tactic transitions",
121
+ "full_name": "add_pow_char_pow",
122
+ "id": "add_pow_char_pow",
123
+ "reason": "import_failure",
124
+ "selection": {
125
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
126
+ "end": [
127
+ 126,
128
+ 54
129
+ ],
130
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
131
+ "full_name": "add_pow_char_pow",
132
+ "start": [
133
+ 124,
134
+ 1
135
+ ],
136
+ "traced_tactics": [],
137
+ "url": "https://github.com/leanprover-community/mathlib4"
138
+ },
139
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
140
+ },
141
+ {
142
+ "error": "theorem has no selected tactic transitions",
143
+ "full_name": "add_pow_prime_eq",
144
+ "id": "add_pow_prime_eq",
145
+ "reason": "import_failure",
146
+ "selection": {
147
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
148
+ "end": [
149
+ 75,
150
+ 40
151
+ ],
152
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
153
+ "full_name": "add_pow_prime_eq",
154
+ "start": [
155
+ 72,
156
+ 1
157
+ ],
158
+ "traced_tactics": [],
159
+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
162
+ },
163
+ {
164
+ "error": "theorem has no selected tactic transitions",
165
+ "full_name": "add_pow_prime_pow_eq",
166
+ "id": "add_pow_prime_pow_eq",
167
+ "reason": "import_failure",
168
+ "selection": {
169
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
170
+ "end": [
171
+ 69,
172
+ 46
173
+ ],
174
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
175
+ "full_name": "add_pow_prime_pow_eq",
176
+ "start": [
177
+ 65,
178
+ 1
179
+ ],
180
+ "traced_tactics": [],
181
+ "url": "https://github.com/leanprover-community/mathlib4"
182
+ },
183
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
184
+ },
185
+ {
186
+ "error": "theorem has no selected tactic transitions",
187
+ "full_name": "exists_add_pow_prime_eq",
188
+ "id": "exists_add_pow_prime_eq",
189
+ "reason": "import_failure",
190
+ "selection": {
191
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
192
+ "end": [
193
+ 85,
194
+ 47
195
+ ],
196
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
197
+ "full_name": "exists_add_pow_prime_eq",
198
+ "start": [
199
+ 83,
200
+ 1
201
+ ],
202
+ "traced_tactics": [],
203
+ "url": "https://github.com/leanprover-community/mathlib4"
204
+ },
205
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
206
+ },
207
+ {
208
+ "error": "theorem has no selected tactic transitions",
209
+ "full_name": "exists_add_pow_prime_pow_eq",
210
+ "id": "exists_add_pow_prime_pow_eq",
211
+ "reason": "import_failure",
212
+ "selection": {
213
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
214
+ "end": [
215
+ 80,
216
+ 53
217
+ ],
218
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
219
+ "full_name": "exists_add_pow_prime_pow_eq",
220
+ "start": [
221
+ 78,
222
+ 1
223
+ ],
224
+ "traced_tactics": [],
225
+ "url": "https://github.com/leanprover-community/mathlib4"
226
+ },
227
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
228
+ },
229
+ {
230
+ "error": "theorem has no selected tactic transitions",
231
+ "full_name": "sub_pow_char",
232
+ "id": "sub_pow_char",
233
+ "reason": "import_failure",
234
+ "selection": {
235
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
236
+ "end": [
237
+ 131,
238
+ 50
239
+ ],
240
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
241
+ "full_name": "sub_pow_char",
242
+ "start": [
243
+ 129,
244
+ 1
245
+ ],
246
+ "traced_tactics": [],
247
+ "url": "https://github.com/leanprover-community/mathlib4"
248
+ },
249
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
250
+ },
251
+ {
252
+ "error": "theorem has no selected tactic transitions",
253
+ "full_name": "sub_pow_char_pow",
254
+ "id": "sub_pow_char_pow",
255
+ "reason": "import_failure",
256
+ "selection": {
257
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
258
+ "end": [
259
+ 136,
260
+ 54
261
+ ],
262
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
263
+ "full_name": "sub_pow_char_pow",
264
+ "start": [
265
+ 134,
266
+ 1
267
+ ],
268
+ "traced_tactics": [],
269
+ "url": "https://github.com/leanprover-community/mathlib4"
270
+ },
271
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
272
+ },
273
+ {
274
+ "error": "no internal trace for tactic 'simp'",
275
+ "full_name": "sub_pow_char_pow_of_commute",
276
+ "id": "sub_pow_char_pow_of_commute",
277
+ "reason": "import_failure",
278
+ "selection": {
279
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
280
+ "end": [
281
+ 116,
282
+ 61
283
+ ],
284
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
285
+ "full_name": "sub_pow_char_pow_of_commute",
286
+ "start": [
287
+ 110,
288
+ 1
289
+ ],
290
+ "traced_tactics": [
291
+ {
292
+ "annotated_tactic": [
293
+ "\"induction n with\\n | zero => simp\\n | succ n n_ih =>\\n rw [<a>pow_succ<\\/a>, <a>pow_mul<\\/a>, <a>pow_mul<\\/a>, <a>pow_mul<\\/a>, n_ih]\\n apply <a>sub_pow_char_of_commute<\\/a>; apply <a>Commute.pow_pow<\\/a> h\"",
294
+ "[{\"full_name\":\"pow_succ\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[657,9],\"def_end_pos\":[657,17]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]},{\"full_name\":\"sub_pow_char_of_commute\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Basic.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,32]},{\"full_name\":\"Commute.pow_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[191,9],\"def_end_pos\":[191,16]}]"
295
+ ],
296
+ "state_after": "no goals",
297
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nn : \u2115\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n",
298
+ "tactic": "induction n with\n| zero => simp\n| succ n n_ih =>\n rw [pow_succ, pow_mul, pow_mul, pow_mul, n_ih]\n apply sub_pow_char_of_commute; apply Commute.pow_pow h"
299
+ },
300
+ {
301
+ "annotated_tactic": [
302
+ "\"simp\"",
303
+ "[]"
304
+ ],
305
+ "state_after": "no goals",
306
+ "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ 0 = x ^ p ^ 0 - y ^ p ^ 0",
307
+ "tactic": "simp"
308
+ },
309
+ {
310
+ "annotated_tactic": [
311
+ "\"rw [<a>pow_succ<\\/a>, <a>pow_mul<\\/a>, <a>pow_mul<\\/a>, <a>pow_mul<\\/a>, n_ih]\"",
312
+ "[{\"full_name\":\"pow_succ\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[657,9],\"def_end_pos\":[657,17]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]}]"
313
+ ],
314
+ "state_after": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p",
315
+ "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x - y) ^ p ^ (n + 1) = x ^ p ^ (n + 1) - y ^ p ^ (n + 1)",
316
+ "tactic": "rw [pow_succ, pow_mul, pow_mul, pow_mul, n_ih]"
317
+ },
318
+ {
319
+ "annotated_tactic": [
320
+ "\"apply <a>sub_pow_char_of_commute<\\/a>\"",
321
+ "[{\"full_name\":\"sub_pow_char_of_commute\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Basic.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,32]}]"
322
+ ],
323
+ "state_after": "case succ.h\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 Commute (x ^ p ^ n) (y ^ p ^ n)",
324
+ "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p",
325
+ "tactic": "apply sub_pow_char_of_commute"
326
+ },
327
+ {
328
+ "annotated_tactic": [
329
+ "\"apply <a>Commute.pow_pow<\\/a> h\"",
330
+ "[{\"full_name\":\"Commute.pow_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[191,9],\"def_end_pos\":[191,16]}]"
331
+ ],
332
+ "state_after": "no goals",
333
+ "state_before": "case succ.h\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 Commute (x ^ p ^ n) (y ^ p ^ n)",
334
+ "tactic": "apply Commute.pow_pow h"
335
+ }
336
+ ],
337
+ "url": "https://github.com/leanprover-community/mathlib4"
338
+ },
339
+ "source_file": "Mathlib/Algebra/CharP/Basic.lean"
340
+ },
341
+ {
342
+ "error": "theorem has no selected tactic transitions",
343
+ "full_name": "charP_of_card_eq_prime",
344
+ "id": "charP_of_card_eq_prime",
345
+ "reason": "import_failure",
346
+ "selection": {
347
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
348
+ "end": [
349
+ 89,
350
+ 72
351
+ ],
352
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
353
+ "full_name": "charP_of_card_eq_prime",
354
+ "start": [
355
+ 86,
356
+ 1
357
+ ],
358
+ "traced_tactics": [],
359
+ "url": "https://github.com/leanprover-community/mathlib4"
360
+ },
361
+ "source_file": "Mathlib/Algebra/CharP/CharAndCard.lean"
362
+ },
363
+ {
364
+ "error": "theorem has no selected tactic transitions",
365
+ "full_name": "isUnit_iff_not_dvd_char",
366
+ "id": "isUnit_iff_not_dvd_char",
367
+ "reason": "import_failure",
368
+ "selection": {
369
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
370
+ "end": [
371
+ 54,
372
+ 97
373
+ ],
374
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
375
+ "full_name": "isUnit_iff_not_dvd_char",
376
+ "start": [
377
+ 52,
378
+ 1
379
+ ],
380
+ "traced_tactics": [],
381
+ "url": "https://github.com/leanprover-community/mathlib4"
382
+ },
383
+ "source_file": "Mathlib/Algebra/CharP/CharAndCard.lean"
384
+ },
385
+ {
386
+ "error": "no internal trace for tactic 'rintro \u27e8r, hr\u27e9'",
387
+ "full_name": "isUnit_iff_not_dvd_char_of_ringChar_ne_zero",
388
+ "id": "isUnit_iff_not_dvd_char_of_ringChar_ne_zero",
389
+ "reason": "import_failure",
390
+ "selection": {
391
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
392
+ "end": [
393
+ 47,
394
+ 45
395
+ ],
396
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
397
+ "full_name": "isUnit_iff_not_dvd_char_of_ringChar_ne_zero",
398
+ "start": [
399
+ 24,
400
+ 1
401
+ ],
402
+ "traced_tactics": [
403
+ {
404
+ "annotated_tactic": [
405
+ "\"have hch := <a>CharP.cast_eq_zero<\\/a> R (<a>ringChar<\\/a> R)\"",
406
+ "[{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]},{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]}]"
407
+ ],
408
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\n\u22a2 IsUnit \u2191p \u2194 \u00acp \u2223 ringChar R",
409
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\n\u22a2 IsUnit \u2191p \u2194 \u00acp \u2223 ringChar R",
410
+ "tactic": "have hch := CharP.cast_eq_zero R (ringChar R)"
411
+ },
412
+ {
413
+ "annotated_tactic": [
414
+ "\"have hp : p.Prime := <a>Fact.out<\\/a>\"",
415
+ "[{\"full_name\":\"Fact.out\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[117,3],\"def_end_pos\":[117,6]}]"
416
+ ],
417
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 IsUnit \u2191p \u2194 \u00acp \u2223 ringChar R",
418
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\n\u22a2 IsUnit \u2191p \u2194 \u00acp \u2223 ringChar R",
419
+ "tactic": "have hp : p.Prime := Fact.out"
420
+ },
421
+ {
422
+ "annotated_tactic": [
423
+ "\"constructor\"",
424
+ "[]"
425
+ ],
426
+ "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 IsUnit \u2191p \u2192 \u00acp \u2223 ringChar R\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 \u00acp \u2223 ringChar R \u2192 IsUnit \u2191p",
427
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 IsUnit \u2191p \u2194 \u00acp \u2223 ringChar R",
428
+ "tactic": "constructor"
429
+ },
430
+ {
431
+ "annotated_tactic": [
432
+ "\"rintro h\\u2081 \\u27e8q, hq\\u27e9\"",
433
+ "[]"
434
+ ],
435
+ "state_after": "case mp.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\n\u22a2 False",
436
+ "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 IsUnit \u2191p \u2192 \u00acp \u2223 ringChar R",
437
+ "tactic": "rintro h\u2081 \u27e8q, hq\u27e9"
438
+ },
439
+ {
440
+ "annotated_tactic": [
441
+ "\"rcases <a>IsUnit.exists_left_inv<\\/a> h\\u2081 with \\u27e8a, ha\\u27e9\"",
442
+ "[{\"full_name\":\"IsUnit.exists_left_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[722,7],\"def_end_pos\":[722,29]}]"
443
+ ],
444
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\n\u22a2 False",
445
+ "state_before": "case mp.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\n\u22a2 False",
446
+ "tactic": "rcases IsUnit.exists_left_inv h\u2081 with \u27e8a, ha\u27e9"
447
+ },
448
+ {
449
+ "annotated_tactic": [
450
+ "\"have h\\u2083 : \\u00ac<a>ringChar<\\/a> R \\u2223 q := by\\n rintro \\u27e8r, hr\\u27e9\\n rw [hr, \\u2190 <a>mul_assoc<\\/a>, <a>mul_comm<\\/a> p, <a>mul_assoc<\\/a>] at hq\\n nth_rw 1 [\\u2190 <a>mul_one<\\/a> (<a>ringChar<\\/a> R)] at hq\\n exact <a>Nat.Prime.not_dvd_one<\\/a> hp \\u27e8r, <a>mul_left_cancel\\u2080<\\/a> hR hq\\u27e9\"",
451
+ "[{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]},{\"full_name\":\"Nat.Prime.not_dvd_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,26]},{\"full_name\":\"mul_left_cancel\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,25]}]"
452
+ ],
453
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\n\u22a2 False",
454
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\n\u22a2 False",
455
+ "tactic": "have h\u2083 : \u00acringChar R \u2223 q := by\n rintro \u27e8r, hr\u27e9\n rw [hr, \u2190 mul_assoc, mul_comm p, mul_assoc] at hq\n nth_rw 1 [\u2190 mul_one (ringChar R)] at hq\n exact Nat.Prime.not_dvd_one hp \u27e8r, mul_left_cancel\u2080 hR hq\u27e9"
456
+ },
457
+ {
458
+ "annotated_tactic": [
459
+ "\"have h\\u2084 := <a>mt<\\/a> (<a>CharP.intCast_eq_zero_iff<\\/a> R (<a>ringChar<\\/a> R) q).<a>mp<\\/a>\"",
460
+ "[{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"CharP.intCast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[102,7],\"def_end_pos\":[102,26]},{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
461
+ ],
462
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\n\u22a2 False",
463
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\n\u22a2 False",
464
+ "tactic": "have h\u2084 := mt (CharP.intCast_eq_zero_iff R (ringChar R) q).mp"
465
+ },
466
+ {
467
+ "annotated_tactic": [
468
+ "\"apply_fun ((\\u2191) : \\u2115 \\u2192 R) at hq\"",
469
+ "[]"
470
+ ],
471
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\nhq : \u2191(ringChar R) = \u2191(p * q)\n\u22a2 False",
472
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\n\u22a2 False",
473
+ "tactic": "apply_fun ((\u2191) : \u2115 \u2192 R) at hq"
474
+ },
475
+ {
476
+ "annotated_tactic": [
477
+ "\"rw [<a>Nat.cast_mul<\\/a>, hch, <a>mul_zero<\\/a>, \\u2190 <a>mul_assoc<\\/a>, ha, <a>one_mul<\\/a>] at hq\"",
478
+ "[{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
479
+ ],
480
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\nhq : 0 = \u2191q\n\u22a2 False",
481
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\nhq : a * \u2191(ringChar R) = a * \u2191(p * q)\n\u22a2 False",
482
+ "tactic": "rw [Nat.cast_mul, hch, mul_zero, \u2190 mul_assoc, ha, one_mul] at hq"
483
+ },
484
+ {
485
+ "annotated_tactic": [
486
+ "\"norm_cast at h\\u2084\"",
487
+ "[]"
488
+ ],
489
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nhq : 0 = \u2191q\nh\u2084 : \u00acringChar R \u2223 q \u2192 \u00ac\u2191q = 0\n\u22a2 False",
490
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nh\u2084 : \u00ac\u2191(ringChar R) \u2223 \u2191q \u2192 \u00ac\u2191\u2191q = 0\nhq : 0 = \u2191q\n\u22a2 False",
491
+ "tactic": "norm_cast at h\u2084"
492
+ },
493
+ {
494
+ "annotated_tactic": [
495
+ "\"exact h\\u2084 h\\u2083 hq.symm\"",
496
+ "[]"
497
+ ],
498
+ "state_after": "no goals",
499
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nh\u2083 : \u00acringChar R \u2223 q\nhq : 0 = \u2191q\nh\u2084 : \u00acringChar R \u2223 q \u2192 \u00ac\u2191q = 0\n\u22a2 False",
500
+ "tactic": "exact h\u2084 h\u2083 hq.symm"
501
+ },
502
+ {
503
+ "annotated_tactic": [
504
+ "\"rintro \\u27e8r, hr\\u27e9\"",
505
+ "[]"
506
+ ],
507
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nr : \u2115\nhr : q = ringChar R * r\n\u22a2 False",
508
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\n\u22a2 \u00acringChar R \u2223 q",
509
+ "tactic": "rintro \u27e8r, hr\u27e9"
510
+ },
511
+ {
512
+ "annotated_tactic": [
513
+ "\"rw [hr, \\u2190 <a>mul_assoc<\\/a>, <a>mul_comm<\\/a> p, <a>mul_assoc<\\/a>] at hq\"",
514
+ "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
515
+ ],
516
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nr : \u2115\nhq : ringChar R = ringChar R * (p * r)\nhr : q = ringChar R * r\n\u22a2 False",
517
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\nhq : ringChar R = p * q\na : R\nha : a * \u2191p = 1\nr : \u2115\nhr : q = ringChar R * r\n\u22a2 False",
518
+ "tactic": "rw [hr, \u2190 mul_assoc, mul_comm p, mul_assoc] at hq"
519
+ },
520
+ {
521
+ "annotated_tactic": [
522
+ "\"nth_rw 1 [\\u2190 <a>mul_one<\\/a> (<a>ringChar<\\/a> R)] at hq\"",
523
+ "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]}]"
524
+ ],
525
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nr : \u2115\nhq : ringChar R * 1 = ringChar R * (p * r)\nhr : q = ringChar R * r\n\u22a2 False",
526
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nr : \u2115\nhq : ringChar R = ringChar R * (p * r)\nhr : q = ringChar R * r\n\u22a2 False",
527
+ "tactic": "nth_rw 1 [\u2190 mul_one (ringChar R)] at hq"
528
+ },
529
+ {
530
+ "annotated_tactic": [
531
+ "\"exact <a>Nat.Prime.not_dvd_one<\\/a> hp \\u27e8r, <a>mul_left_cancel\\u2080<\\/a> hR hq\\u27e9\"",
532
+ "[{\"full_name\":\"Nat.Prime.not_dvd_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,26]},{\"full_name\":\"mul_left_cancel\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,25]}]"
533
+ ],
534
+ "state_after": "no goals",
535
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh\u2081 : IsUnit \u2191p\nq : \u2115\na : R\nha : a * \u2191p = 1\nr : \u2115\nhq : ringChar R * 1 = ringChar R * (p * r)\nhr : q = ringChar R * r\n\u22a2 False",
536
+ "tactic": "exact Nat.Prime.not_dvd_one hp \u27e8r, mul_left_cancel\u2080 hR hq\u27e9"
537
+ },
538
+ {
539
+ "annotated_tactic": [
540
+ "\"intro h\"",
541
+ "[]"
542
+ ],
543
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\n\u22a2 IsUnit \u2191p",
544
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\n\u22a2 \u00acp \u2223 ringChar R \u2192 IsUnit \u2191p",
545
+ "tactic": "intro h"
546
+ },
547
+ {
548
+ "annotated_tactic": [
549
+ "\"rcases (hp.coprime_iff_not_dvd.mpr h).<a>isCoprime<\\/a> with \\u27e8a, b, hab\\u27e9\"",
550
+ "[{\"full_name\":\"Nat.Coprime.isCoprime\",\"def_path\":\"Mathlib\\/RingTheory\\/Coprime\\/Lemmas.lean\",\"def_pos\":[46,31],\"def_end_pos\":[46,52]}]"
551
+ ],
552
+ "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : a * \u2191p + b * \u2191(ringChar R) = 1\n\u22a2 IsUnit \u2191p",
553
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\n\u22a2 IsUnit \u2191p",
554
+ "tactic": "rcases (hp.coprime_iff_not_dvd.mpr h).isCoprime with \u27e8a, b, hab\u27e9"
555
+ },
556
+ {
557
+ "annotated_tactic": [
558
+ "\"apply_fun ((\\u2191) : \\u2124 \\u2192 R) at hab\"",
559
+ "[]"
560
+ ],
561
+ "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191(a * \u2191p + b * \u2191(ringChar R)) = \u21911\n\u22a2 IsUnit \u2191p",
562
+ "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : a * \u2191p + b * \u2191(ringChar R) = 1\n\u22a2 IsUnit \u2191p",
563
+ "tactic": "apply_fun ((\u2191) : \u2124 \u2192 R) at hab"
564
+ },
565
+ {
566
+ "annotated_tactic": [
567
+ "\"push_cast at hab\"",
568
+ "[]"
569
+ ],
570
+ "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191a * \u2191p + \u2191b * \u2191(ringChar R) = 1\n\u22a2 IsUnit \u2191p",
571
+ "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191(a * \u2191p + b * \u2191(ringChar R)) = \u21911\n\u22a2 IsUnit \u2191p",
572
+ "tactic": "push_cast at hab"
573
+ },
574
+ {
575
+ "annotated_tactic": [
576
+ "\"rw [hch, <a>mul_zero<\\/a>, <a>add_zero<\\/a>, <a>mul_comm<\\/a>] at hab\"",
577
+ "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
578
+ ],
579
+ "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191p * \u2191a = 1\n\u22a2 IsUnit \u2191p",
580
+ "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191a * \u2191p + \u2191b * \u2191(ringChar R) = 1\n\u22a2 IsUnit \u2191p",
581
+ "tactic": "rw [hch, mul_zero, add_zero, mul_comm] at hab"
582
+ },
583
+ {
584
+ "annotated_tactic": [
585
+ "\"exact <a>isUnit_of_mul_eq_one<\\/a> (p : R) a hab\"",
586
+ "[{\"full_name\":\"isUnit_of_mul_eq_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[701,9],\"def_end_pos\":[701,29]}]"
587
+ ],
588
+ "state_after": "no goals",
589
+ "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhR : ringChar R \u2260 0\nhch : \u2191(ringChar R) = 0\nhp : Nat.Prime p\nh : \u00acp \u2223 ringChar R\na b : \u2124\nhab : \u2191p * \u2191a = 1\n\u22a2 IsUnit \u2191p",
590
+ "tactic": "exact isUnit_of_mul_eq_one (p : R) a hab"
591
+ }
592
+ ],
593
+ "url": "https://github.com/leanprover-community/mathlib4"
594
+ },
595
+ "source_file": "Mathlib/Algebra/CharP/CharAndCard.lean"
596
+ },
597
+ {
598
+ "error": "theorem has no selected tactic transitions",
599
+ "full_name": "not_isUnit_prime_of_dvd_card",
600
+ "id": "not_isUnit_prime_of_dvd_card",
601
+ "reason": "import_failure",
602
+ "selection": {
603
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
604
+ "end": [
605
+ 83,
606
+ 69
607
+ ],
608
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
609
+ "full_name": "not_isUnit_prime_of_dvd_card",
610
+ "start": [
611
+ 80,
612
+ 1
613
+ ],
614
+ "traced_tactics": [],
615
+ "url": "https://github.com/leanprover-community/mathlib4"
616
+ },
617
+ "source_file": "Mathlib/Algebra/CharP/CharAndCard.lean"
618
+ }
619
+ ],
620
+ "input_proofs": 26,
621
+ "input_units": 2,
622
+ "proofs": 9,
623
+ "published_proofs": 7,
624
+ "published_transitions": 17,
625
+ "replay_failures": [],
626
+ "transitions": 32,
627
+ "validation_failures": [
628
+ {
629
+ "errors": [
630
+ "internal transition boundary mismatch",
631
+ "internal transition boundary mismatch"
632
+ ],
633
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
634
+ "proof_id": "mathlib:Commute.add_pow_prime_pow_eq",
635
+ "provenance": {
636
+ "selection": {
637
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
638
+ "end": [
639
+ 40,
640
+ 65
641
+ ],
642
+ "file_path": "Mathlib/Algebra/CharP/Basic.lean",
643
+ "full_name": "Commute.add_pow_prime_pow_eq",
644
+ "start": [
645
+ 28,
646
+ 11
647
+ ],
648
+ "traced_tactics": [
649
+ {
650
+ "annotated_tactic": [
651
+ "\"trans x ^ p ^ n + y ^ p ^ n + \\u2211 k \\u2208 <a>Ioo<\\/a> 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * (p ^ n).<a>choose<\\/a> k\"",
652
+ "[{\"full_name\":\"Finset.Ioo\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[308,5],\"def_end_pos\":[308,8]},{\"full_name\":\"Nat.choose\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Choose\\/Basic.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,11]}]"
653
+ ],
654
+ "state_after": "R : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k)\n\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 x ^ p ^ n + y ^ p ^ n + \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n x ^ p ^ n + y ^ p ^ n + \u2191p * \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k / p)",
655
+ "state_before": "R : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + \u2191p * \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k / p)",
656
+ "tactic": "trans x ^ p ^ n + y ^ p ^ n + \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * (p ^ n).choose k"
657
+ },
658
+ {
659
+ "annotated_tactic": [
660
+ "\"simp_rw [h.add_pow, \\u2190 <a>Nat.Ico_zero_eq_range<\\/a>, <a>Nat.Ico_succ_right<\\/a>, <a>Icc_eq_cons_Ico<\\/a> (<a>zero_le<\\/a> _),\\n <a>Finset.sum_cons<\\/a>, <a>Ico_eq_cons_Ioo<\\/a> (<a>pow_pos<\\/a> hp.pos _), <a>Finset.sum_cons<\\/a>, <a>tsub_self<\\/a>, <a>tsub_zero<\\/a>,\\n <a>pow_zero<\\/a>, <a>Nat.choose_zero_right<\\/a>, <a>Nat.choose_self<\\/a>, <a>Nat.cast_one<\\/a>, <a>mul_one<\\/a>, <a>one_mul<\\/a>, \\u2190 <a>add_assoc<\\/a>]\"",
661
+ "[{\"full_name\":\"Nat.Ico_zero_eq_range\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Nat.lean\",\"def_pos\":[67,9],\"def_end_pos\":[67,26]},{\"full_name\":\"Nat.Ico_succ_right\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Nat.lean\",\"def_pos\":[153,9],\"def_end_pos\":[153,23]},{\"full_name\":\"Finset.Icc_eq_cons_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[625,9],\"def_end_pos\":[625,24]},{\"full_name\":\"zero_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[204,30],\"def_end_pos\":[204,37]},{\"full_name\":\"Finset.sum_cons\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[344,3],\"def_end_pos\":[344,14]},{\"full_name\":\"Finset.Ico_eq_cons_Ioo\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[640,9],\"def_end_pos\":[640,24]},{\"full_name\":\"pow_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,16]},{\"full_name\":\"Finset.sum_cons\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[344,3],\"def_end_pos\":[344,14]},{\"full_name\":\"tsub_self\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[342,9],\"def_end_pos\":[342,18]},{\"full_name\":\"tsub_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Defs.lean\",\"def_pos\":[448,9],\"def_end_pos\":[448,18]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]},{\"full_name\":\"Nat.choose_zero_right\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Choose\\/Basic.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,26]},{\"full_name\":\"Nat.choose_self\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Choose\\/Basic.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,20]},{\"full_name\":\"Nat.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"add_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[288,3],\"def_end_pos\":[288,14]}]"
662
+ ],
663
+ "state_after": "no goals",
664
+ "state_before": "R : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k)",
665
+ "tactic": "simp_rw [h.add_pow, \u2190 Nat.Ico_zero_eq_range, Nat.Ico_succ_right, Icc_eq_cons_Ico (zero_le _),\n Finset.sum_cons, Ico_eq_cons_Ioo (pow_pos hp.pos _), Finset.sum_cons, tsub_self, tsub_zero,\n pow_zero, Nat.choose_zero_right, Nat.choose_self, Nat.cast_one, mul_one, one_mul, \u2190 add_assoc]"
666
+ },
667
+ {
668
+ "annotated_tactic": [
669
+ "\"congr 1\"",
670
+ "[]"
671
+ ],
672
+ "state_after": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n \u2191p * \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k / p)",
673
+ "state_before": "R : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 x ^ p ^ n + y ^ p ^ n + \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n x ^ p ^ n + y ^ p ^ n + \u2191p * \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k / p)",
674
+ "tactic": "congr 1"
675
+ },
676
+ {
677
+ "annotated_tactic": [
678
+ "\"simp_rw [<a>Finset.mul_sum<\\/a>, <a>Nat.cast_comm<\\/a>, <a>mul_assoc<\\/a> _ _ (p : R), \\u2190 <a>Nat.cast_mul<\\/a>]\"",
679
+ "[{\"full_name\":\"Finset.mul_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Ring.lean\",\"def_pos\":[61,7],\"def_end_pos\":[61,14]},{\"full_name\":\"Nat.cast_comm\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[33,9],\"def_end_pos\":[33,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]}]"
680
+ ],
681
+ "state_after": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n \u2211 x_1 \u2208 Ioo 0 (p ^ n), x ^ x_1 * y ^ (p ^ n - x_1) * \u2191((p ^ n).choose x_1 / p * p)",
682
+ "state_before": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n \u2191p * \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k / p)",
683
+ "tactic": "simp_rw [Finset.mul_sum, Nat.cast_comm, mul_assoc _ _ (p : R), \u2190 Nat.cast_mul]"
684
+ },
685
+ {
686
+ "annotated_tactic": [
687
+ "\"refine <a>Finset.sum_congr<\\/a> <a>rfl<\\/a> fun i hi => ?_\"",
688
+ "[{\"full_name\":\"Finset.sum_congr\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[420,3],\"def_end_pos\":[420,14]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
689
+ ],
690
+ "state_after": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : i \u2208 Ioo 0 (p ^ n)\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i) = x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i / p * p)",
691
+ "state_before": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 k \u2208 Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * \u2191((p ^ n).choose k) =\n \u2211 x_1 \u2208 Ioo 0 (p ^ n), x ^ x_1 * y ^ (p ^ n - x_1) * \u2191((p ^ n).choose x_1 / p * p)",
692
+ "tactic": "refine Finset.sum_congr rfl fun i hi => ?_"
693
+ },
694
+ {
695
+ "annotated_tactic": [
696
+ "\"rw [<a>mem_Ioo<\\/a>] at hi\"",
697
+ "[{\"full_name\":\"Finset.mem_Ioo\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[328,9],\"def_end_pos\":[328,16]}]"
698
+ ],
699
+ "state_after": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : 0 < i \u2227 i < p ^ n\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i) = x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i / p * p)",
700
+ "state_before": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : i \u2208 Ioo 0 (p ^ n)\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i) = x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i / p * p)",
701
+ "tactic": "rw [mem_Ioo] at hi"
702
+ },
703
+ {
704
+ "annotated_tactic": [
705
+ "\"rw [<a>Nat.div_mul_cancel<\\/a> (hp.dvd_choose_pow hi.1.<a>ne'<\\/a> hi.2.<a>ne<\\/a>)]\"",
706
+ "[{\"full_name\":\"Nat.div_mul_cancel\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Dvd.lean\",\"def_pos\":[91,19],\"def_end_pos\":[91,33]},{\"full_name\":\"LT.lt.ne'\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[306,9],\"def_end_pos\":[306,12]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
707
+ ],
708
+ "state_after": "no goals",
709
+ "state_before": "case e_a\nR : Type u_1\ninst\u271d : Semiring R\np : \u2115\nx y : R\nhp : Nat.Prime p\nh : Commute x y\nn i : \u2115\nhi : 0 < i \u2227 i < p ^ n\n\u22a2 x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i) = x ^ i * y ^ (p ^ n - i) * \u2191((p ^ n).choose i / p * p)",
710
+ "tactic": "rw [Nat.div_mul_cancel (hp.dvd_choose_pow hi.1.ne' hi.2.ne)]"
711
+ }
712
+ ],
713
+ "url": "https://github.com/leanprover-community/mathlib4"
714
+ }
715
+ },
716
+ "theorem_name": "Commute.add_pow_prime_pow_eq"
717
+ },
718
+ {
719
+ "errors": [
720
+ "internal transition boundary mismatch"
721
+ ],
722
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
723
+ "proof_id": "mathlib:prime_dvd_char_iff_dvd_card",
724
+ "provenance": {
725
+ "selection": {
726
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
727
+ "end": [
728
+ 75,
729
+ 99
730
+ ],
731
+ "file_path": "Mathlib/Algebra/CharP/CharAndCard.lean",
732
+ "full_name": "prime_dvd_char_iff_dvd_card",
733
+ "start": [
734
+ 59,
735
+ 1
736
+ ],
737
+ "traced_tactics": [
738
+ {
739
+ "annotated_tactic": [
740
+ "\"refine\\n \\u27e8fun h =>\\n h.trans <|\\n Int.natCast_dvd_natCast.mp <|\\n (<a>CharP.intCast_eq_zero_iff<\\/a> R (<a>ringChar<\\/a> R) (<a>Fintype.card<\\/a> R)).<a>mp<\\/a> <|\\n mod_cast <a>Nat.cast_card_eq_zero<\\/a> R,\\n fun h => ?_\\u27e9\"",
741
+ "[{\"full_name\":\"CharP.intCast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[102,7],\"def_end_pos\":[102,26]},{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]},{\"full_name\":\"Fintype.card\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[63,5],\"def_end_pos\":[63,9]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]},{\"full_name\":\"Nat.cast_card_eq_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[1315,7],\"def_end_pos\":[1315,28]}]"
742
+ ],
743
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\n\u22a2 p \u2223 ringChar R",
744
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 p \u2223 ringChar R \u2194 p \u2223 Fintype.card R",
745
+ "tactic": "refine\n \u27e8fun h =>\n h.trans <|\n Int.natCast_dvd_natCast.mp <|\n (CharP.intCast_eq_zero_iff R (ringChar R) (Fintype.card R)).mp <|\n mod_cast Nat.cast_card_eq_zero R,\n fun h => ?_\u27e9"
746
+ },
747
+ {
748
+ "annotated_tactic": [
749
+ "\"by_contra h\\u2080\"",
750
+ "[]"
751
+ ],
752
+ "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\n\u22a2 False",
753
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\n\u22a2 p \u2223 ringChar R",
754
+ "tactic": "by_contra h\u2080"
755
+ },
756
+ {
757
+ "annotated_tactic": [
758
+ "\"rcases <a>exists_prime_addOrderOf_dvd_card<\\/a> p h with \\u27e8r, hr\\u27e9\"",
759
+ "[{\"full_name\":\"exists_prime_addOrderOf_dvd_card\",\"def_path\":\"Mathlib\\/GroupTheory\\/Perm\\/Cycle\\/Type.lean\",\"def_pos\":[509,9],\"def_end_pos\":[509,48]}]"
760
+ ],
761
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\n\u22a2 False",
762
+ "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\n\u22a2 False",
763
+ "tactic": "rcases exists_prime_addOrderOf_dvd_card p h with \u27e8r, hr\u27e9"
764
+ },
765
+ {
766
+ "annotated_tactic": [
767
+ "\"have hr\\u2081 := <a>addOrderOf_nsmul_eq_zero<\\/a> r\"",
768
+ "[{\"full_name\":\"addOrderOf_nsmul_eq_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[176,15],\"def_end_pos\":[176,39]}]"
769
+ ],
770
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nhr\u2081 : addOrderOf r \u2022 r = 0\n\u22a2 False",
771
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\n\u22a2 False",
772
+ "tactic": "have hr\u2081 := addOrderOf_nsmul_eq_zero r"
773
+ },
774
+ {
775
+ "annotated_tactic": [
776
+ "\"rw [hr, <a>nsmul_eq_mul<\\/a>] at hr\\u2081\"",
777
+ "[{\"full_name\":\"nsmul_eq_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[81,15],\"def_end_pos\":[81,34]}]"
778
+ ],
779
+ "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nhr\u2081 : \u2191p * r = 0\n\u22a2 False",
780
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nhr\u2081 : addOrderOf r \u2022 r = 0\n\u22a2 False",
781
+ "tactic": "rw [hr, nsmul_eq_mul] at hr\u2081"
782
+ },
783
+ {
784
+ "annotated_tactic": [
785
+ "\"rcases <a>IsUnit.exists_left_inv<\\/a> ((<a>isUnit_iff_not_dvd_char<\\/a> R p).<a>mpr<\\/a> h\\u2080) with \\u27e8u, hu\\u27e9\"",
786
+ "[{\"full_name\":\"IsUnit.exists_left_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[722,7],\"def_end_pos\":[722,29]},{\"full_name\":\"isUnit_iff_not_dvd_char\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/CharAndCard.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,32]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
787
+ ],
788
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nhr\u2081 : \u2191p * r = 0\nu : R\nhu : u * \u2191p = 1\n\u22a2 False",
789
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nhr\u2081 : \u2191p * r = 0\n\u22a2 False",
790
+ "tactic": "rcases IsUnit.exists_left_inv ((isUnit_iff_not_dvd_char R p).mpr h\u2080) with \u27e8u, hu\u27e9"
791
+ },
792
+ {
793
+ "annotated_tactic": [
794
+ "\"rw [<a>mul_zero<\\/a>, \\u2190 <a>mul_assoc<\\/a>, hu, <a>one_mul<\\/a>] at hr\\u2081\"",
795
+ "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
796
+ ],
797
+ "state_after": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nu : R\nhu : u * \u2191p = 1\nhr\u2081 : r = 0\n\u22a2 False",
798
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nu : R\nhu : u * \u2191p = 1\nhr\u2081 : u * (\u2191p * r) = u * 0\n\u22a2 False",
799
+ "tactic": "rw [mul_zero, \u2190 mul_assoc, hu, one_mul] at hr\u2081"
800
+ },
801
+ {
802
+ "annotated_tactic": [
803
+ "\"exact <a>mt<\\/a> AddMonoid.addOrderOf_eq_one_iff.mpr (<a>ne_of_eq_of_ne<\\/a> hr (<a>Nat.Prime.ne_one<\\/a> <a>Fact.out<\\/a>)) hr\\u2081\"",
804
+ "[{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"ne_of_eq_of_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[533,7],\"def_end_pos\":[533,21]},{\"full_name\":\"Nat.Prime.ne_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,21]},{\"full_name\":\"Fact.out\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[117,3],\"def_end_pos\":[117,6]}]"
805
+ ],
806
+ "state_after": "no goals",
807
+ "state_before": "case intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : p \u2223 Fintype.card R\nh\u2080 : \u00acp \u2223 ringChar R\nr : R\nhr : addOrderOf r = p\nu : R\nhu : u * \u2191p = 1\nhr\u2081 : r = 0\n\u22a2 False",
808
+ "tactic": "exact mt AddMonoid.addOrderOf_eq_one_iff.mpr (ne_of_eq_of_ne hr (Nat.Prime.ne_one Fact.out)) hr\u2081"
809
+ }
810
+ ],
811
+ "url": "https://github.com/leanprover-community/mathlib4"
812
+ }
813
+ },
814
+ "theorem_name": "prime_dvd_char_iff_dvd_card"
815
+ }
816
+ ],
817
+ "validation_rejected_proofs": 2,
818
+ "zero_transition_proofs": 0
819
+ },
820
+ "shards": [
821
+ {
822
+ "path": "data/train/mathlib-000079-0000-d96fa02282637c0e.parquet",
823
+ "sha256": "d96fa02282637c0e6da9a9261bd4ff29c013f9e7f032f8d163d0b7f4ecbac6eb",
824
+ "bytes": 64322,
825
+ "rows": 15,
826
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
827
+ },
828
+ {
829
+ "path": "data/internal_test/mathlib-000079-0000-2f4497b15123792a.parquet",
830
+ "sha256": "2f4497b15123792adf496e813b9264e55ac497c98e2ccde144afe396c8975a9f",
831
+ "bytes": 45526,
832
+ "rows": 2,
833
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
834
+ }
835
+ ],
836
+ "proof_ids": [
837
+ "mathlib:CharP.char_ne_zero_of_finite",
838
+ "mathlib:CharP.neg_one_pow_char",
839
+ "mathlib:CharP.neg_one_pow_char_pow",
840
+ "mathlib:Commute.add_pow_prime_eq",
841
+ "mathlib:add_pow_char_of_commute",
842
+ "mathlib:add_pow_char_pow_of_commute",
843
+ "mathlib:sub_pow_char_of_commute"
844
+ ],
845
+ "proof_keys": [
846
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"CharP.char_ne_zero_of_finite\"]",
847
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"CharP.neg_one_pow_char\"]",
848
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"CharP.neg_one_pow_char_pow\"]",
849
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"Commute.add_pow_prime_eq\"]",
850
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"add_pow_char_of_commute\"]",
851
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"add_pow_char_pow_of_commute\"]",
852
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Basic.lean\",null,null,\"sub_pow_char_of_commute\"]"
853
+ ]
854
+ }
manifests/batches/mathlib/000080.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/000081.json ADDED
@@ -0,0 +1,365 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000081",
4
+ "batch_fingerprint": "e8c68764ee3e341bf4039bcbda2d220848da116db6d77ba240a88f67a01c5e88",
5
+ "evidence": {
6
+ "batch_fingerprint": "e8c68764ee3e341bf4039bcbda2d220848da116db6d77ba240a88f67a01c5e88",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic \"rw [Nat.mul_div_cancel' (Nat.ord_proj_dvd q r)]\"",
11
+ "full_name": "charP_zero_or_prime_power",
12
+ "id": "charP_zero_or_prime_power",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 67,
18
+ 30
19
+ ],
20
+ "file_path": "Mathlib/Algebra/CharP/LocalRing.lean",
21
+ "full_name": "charP_zero_or_prime_power",
22
+ "start": [
23
+ 25,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"apply <a>or_iff_not_imp_left<\\/a>.2\"",
30
+ "[{\"full_name\":\"Classical.or_iff_not_imp_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,28]}]"
31
+ ],
32
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\n\u22a2 \u00acq = 0 \u2192 IsPrimePow q",
33
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\n\u22a2 q = 0 \u2228 IsPrimePow q",
34
+ "tactic": "apply or_iff_not_imp_left.2"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"intro q_pos\"",
39
+ "[]"
40
+ ],
41
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\n\u22a2 IsPrimePow q",
42
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\n\u22a2 \u00acq = 0 \u2192 IsPrimePow q",
43
+ "tactic": "intro q_pos"
44
+ },
45
+ {
46
+ "annotated_tactic": [
47
+ "\"let K := <a>LocalRing.ResidueField<\\/a> R\"",
48
+ "[{\"full_name\":\"LocalRing.ResidueField\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[352,5],\"def_end_pos\":[352,17]}]"
49
+ ],
50
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\n\u22a2 IsPrimePow q",
51
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\n\u22a2 IsPrimePow q",
52
+ "tactic": "let K := LocalRing.ResidueField R"
53
+ },
54
+ {
55
+ "annotated_tactic": [
56
+ "\"haveI RM_char := <a>ringChar.charP<\\/a> K\"",
57
+ "[{\"full_name\":\"ringChar.charP\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[182,10],\"def_end_pos\":[182,15]}]"
58
+ ],
59
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\n\u22a2 IsPrimePow q",
60
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\n\u22a2 IsPrimePow q",
61
+ "tactic": "haveI RM_char := ringChar.charP K"
62
+ },
63
+ {
64
+ "annotated_tactic": [
65
+ "\"let r := <a>ringChar<\\/a> K\"",
66
+ "[{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]}]"
67
+ ],
68
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\n\u22a2 IsPrimePow q",
69
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\n\u22a2 IsPrimePow q",
70
+ "tactic": "let r := ringChar K"
71
+ },
72
+ {
73
+ "annotated_tactic": [
74
+ "\"let n := q.factorization r\"",
75
+ "[]"
76
+ ],
77
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\n\u22a2 IsPrimePow q",
78
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\n\u22a2 IsPrimePow q",
79
+ "tactic": "let n := q.factorization r"
80
+ },
81
+ {
82
+ "annotated_tactic": [
83
+ "\"cases' <a>CharP.char_is_prime_or_zero<\\/a> K r with r_prime r_zero\"",
84
+ "[{\"full_name\":\"CharP.char_is_prime_or_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[285,7],\"def_end_pos\":[285,28]}]"
85
+ ],
86
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\n\u22a2 IsPrimePow q\n\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\n\u22a2 IsPrimePow q",
87
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\n\u22a2 IsPrimePow q",
88
+ "tactic": "cases' CharP.char_is_prime_or_zero K r with r_prime r_zero"
89
+ },
90
+ {
91
+ "annotated_tactic": [
92
+ "\"let a := q \\/ r ^ n\"",
93
+ "[]"
94
+ ],
95
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\n\u22a2 IsPrimePow q",
96
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\n\u22a2 IsPrimePow q",
97
+ "tactic": "let a := q / r ^ n"
98
+ },
99
+ {
100
+ "annotated_tactic": [
101
+ "\"have q_eq_a_mul_rn : q = r ^ n * a := by rw [<a>Nat.mul_div_cancel'<\\/a> (<a>Nat.ord_proj_dvd<\\/a> q r)]\"",
102
+ "[{\"full_name\":\"Nat.mul_div_cancel'\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Dvd.lean\",\"def_pos\":[87,19],\"def_end_pos\":[87,34]},{\"full_name\":\"Nat.ord_proj_dvd\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Factorization\\/Basic.lean\",\"def_pos\":[340,9],\"def_end_pos\":[340,21]}]"
103
+ ],
104
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\n\u22a2 IsPrimePow q",
105
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\n\u22a2 IsPrimePow q",
106
+ "tactic": "have q_eq_a_mul_rn : q = r ^ n * a := by rw [Nat.mul_div_cancel' (Nat.ord_proj_dvd q r)]"
107
+ },
108
+ {
109
+ "annotated_tactic": [
110
+ "\"have r_ne_dvd_a := <a>Nat.not_dvd_ord_compl<\\/a> r_prime q_pos\"",
111
+ "[{\"full_name\":\"Nat.not_dvd_ord_compl\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Factorization\\/Basic.lean\",\"def_pos\":[490,9],\"def_end_pos\":[490,26]}]"
112
+ ],
113
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\n\u22a2 IsPrimePow q",
114
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\n\u22a2 IsPrimePow q",
115
+ "tactic": "have r_ne_dvd_a := Nat.not_dvd_ord_compl r_prime q_pos"
116
+ },
117
+ {
118
+ "annotated_tactic": [
119
+ "\"have rn_dvd_q : r ^ n \\u2223 q := \\u27e8a, q_eq_a_mul_rn\\u27e9\"",
120
+ "[]"
121
+ ],
122
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q",
123
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\n\u22a2 IsPrimePow q",
124
+ "tactic": "have rn_dvd_q : r ^ n \u2223 q := \u27e8a, q_eq_a_mul_rn\u27e9"
125
+ },
126
+ {
127
+ "annotated_tactic": [
128
+ "\"rw [<a>mul_comm<\\/a>] at q_eq_a_mul_rn\"",
129
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
130
+ ],
131
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q",
132
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q",
133
+ "tactic": "rw [mul_comm] at q_eq_a_mul_rn"
134
+ },
135
+ {
136
+ "annotated_tactic": [
137
+ "\"have a_unit : <a>IsUnit<\\/a> (a : R) := by\\n by_contra g\\n rw [\\u2190 <a>mem_nonunits_iff<\\/a>] at g\\n rw [\\u2190 <a>LocalRing.mem_maximalIdeal<\\/a>] at g\\n have a_cast_zero := <a>Ideal.Quotient.eq_zero_iff_mem<\\/a>.2 g\\n rw [<a>map_natCast<\\/a>] at a_cast_zero\\n have r_dvd_a := (<a>ringChar.spec<\\/a> K a).1 a_cast_zero\\n exact <a>absurd<\\/a> r_dvd_a r_ne_dvd_a\"",
138
+ "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"mem_nonunits_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[897,9],\"def_end_pos\":[897,25]},{\"full_name\":\"LocalRing.mem_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,25]},{\"full_name\":\"Ideal.Quotient.eq_zero_iff_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,24]},{\"full_name\":\"map_natCast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,20]},{\"full_name\":\"ringChar.spec\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[173,7],\"def_end_pos\":[173,11]},{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
139
+ ],
140
+ "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\n\u22a2 IsPrimePow q",
141
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q",
142
+ "tactic": "have a_unit : IsUnit (a : R) := by\n by_contra g\n rw [\u2190 mem_nonunits_iff] at g\n rw [\u2190 LocalRing.mem_maximalIdeal] at g\n have a_cast_zero := Ideal.Quotient.eq_zero_iff_mem.2 g\n rw [map_natCast] at a_cast_zero\n have r_dvd_a := (ringChar.spec K a).1 a_cast_zero\n exact absurd r_dvd_a r_ne_dvd_a"
143
+ },
144
+ {
145
+ "annotated_tactic": [
146
+ "\"cases' a_unit.exists_left_inv with a_inv h_inv_mul_a\"",
147
+ "[]"
148
+ ],
149
+ "state_after": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 IsPrimePow q",
150
+ "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\n\u22a2 IsPrimePow q",
151
+ "tactic": "cases' a_unit.exists_left_inv with a_inv h_inv_mul_a"
152
+ },
153
+ {
154
+ "annotated_tactic": [
155
+ "\"have rn_cast_zero : \\u2191(r ^ n) = (0 : R) := by\\n rw [\\u2190 @<a>mul_one<\\/a> R _ \\u2191(r ^ n), <a>mul_comm<\\/a>, \\u2190 <a>Classical.choose_spec<\\/a> a_unit.exists_left_inv,\\n <a>mul_assoc<\\/a>, \\u2190 <a>Nat.cast_mul<\\/a>, \\u2190 q_eq_a_mul_rn, <a>CharP.cast_eq_zero<\\/a> R q]\\n simp\"",
156
+ "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]}]"
157
+ ],
158
+ "state_after": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\n\u22a2 IsPrimePow q",
159
+ "state_before": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 IsPrimePow q",
160
+ "tactic": "have rn_cast_zero : \u2191(r ^ n) = (0 : R) := by\n rw [\u2190 @mul_one R _ \u2191(r ^ n), mul_comm, \u2190 Classical.choose_spec a_unit.exists_left_inv,\n mul_assoc, \u2190 Nat.cast_mul, \u2190 q_eq_a_mul_rn, CharP.cast_eq_zero R q]\n simp"
161
+ },
162
+ {
163
+ "annotated_tactic": [
164
+ "\"have q_eq_rn := <a>Nat.dvd_antisymm<\\/a> ((<a>CharP.cast_eq_zero_iff<\\/a> R q (r ^ n)).<a>mp<\\/a> rn_cast_zero) rn_dvd_q\"",
165
+ "[{\"full_name\":\"Nat.dvd_antisymm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Dvd.lean\",\"def_pos\":[57,19],\"def_end_pos\":[57,31]},{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
166
+ ],
167
+ "state_after": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\n\u22a2 IsPrimePow q",
168
+ "state_before": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\n\u22a2 IsPrimePow q",
169
+ "tactic": "have q_eq_rn := Nat.dvd_antisymm ((CharP.cast_eq_zero_iff R q (r ^ n)).mp rn_cast_zero) rn_dvd_q"
170
+ },
171
+ {
172
+ "annotated_tactic": [
173
+ "\"have n_pos : n \\u2260 0 := fun n_zero =>\\n <a>absurd<\\/a> (by simpa [n_zero] using q_eq_rn) (<a>CharP.char_ne_one<\\/a> R q)\"",
174
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]},{\"full_name\":\"CharP.char_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[253,7],\"def_end_pos\":[253,18]}]"
175
+ ],
176
+ "state_after": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\nn_pos : n \u2260 0\n\u22a2 IsPrimePow q",
177
+ "state_before": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\n\u22a2 IsPrimePow q",
178
+ "tactic": "have n_pos : n \u2260 0 := fun n_zero =>\n absurd (by simpa [n_zero] using q_eq_rn) (CharP.char_ne_one R q)"
179
+ },
180
+ {
181
+ "annotated_tactic": [
182
+ "\"exact \\u27e8r, \\u27e8n, \\u27e8r_prime.prime, \\u27e8pos_iff_ne_zero.mpr n_pos, q_eq_rn.symm\\u27e9\\u27e9\\u27e9\\u27e9\"",
183
+ "[]"
184
+ ],
185
+ "state_after": "no goals",
186
+ "state_before": "case inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\nn_pos : n \u2260 0\n\u22a2 IsPrimePow q",
187
+ "tactic": "exact \u27e8r, \u27e8n, \u27e8r_prime.prime, \u27e8pos_iff_ne_zero.mpr n_pos, q_eq_rn.symm\u27e9\u27e9\u27e9\u27e9"
188
+ },
189
+ {
190
+ "annotated_tactic": [
191
+ "\"rw [<a>Nat.mul_div_cancel'<\\/a> (<a>Nat.ord_proj_dvd<\\/a> q r)]\"",
192
+ "[{\"full_name\":\"Nat.mul_div_cancel'\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Dvd.lean\",\"def_pos\":[87,19],\"def_end_pos\":[87,34]},{\"full_name\":\"Nat.ord_proj_dvd\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Factorization\\/Basic.lean\",\"def_pos\":[340,9],\"def_end_pos\":[340,21]}]"
193
+ ],
194
+ "state_after": "no goals",
195
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\n\u22a2 q = r ^ n * a",
196
+ "tactic": "rw [Nat.mul_div_cancel' (Nat.ord_proj_dvd q r)]"
197
+ },
198
+ {
199
+ "annotated_tactic": [
200
+ "\"by_contra g\"",
201
+ "[]"
202
+ ],
203
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u00acIsUnit \u2191a\n\u22a2 False",
204
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsUnit \u2191a",
205
+ "tactic": "by_contra g"
206
+ },
207
+ {
208
+ "annotated_tactic": [
209
+ "\"rw [\\u2190 <a>mem_nonunits_iff<\\/a>] at g\"",
210
+ "[{\"full_name\":\"mem_nonunits_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[897,9],\"def_end_pos\":[897,25]}]"
211
+ ],
212
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 nonunits R\n\u22a2 False",
213
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u00acIsUnit \u2191a\n\u22a2 False",
214
+ "tactic": "rw [\u2190 mem_nonunits_iff] at g"
215
+ },
216
+ {
217
+ "annotated_tactic": [
218
+ "\"rw [\\u2190 <a>LocalRing.mem_maximalIdeal<\\/a>] at g\"",
219
+ "[{\"full_name\":\"LocalRing.mem_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,25]}]"
220
+ ],
221
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\n\u22a2 False",
222
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 nonunits R\n\u22a2 False",
223
+ "tactic": "rw [\u2190 LocalRing.mem_maximalIdeal] at g"
224
+ },
225
+ {
226
+ "annotated_tactic": [
227
+ "\"have a_cast_zero := <a>Ideal.Quotient.eq_zero_iff_mem<\\/a>.2 g\"",
228
+ "[{\"full_name\":\"Ideal.Quotient.eq_zero_iff_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,24]}]"
229
+ ],
230
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : (Ideal.Quotient.mk (LocalRing.maximalIdeal R)) \u2191a = 0\n\u22a2 False",
231
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\n\u22a2 False",
232
+ "tactic": "have a_cast_zero := Ideal.Quotient.eq_zero_iff_mem.2 g"
233
+ },
234
+ {
235
+ "annotated_tactic": [
236
+ "\"rw [<a>map_natCast<\\/a>] at a_cast_zero\"",
237
+ "[{\"full_name\":\"map_natCast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,20]}]"
238
+ ],
239
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\n\u22a2 False",
240
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : (Ideal.Quotient.mk (LocalRing.maximalIdeal R)) \u2191a = 0\n\u22a2 False",
241
+ "tactic": "rw [map_natCast] at a_cast_zero"
242
+ },
243
+ {
244
+ "annotated_tactic": [
245
+ "\"have r_dvd_a := (<a>ringChar.spec<\\/a> K a).1 a_cast_zero\"",
246
+ "[{\"full_name\":\"ringChar.spec\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[173,7],\"def_end_pos\":[173,11]}]"
247
+ ],
248
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\nr_dvd_a : ringChar K \u2223 a\n\u22a2 False",
249
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\n\u22a2 False",
250
+ "tactic": "have r_dvd_a := (ringChar.spec K a).1 a_cast_zero"
251
+ },
252
+ {
253
+ "annotated_tactic": [
254
+ "\"exact <a>absurd<\\/a> r_dvd_a r_ne_dvd_a\"",
255
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
256
+ ],
257
+ "state_after": "no goals",
258
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\nr_dvd_a : ringChar K \u2223 a\n\u22a2 False",
259
+ "tactic": "exact absurd r_dvd_a r_ne_dvd_a"
260
+ },
261
+ {
262
+ "annotated_tactic": [
263
+ "\"rw [\\u2190 @<a>mul_one<\\/a> R _ \\u2191(r ^ n), <a>mul_comm<\\/a>, \\u2190 <a>Classical.choose_spec<\\/a> a_unit.exists_left_inv,\\n <a>mul_assoc<\\/a>, \\u2190 <a>Nat.cast_mul<\\/a>, \\u2190 q_eq_a_mul_rn, <a>CharP.cast_eq_zero<\\/a> R q]\"",
264
+ "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]}]"
265
+ ],
266
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 Classical.choose \u22ef * 0 = 0",
267
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 \u2191(r ^ n) = 0",
268
+ "tactic": "rw [\u2190 @mul_one R _ \u2191(r ^ n), mul_comm, \u2190 Classical.choose_spec a_unit.exists_left_inv,\n mul_assoc, \u2190 Nat.cast_mul, \u2190 q_eq_a_mul_rn, CharP.cast_eq_zero R q]"
269
+ },
270
+ {
271
+ "annotated_tactic": [
272
+ "\"simp\"",
273
+ "[]"
274
+ ],
275
+ "state_after": "no goals",
276
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 Classical.choose \u22ef * 0 = 0",
277
+ "tactic": "simp"
278
+ },
279
+ {
280
+ "annotated_tactic": [
281
+ "\"simpa [n_zero] using q_eq_rn\"",
282
+ "[]"
283
+ ],
284
+ "state_after": "no goals",
285
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ q.factorization r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\nn_zero : n = 0\n\u22a2 q = 1",
286
+ "tactic": "simpa [n_zero] using q_eq_rn"
287
+ },
288
+ {
289
+ "annotated_tactic": [
290
+ "\"haveI K_char_p_0 := <a>ringChar.of_eq<\\/a> r_zero\"",
291
+ "[{\"full_name\":\"ringChar.of_eq\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[188,7],\"def_end_pos\":[188,12]}]"
292
+ ],
293
+ "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\n\u22a2 IsPrimePow q",
294
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\n\u22a2 IsPrimePow q",
295
+ "tactic": "haveI K_char_p_0 := ringChar.of_eq r_zero"
296
+ },
297
+ {
298
+ "annotated_tactic": [
299
+ "\"haveI K_char_zero : <a>CharZero<\\/a> K := <a>CharP.charP_to_charZero<\\/a> K\"",
300
+ "[{\"full_name\":\"CharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,15]},{\"full_name\":\"CharP.charP_to_charZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,24]}]"
301
+ ],
302
+ "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\n\u22a2 IsPrimePow q",
303
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\n\u22a2 IsPrimePow q",
304
+ "tactic": "haveI K_char_zero : CharZero K := CharP.charP_to_charZero K"
305
+ },
306
+ {
307
+ "annotated_tactic": [
308
+ "\"haveI R_char_zero := <a>RingHom.charZero<\\/a> (<a>LocalRing.residue<\\/a> R)\"",
309
+ "[{\"full_name\":\"RingHom.charZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Lemmas.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,25]},{\"full_name\":\"LocalRing.residue\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[368,5],\"def_end_pos\":[368,12]}]"
310
+ ],
311
+ "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\n\u22a2 IsPrimePow q",
312
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\n\u22a2 IsPrimePow q",
313
+ "tactic": "haveI R_char_zero := RingHom.charZero (LocalRing.residue R)"
314
+ },
315
+ {
316
+ "annotated_tactic": [
317
+ "\"have q_zero := <a>CharP.eq<\\/a> R char_R_q (<a>CharP.ofCharZero<\\/a> R)\"",
318
+ "[{\"full_name\":\"CharP.eq\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[75,7],\"def_end_pos\":[75,9]},{\"full_name\":\"CharP.ofCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[80,10],\"def_end_pos\":[80,20]}]"
319
+ ],
320
+ "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\nq_zero : q = 0\n\u22a2 IsPrimePow q",
321
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\n\u22a2 IsPrimePow q",
322
+ "tactic": "have q_zero := CharP.eq R char_R_q (CharP.ofCharZero R)"
323
+ },
324
+ {
325
+ "annotated_tactic": [
326
+ "\"exact <a>absurd<\\/a> q_zero q_pos\"",
327
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
328
+ ],
329
+ "state_after": "no goals",
330
+ "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : \u2115 := q.factorization r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\nq_zero : q = 0\n\u22a2 IsPrimePow q",
331
+ "tactic": "exact absurd q_zero q_pos"
332
+ }
333
+ ],
334
+ "url": "https://github.com/leanprover-community/mathlib4"
335
+ },
336
+ "source_file": "Mathlib/Algebra/CharP/LocalRing.lean"
337
+ }
338
+ ],
339
+ "input_proofs": 2,
340
+ "input_units": 2,
341
+ "proofs": 1,
342
+ "published_proofs": 1,
343
+ "published_transitions": 2,
344
+ "replay_failures": [],
345
+ "transitions": 2,
346
+ "validation_failures": [],
347
+ "validation_rejected_proofs": 0,
348
+ "zero_transition_proofs": 0
349
+ },
350
+ "shards": [
351
+ {
352
+ "path": "data/train/mathlib-000081-0000-ee4903fcdab6460c.parquet",
353
+ "sha256": "ee4903fcdab6460c61380d9d5b0e0ab9cbe4460e8b5d4795bcb621892aae3bfd",
354
+ "bytes": 38429,
355
+ "rows": 2,
356
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
357
+ }
358
+ ],
359
+ "proof_ids": [
360
+ "mathlib:not_ringChar_dvd_of_invertible"
361
+ ],
362
+ "proof_keys": [
363
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Invertible.lean\",null,null,\"not_ringChar_dvd_of_invertible\"]"
364
+ ]
365
+ }
manifests/batches/mathlib/000082.json ADDED
@@ -0,0 +1,994 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000082",
4
+ "batch_fingerprint": "86d5b52638ebb6e17c35c3885307a65673cb9d4688375c4102c55cc7cc0ad4af",
5
+ "evidence": {
6
+ "batch_fingerprint": "86d5b52638ebb6e17c35c3885307a65673cb9d4688375c4102c55cc7cc0ad4af",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "EqualCharZero.iff_not_mixedCharZero",
12
+ "id": "EqualCharZero.iff_not_mixedCharZero",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 276,
18
+ 51
19
+ ],
20
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
21
+ "full_name": "EqualCharZero.iff_not_mixedCharZero",
22
+ "start": [
23
+ 274,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Algebra/CharP/MixedCharZero.lean"
30
+ },
31
+ {
32
+ "error": "no internal trace for tactic 'simpa only [\u2190 Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]'",
33
+ "full_name": "EqualCharZero.of_algebraRat",
34
+ "id": "EqualCharZero.of_algebraRat",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 166,
40
+ 71
41
+ ],
42
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
43
+ "full_name": "EqualCharZero.of_algebraRat",
44
+ "start": [
45
+ 158,
46
+ 1
47
+ ],
48
+ "traced_tactics": [
49
+ {
50
+ "annotated_tactic": [
51
+ "\"intro I hI\"",
52
+ "[]"
53
+ ],
54
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\n\u22a2 CharZero (R \u29f8 I)",
55
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
56
+ "tactic": "intro I hI"
57
+ },
58
+ {
59
+ "annotated_tactic": [
60
+ "\"constructor\"",
61
+ "[]"
62
+ ],
63
+ "state_after": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\n\u22a2 Function.Injective Nat.cast",
64
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\n\u22a2 CharZero (R \u29f8 I)",
65
+ "tactic": "constructor"
66
+ },
67
+ {
68
+ "annotated_tactic": [
69
+ "\"intro a b h_ab\"",
70
+ "[]"
71
+ ],
72
+ "state_after": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\na b : \u2115\nh_ab : \u2191a = \u2191b\n\u22a2 a = b",
73
+ "state_before": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\n\u22a2 Function.Injective Nat.cast",
74
+ "tactic": "intro a b h_ab"
75
+ },
76
+ {
77
+ "annotated_tactic": [
78
+ "\"contrapose! hI\"",
79
+ "[]"
80
+ ],
81
+ "state_after": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 I = \u22a4",
82
+ "state_before": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\nhI : I \u2260 \u22a4\na b : \u2115\nh_ab : \u2191a = \u2191b\n\u22a2 a = b",
83
+ "tactic": "contrapose! hI"
84
+ },
85
+ {
86
+ "annotated_tactic": [
87
+ "\"refine I.eq_top_of_isUnit_mem ?_ (<a>IsUnit.map<\\/a> (<a>algebraMap<\\/a> \\u211a R) (<a>IsUnit.mk0<\\/a> (a - b : \\u211a) ?_))\"",
88
+ "[{\"full_name\":\"IsUnit.map\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units\\/Hom.lean\",\"def_pos\":[198,9],\"def_end_pos\":[198,12]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"IsUnit.mk0\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[249,9],\"def_end_pos\":[249,19]}]"
89
+ ],
90
+ "state_after": "case cast_injective.refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 (algebraMap \u211a R) (\u2191a - \u2191b) \u2208 I\n\ncase cast_injective.refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 \u2191a - \u2191b \u2260 0",
91
+ "state_before": "case cast_injective\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 I = \u22a4",
92
+ "tactic": "refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap \u211a R) (IsUnit.mk0 (a - b : \u211a) ?_))"
93
+ },
94
+ {
95
+ "annotated_tactic": [
96
+ "\"simpa only [<a>Ne<\\/a>, <a>sub_eq_zero<\\/a>] using (@<a>Nat.cast_injective<\\/a> \\u211a _ _).<a>ne<\\/a> hI\"",
97
+ "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"Nat.cast_injective\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[69,9],\"def_end_pos\":[69,23]},{\"full_name\":\"Function.Injective.ne\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[97,9],\"def_end_pos\":[97,21]}]"
98
+ ],
99
+ "state_after": "no goals",
100
+ "state_before": "case cast_injective.refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 \u2191a - \u2191b \u2260 0",
101
+ "tactic": "simpa only [Ne, sub_eq_zero] using (@Nat.cast_injective \u211a _ _).ne hI"
102
+ },
103
+ {
104
+ "annotated_tactic": [
105
+ "\"simpa only [\\u2190 <a>Ideal.Quotient.eq_zero_iff_mem<\\/a>, <a>map_sub<\\/a>, <a>sub_eq_zero<\\/a>, <a>map_natCast<\\/a>]\"",
106
+ "[{\"full_name\":\"Ideal.Quotient.eq_zero_iff_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,24]},{\"full_name\":\"map_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[461,3],\"def_end_pos\":[461,14]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"map_natCast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,20]}]"
107
+ ],
108
+ "state_after": "no goals",
109
+ "state_before": "case cast_injective.refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra \u211a R\nI : Ideal R\na b : \u2115\nh_ab : \u2191a = \u2191b\nhI : a \u2260 b\n\u22a2 (algebraMap \u211a R) (\u2191a - \u2191b) \u2208 I",
110
+ "tactic": "simpa only [\u2190 Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]"
111
+ }
112
+ ],
113
+ "url": "https://github.com/leanprover-community/mathlib4"
114
+ },
115
+ "source_file": "Mathlib/Algebra/CharP/MixedCharZero.lean"
116
+ },
117
+ {
118
+ "error": "no internal trace for tactic 'cases p with\\n| zero => exact hp\\n| succ p =>\\n have h_mixed : MixedCharZero R p.succ := \u27e8\u27e8I, \u27e8hI_ne_top, hp\u27e9\u27e9\u27e9\\n exact absurd h_mixed (h p.succ p.succ_pos)'",
119
+ "full_name": "EqualCharZero.of_not_mixedCharZero",
120
+ "id": "EqualCharZero.of_not_mixedCharZero",
121
+ "reason": "import_failure",
122
+ "selection": {
123
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
124
+ "end": [
125
+ 257,
126
+ 49
127
+ ],
128
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
129
+ "full_name": "EqualCharZero.of_not_mixedCharZero",
130
+ "start": [
131
+ 247,
132
+ 1
133
+ ],
134
+ "traced_tactics": [
135
+ {
136
+ "annotated_tactic": [
137
+ "\"intro I hI_ne_top\"",
138
+ "[]"
139
+ ],
140
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\n\u22a2 CharZero (R \u29f8 I)",
141
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
142
+ "tactic": "intro I hI_ne_top"
143
+ },
144
+ {
145
+ "annotated_tactic": [
146
+ "\"suffices <a>CharP<\\/a> (R \\u29f8 I) 0 from <a>CharP.charP_to_charZero<\\/a> _\"",
147
+ "[{\"full_name\":\"CharP\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[39,7],\"def_end_pos\":[39,19]},{\"full_name\":\"CharP.charP_to_charZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,24]}]"
148
+ ],
149
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\n\u22a2 CharP (R \u29f8 I) 0",
150
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\n\u22a2 CharZero (R \u29f8 I)",
151
+ "tactic": "suffices CharP (R \u29f8 I) 0 from CharP.charP_to_charZero _"
152
+ },
153
+ {
154
+ "annotated_tactic": [
155
+ "\"cases <a>CharP.exists<\\/a> (R \\u29f8 I) with\\n | <a>intro<\\/a> p hp =>\\n cases p with\\n | zero => exact hp\\n | succ p =>\\n have h_mixed : <a>MixedCharZero<\\/a> R p.succ := \\u27e8\\u27e8I, \\u27e8hI_ne_top, hp\\u27e9\\u27e9\\u27e9\\n exact <a>absurd<\\/a> h_mixed (h p.succ p.succ_pos)\"",
156
+ "[{\"full_name\":\"CharP.exists\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[131,7],\"def_end_pos\":[131,15]},{\"full_name\":\"Exists.intro\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[233,5],\"def_end_pos\":[233,10]},{\"full_name\":\"MixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[74,7],\"def_end_pos\":[74,20]},{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
157
+ ],
158
+ "state_after": "no goals",
159
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\n\u22a2 CharP (R \u29f8 I) 0",
160
+ "tactic": "cases CharP.exists (R \u29f8 I) with\n| intro p hp =>\n cases p with\n | zero => exact hp\n | succ p =>\n have h_mixed : MixedCharZero R p.succ := \u27e8\u27e8I, \u27e8hI_ne_top, hp\u27e9\u27e9\u27e9\n exact absurd h_mixed (h p.succ p.succ_pos)"
161
+ },
162
+ {
163
+ "annotated_tactic": [
164
+ "\"cases p with\\n | zero => exact hp\\n | succ p =>\\n have h_mixed : <a>MixedCharZero<\\/a> R p.succ := \\u27e8\\u27e8I, \\u27e8hI_ne_top, hp\\u27e9\\u27e9\\u27e9\\n exact <a>absurd<\\/a> h_mixed (h p.succ p.succ_pos)\"",
165
+ "[{\"full_name\":\"MixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[74,7],\"def_end_pos\":[74,20]},{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
166
+ ],
167
+ "state_after": "no goals",
168
+ "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\np : \u2115\nhp : CharP (R \u29f8 I) p\n\u22a2 CharP (R \u29f8 I) 0",
169
+ "tactic": "cases p with\n| zero => exact hp\n| succ p =>\n have h_mixed : MixedCharZero R p.succ := \u27e8\u27e8I, \u27e8hI_ne_top, hp\u27e9\u27e9\u27e9\n exact absurd h_mixed (h p.succ p.succ_pos)"
170
+ },
171
+ {
172
+ "annotated_tactic": [
173
+ "\"exact hp\"",
174
+ "[]"
175
+ ],
176
+ "state_after": "no goals",
177
+ "state_before": "case intro.zero\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nhp : CharP (R \u29f8 I) 0\n\u22a2 CharP (R \u29f8 I) 0",
178
+ "tactic": "exact hp"
179
+ },
180
+ {
181
+ "annotated_tactic": [
182
+ "\"have h_mixed : <a>MixedCharZero<\\/a> R p.succ := \\u27e8\\u27e8I, \\u27e8hI_ne_top, hp\\u27e9\\u27e9\\u27e9\"",
183
+ "[{\"full_name\":\"MixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[74,7],\"def_end_pos\":[74,20]}]"
184
+ ],
185
+ "state_after": "case intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\np : \u2115\nhp : CharP (R \u29f8 I) (p + 1)\nh_mixed : MixedCharZero R p.succ\n\u22a2 CharP (R \u29f8 I) 0",
186
+ "state_before": "case intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\np : \u2115\nhp : CharP (R \u29f8 I) (p + 1)\n\u22a2 CharP (R \u29f8 I) 0",
187
+ "tactic": "have h_mixed : MixedCharZero R p.succ := \u27e8\u27e8I, \u27e8hI_ne_top, hp\u27e9\u27e9\u27e9"
188
+ },
189
+ {
190
+ "annotated_tactic": [
191
+ "\"exact <a>absurd<\\/a> h_mixed (h p.succ p.succ_pos)\"",
192
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]}]"
193
+ ],
194
+ "state_after": "no goals",
195
+ "state_before": "case intro.succ\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nh : \u2200 p > 0, \u00acMixedCharZero R p\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\np : \u2115\nhp : CharP (R \u29f8 I) (p + 1)\nh_mixed : MixedCharZero R p.succ\n\u22a2 CharP (R \u29f8 I) 0",
196
+ "tactic": "exact absurd h_mixed (h p.succ p.succ_pos)"
197
+ }
198
+ ],
199
+ "url": "https://github.com/leanprover-community/mathlib4"
200
+ },
201
+ "source_file": "Mathlib/Algebra/CharP/MixedCharZero.lean"
202
+ },
203
+ {
204
+ "error": "no internal trace for tactic 'intro g'",
205
+ "full_name": "MixedCharZero.reduce_to_maximal_ideal",
206
+ "id": "MixedCharZero.reduce_to_maximal_ideal",
207
+ "reason": "import_failure",
208
+ "selection": {
209
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
210
+ "end": [
211
+ 133,
212
+ 51
213
+ ],
214
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
215
+ "full_name": "MixedCharZero.reduce_to_maximal_ideal",
216
+ "start": [
217
+ 112,
218
+ 1
219
+ ],
220
+ "traced_tactics": [
221
+ {
222
+ "annotated_tactic": [
223
+ "\"constructor\"",
224
+ "[]"
225
+ ],
226
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\n\u22a2 (\u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p) \u2192 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p\n\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\n\u22a2 (\u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p) \u2192 \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
227
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\n\u22a2 (\u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p) \u2194 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
228
+ "tactic": "constructor"
229
+ },
230
+ {
231
+ "annotated_tactic": [
232
+ "\"intro g\"",
233
+ "[]"
234
+ ],
235
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\ng : \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
236
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\n\u22a2 (\u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p) \u2192 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
237
+ "tactic": "intro g"
238
+ },
239
+ {
240
+ "annotated_tactic": [
241
+ "\"rcases g with \\u27e8I, \\u27e8hI_not_top, _\\u27e9\\u27e9\"",
242
+ "[]"
243
+ ],
244
+ "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
245
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\ng : \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
246
+ "tactic": "rcases g with \u27e8I, \u27e8hI_not_top, _\u27e9\u27e9"
247
+ },
248
+ {
249
+ "annotated_tactic": [
250
+ "\"rcases <a>Ideal.exists_le_maximal<\\/a> I hI_not_top with \\u27e8M, \\u27e8hM_max, hM_ge\\u27e9\\u27e9\"",
251
+ "[{\"full_name\":\"Ideal.exists_le_maximal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,26]}]"
252
+ ],
253
+ "state_after": "case mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
254
+ "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
255
+ "tactic": "rcases Ideal.exists_le_maximal I hI_not_top with \u27e8M, \u27e8hM_max, hM_ge\u27e9\u27e9"
256
+ },
257
+ {
258
+ "annotated_tactic": [
259
+ "\"use M\"",
260
+ "[]"
261
+ ],
262
+ "state_after": "case h\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 M.IsMaximal \u2227 CharP (R \u29f8 M) p",
263
+ "state_before": "case mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 \u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p",
264
+ "tactic": "use M"
265
+ },
266
+ {
267
+ "annotated_tactic": [
268
+ "\"constructor\"",
269
+ "[]"
270
+ ],
271
+ "state_after": "case h.left\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 M.IsMaximal\n\ncase h.right\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 CharP (R \u29f8 M) p",
272
+ "state_before": "case h\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 M.IsMaximal \u2227 CharP (R \u29f8 M) p",
273
+ "tactic": "constructor"
274
+ },
275
+ {
276
+ "annotated_tactic": [
277
+ "\"exact hM_max\"",
278
+ "[]"
279
+ ],
280
+ "state_after": "no goals",
281
+ "state_before": "case h.left\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 M.IsMaximal",
282
+ "tactic": "exact hM_max"
283
+ },
284
+ {
285
+ "annotated_tactic": [
286
+ "\"cases <a>CharP.exists<\\/a> (R \\u29f8 M) with\\n | <a>intro<\\/a> r hr =>\\n convert hr\\n have r_dvd_p : r \\u2223 p := by\\n rw [\\u2190 <a>CharP.cast_eq_zero_iff<\\/a> (R \\u29f8 M) r p]\\n convert <a>congr_arg<\\/a> (<a>Ideal.Quotient.factor<\\/a> I M hM_ge) (<a>CharP.cast_eq_zero<\\/a> (R \\u29f8 I) p)\\n symm\\n apply (<a>Nat.Prime.eq_one_or_self_of_dvd<\\/a> hp r r_dvd_p).<a>resolve_left<\\/a>\\n exact <a>CharP.char_ne_one<\\/a> (R \\u29f8 M) r\"",
287
+ "[{\"full_name\":\"CharP.exists\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[131,7],\"def_end_pos\":[131,15]},{\"full_name\":\"Exists.intro\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[233,5],\"def_end_pos\":[233,10]},{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]},{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.Quotient.factor\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[281,5],\"def_end_pos\":[281,11]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]},{\"full_name\":\"Nat.Prime.eq_one_or_self_of_dvd\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,36]},{\"full_name\":\"Or.resolve_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,24]},{\"full_name\":\"CharP.char_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[253,7],\"def_end_pos\":[253,18]}]"
288
+ ],
289
+ "state_after": "no goals",
290
+ "state_before": "case h.right\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\n\u22a2 CharP (R \u29f8 M) p",
291
+ "tactic": "cases CharP.exists (R \u29f8 M) with\n| intro r hr =>\n convert hr\n have r_dvd_p : r \u2223 p := by\n rw [\u2190 CharP.cast_eq_zero_iff (R \u29f8 M) r p]\n convert congr_arg (Ideal.Quotient.factor I M hM_ge) (CharP.cast_eq_zero (R \u29f8 I) p)\n symm\n apply (Nat.Prime.eq_one_or_self_of_dvd hp r r_dvd_p).resolve_left\n exact CharP.char_ne_one (R \u29f8 M) r"
292
+ },
293
+ {
294
+ "annotated_tactic": [
295
+ "\"convert hr\"",
296
+ "[]"
297
+ ],
298
+ "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 p = r",
299
+ "state_before": "case h.right.intro\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 CharP (R \u29f8 M) p",
300
+ "tactic": "convert hr"
301
+ },
302
+ {
303
+ "annotated_tactic": [
304
+ "\"have r_dvd_p : r \\u2223 p := by\\n rw [\\u2190 <a>CharP.cast_eq_zero_iff<\\/a> (R \\u29f8 M) r p]\\n convert <a>congr_arg<\\/a> (<a>Ideal.Quotient.factor<\\/a> I M hM_ge) (<a>CharP.cast_eq_zero<\\/a> (R \\u29f8 I) p)\"",
305
+ "[{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]},{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.Quotient.factor\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[281,5],\"def_end_pos\":[281,11]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]}]"
306
+ ],
307
+ "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 p = r",
308
+ "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 p = r",
309
+ "tactic": "have r_dvd_p : r \u2223 p := by\n rw [\u2190 CharP.cast_eq_zero_iff (R \u29f8 M) r p]\n convert congr_arg (Ideal.Quotient.factor I M hM_ge) (CharP.cast_eq_zero (R \u29f8 I) p)"
310
+ },
311
+ {
312
+ "annotated_tactic": [
313
+ "\"symm\"",
314
+ "[]"
315
+ ],
316
+ "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 r = p",
317
+ "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 p = r",
318
+ "tactic": "symm"
319
+ },
320
+ {
321
+ "annotated_tactic": [
322
+ "\"apply (<a>Nat.Prime.eq_one_or_self_of_dvd<\\/a> hp r r_dvd_p).<a>resolve_left<\\/a>\"",
323
+ "[{\"full_name\":\"Nat.Prime.eq_one_or_self_of_dvd\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,36]},{\"full_name\":\"Or.resolve_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,24]}]"
324
+ ],
325
+ "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 \u00acr = 1",
326
+ "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 r = p",
327
+ "tactic": "apply (Nat.Prime.eq_one_or_self_of_dvd hp r r_dvd_p).resolve_left"
328
+ },
329
+ {
330
+ "annotated_tactic": [
331
+ "\"exact <a>CharP.char_ne_one<\\/a> (R \\u29f8 M) r\"",
332
+ "[{\"full_name\":\"CharP.char_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[253,7],\"def_end_pos\":[253,18]}]"
333
+ ],
334
+ "state_after": "no goals",
335
+ "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\nr_dvd_p : r \u2223 p\n\u22a2 \u00acr = 1",
336
+ "tactic": "exact CharP.char_ne_one (R \u29f8 M) r"
337
+ },
338
+ {
339
+ "annotated_tactic": [
340
+ "\"rw [\\u2190 <a>CharP.cast_eq_zero_iff<\\/a> (R \\u29f8 M) r p]\"",
341
+ "[{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]}]"
342
+ ],
343
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 \u2191p = 0",
344
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 r \u2223 p",
345
+ "tactic": "rw [\u2190 CharP.cast_eq_zero_iff (R \u29f8 M) r p]"
346
+ },
347
+ {
348
+ "annotated_tactic": [
349
+ "\"convert <a>congr_arg<\\/a> (<a>Ideal.Quotient.factor<\\/a> I M hM_ge) (<a>CharP.cast_eq_zero<\\/a> (R \\u29f8 I) p)\"",
350
+ "[{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.Quotient.factor\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[281,5],\"def_end_pos\":[281,11]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]}]"
351
+ ],
352
+ "state_after": "no goals",
353
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_not_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) p\nM : Ideal R\nhM_max : M.IsMaximal\nhM_ge : I \u2264 M\nr : \u2115\nhr : CharP (R \u29f8 M) r\n\u22a2 \u2191p = 0",
354
+ "tactic": "convert congr_arg (Ideal.Quotient.factor I M hM_ge) (CharP.cast_eq_zero (R \u29f8 I) p)"
355
+ },
356
+ {
357
+ "annotated_tactic": [
358
+ "\"intro \\u27e8I, hI_max, h_charP\\u27e9\"",
359
+ "[]"
360
+ ],
361
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_max : I.IsMaximal\nh_charP : CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
362
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\n\u22a2 (\u2203 I, I.IsMaximal \u2227 CharP (R \u29f8 I) p) \u2192 \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
363
+ "tactic": "intro \u27e8I, hI_max, h_charP\u27e9"
364
+ },
365
+ {
366
+ "annotated_tactic": [
367
+ "\"use I\"",
368
+ "[]"
369
+ ],
370
+ "state_after": "case h\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_max : I.IsMaximal\nh_charP : CharP (R \u29f8 I) p\n\u22a2 I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
371
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_max : I.IsMaximal\nh_charP : CharP (R \u29f8 I) p\n\u22a2 \u2203 I, I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
372
+ "tactic": "use I"
373
+ },
374
+ {
375
+ "annotated_tactic": [
376
+ "\"exact \\u27e8<a>Ideal.IsMaximal.ne_top<\\/a> hI_max, h_charP\\u27e9\"",
377
+ "[{\"full_name\":\"Ideal.IsMaximal.ne_top\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[304,9],\"def_end_pos\":[304,25]}]"
378
+ ],
379
+ "state_after": "no goals",
380
+ "state_before": "case h\nR : Type u_1\ninst\u271d : CommRing R\np : \u2115\nhp : Nat.Prime p\nI : Ideal R\nhI_max : I.IsMaximal\nh_charP : CharP (R \u29f8 I) p\n\u22a2 I \u2260 \u22a4 \u2227 CharP (R \u29f8 I) p",
381
+ "tactic": "exact \u27e8Ideal.IsMaximal.ne_top hI_max, h_charP\u27e9"
382
+ }
383
+ ],
384
+ "url": "https://github.com/leanprover-community/mathlib4"
385
+ },
386
+ "source_file": "Mathlib/Algebra/CharP/MixedCharZero.lean"
387
+ },
388
+ {
389
+ "error": "no internal trace for tactic 'have q_zero :=\\n congr_arg (Ideal.Quotient.factor I M h_IM) (CharP.cast_eq_zero (R \u29f8 I) q)'",
390
+ "full_name": "MixedCharZero.reduce_to_p_prime",
391
+ "id": "MixedCharZero.reduce_to_p_prime",
392
+ "reason": "import_failure",
393
+ "selection": {
394
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
395
+ "end": [
396
+ 105,
397
+ 51
398
+ ],
399
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
400
+ "full_name": "MixedCharZero.reduce_to_p_prime",
401
+ "start": [
402
+ 85,
403
+ 1
404
+ ],
405
+ "traced_tactics": [
406
+ {
407
+ "annotated_tactic": [
408
+ "\"constructor\"",
409
+ "[]"
410
+ ],
411
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\n\u22a2 (\u2200 p > 0, MixedCharZero R p \u2192 P) \u2192 \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\n\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\n\u22a2 (\u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P) \u2192 \u2200 p > 0, MixedCharZero R p \u2192 P",
412
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\n\u22a2 (\u2200 p > 0, MixedCharZero R p \u2192 P) \u2194 \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P",
413
+ "tactic": "constructor"
414
+ },
415
+ {
416
+ "annotated_tactic": [
417
+ "\"intro h q q_prime q_mixedChar\"",
418
+ "[]"
419
+ ],
420
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 p > 0, MixedCharZero R p \u2192 P\nq : \u2115\nq_prime : Nat.Prime q\nq_mixedChar : MixedCharZero R q\n\u22a2 P",
421
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\n\u22a2 (\u2200 p > 0, MixedCharZero R p \u2192 P) \u2192 \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P",
422
+ "tactic": "intro h q q_prime q_mixedChar"
423
+ },
424
+ {
425
+ "annotated_tactic": [
426
+ "\"exact h q (<a>Nat.Prime.pos<\\/a> q_prime) q_mixedChar\"",
427
+ "[{\"full_name\":\"Nat.Prime.pos\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[65,9],\"def_end_pos\":[65,18]}]"
428
+ ],
429
+ "state_after": "no goals",
430
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 p > 0, MixedCharZero R p \u2192 P\nq : \u2115\nq_prime : Nat.Prime q\nq_mixedChar : MixedCharZero R q\n\u22a2 P",
431
+ "tactic": "exact h q (Nat.Prime.pos q_prime) q_mixedChar"
432
+ },
433
+ {
434
+ "annotated_tactic": [
435
+ "\"intro h q q_pos q_mixedChar\"",
436
+ "[]"
437
+ ],
438
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\n\u22a2 P",
439
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\n\u22a2 (\u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P) \u2192 \u2200 p > 0, MixedCharZero R p \u2192 P",
440
+ "tactic": "intro h q q_pos q_mixedChar"
441
+ },
442
+ {
443
+ "annotated_tactic": [
444
+ "\"rcases q_mixedChar.charP_quotient with \\u27e8I, hI_ne_top, _\\u27e9\"",
445
+ "[]"
446
+ ],
447
+ "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\n\u22a2 P",
448
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\n\u22a2 P",
449
+ "tactic": "rcases q_mixedChar.charP_quotient with \u27e8I, hI_ne_top, _\u27e9"
450
+ },
451
+ {
452
+ "annotated_tactic": [
453
+ "\"rcases <a>Ideal.exists_le_maximal<\\/a> I hI_ne_top with \\u27e8M, hM_max, h_IM\\u27e9\"",
454
+ "[{\"full_name\":\"Ideal.exists_le_maximal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,26]}]"
455
+ ],
456
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\n\u22a2 P",
457
+ "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\n\u22a2 P",
458
+ "tactic": "rcases Ideal.exists_le_maximal I hI_ne_top with \u27e8M, hM_max, h_IM\u27e9"
459
+ },
460
+ {
461
+ "annotated_tactic": [
462
+ "\"let r := <a>ringChar<\\/a> (R \\u29f8 M)\"",
463
+ "[{\"full_name\":\"ringChar\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[167,19],\"def_end_pos\":[167,27]}]"
464
+ ],
465
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\n\u22a2 P",
466
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\n\u22a2 P",
467
+ "tactic": "let r := ringChar (R \u29f8 M)"
468
+ },
469
+ {
470
+ "annotated_tactic": [
471
+ "\"have r_pos : r \\u2260 0 := by\\n have q_zero :=\\n <a>congr_arg<\\/a> (<a>Ideal.Quotient.factor<\\/a> I M h_IM) (<a>CharP.cast_eq_zero<\\/a> (R \\u29f8 I) q)\\n simp only [<a>map_natCast<\\/a>, <a>map_zero<\\/a>] at q_zero\\n apply <a>ne_zero_of_dvd_ne_zero<\\/a> (<a>ne_of_gt<\\/a> q_pos)\\n exact (<a>CharP.cast_eq_zero_iff<\\/a> (R \\u29f8 M) r q).<a>mp<\\/a> q_zero\"",
472
+ "[{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.Quotient.factor\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[281,5],\"def_end_pos\":[281,11]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]},{\"full_name\":\"map_natCast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,20]},{\"full_name\":\"map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[203,3],\"def_end_pos\":[203,14]},{\"full_name\":\"ne_zero_of_dvd_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,31]},{\"full_name\":\"ne_of_gt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,17]},{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
473
+ ],
474
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\n\u22a2 P",
475
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\n\u22a2 P",
476
+ "tactic": "have r_pos : r \u2260 0 := by\n have q_zero :=\n congr_arg (Ideal.Quotient.factor I M h_IM) (CharP.cast_eq_zero (R \u29f8 I) q)\n simp only [map_natCast, map_zero] at q_zero\n apply ne_zero_of_dvd_ne_zero (ne_of_gt q_pos)\n exact (CharP.cast_eq_zero_iff (R \u29f8 M) r q).mp q_zero"
477
+ },
478
+ {
479
+ "annotated_tactic": [
480
+ "\"have r_prime : <a>Nat.Prime<\\/a> r :=\\n <a>or_iff_not_imp_right<\\/a>.1 (<a>CharP.char_is_prime_or_zero<\\/a> (R \\u29f8 M) r) r_pos\"",
481
+ "[{\"full_name\":\"Nat.Prime\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[45,5],\"def_end_pos\":[45,10]},{\"full_name\":\"Classical.or_iff_not_imp_right\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,29]},{\"full_name\":\"CharP.char_is_prime_or_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[285,7],\"def_end_pos\":[285,28]}]"
482
+ ],
483
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\n\u22a2 P",
484
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\n\u22a2 P",
485
+ "tactic": "have r_prime : Nat.Prime r :=\n or_iff_not_imp_right.1 (CharP.char_is_prime_or_zero (R \u29f8 M) r) r_pos"
486
+ },
487
+ {
488
+ "annotated_tactic": [
489
+ "\"apply h r r_prime\"",
490
+ "[]"
491
+ ],
492
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\n\u22a2 MixedCharZero R r",
493
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\n\u22a2 P",
494
+ "tactic": "apply h r r_prime"
495
+ },
496
+ {
497
+ "annotated_tactic": [
498
+ "\"have : <a>CharZero<\\/a> R := q_mixedChar.toCharZero\"",
499
+ "[{\"full_name\":\"CharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,15]}]"
500
+ ],
501
+ "state_after": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\nthis : CharZero R\n\u22a2 MixedCharZero R r",
502
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\n\u22a2 MixedCharZero R r",
503
+ "tactic": "have : CharZero R := q_mixedChar.toCharZero"
504
+ },
505
+ {
506
+ "annotated_tactic": [
507
+ "\"exact \\u27e8\\u27e8M, hM_max.ne_top, <a>ringChar.of_eq<\\/a> <a>rfl<\\/a>\\u27e9\\u27e9\"",
508
+ "[{\"full_name\":\"ringChar.of_eq\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[188,7],\"def_end_pos\":[188,12]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
509
+ ],
510
+ "state_after": "no goals",
511
+ "state_before": "case mpr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nr_pos : r \u2260 0\nr_prime : Nat.Prime r\nthis : CharZero R\n\u22a2 MixedCharZero R r",
512
+ "tactic": "exact \u27e8\u27e8M, hM_max.ne_top, ringChar.of_eq rfl\u27e9\u27e9"
513
+ },
514
+ {
515
+ "annotated_tactic": [
516
+ "\"have q_zero :=\\n <a>congr_arg<\\/a> (<a>Ideal.Quotient.factor<\\/a> I M h_IM) (<a>CharP.cast_eq_zero<\\/a> (R \\u29f8 I) q)\"",
517
+ "[{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.Quotient.factor\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[281,5],\"def_end_pos\":[281,11]},{\"full_name\":\"CharP.cast_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,27]}]"
518
+ ],
519
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : (Ideal.Quotient.factor I M h_IM) \u2191q = (Ideal.Quotient.factor I M h_IM) 0\n\u22a2 r \u2260 0",
520
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\n\u22a2 r \u2260 0",
521
+ "tactic": "have q_zero :=\n congr_arg (Ideal.Quotient.factor I M h_IM) (CharP.cast_eq_zero (R \u29f8 I) q)"
522
+ },
523
+ {
524
+ "annotated_tactic": [
525
+ "\"simp only [<a>map_natCast<\\/a>, <a>map_zero<\\/a>] at q_zero\"",
526
+ "[{\"full_name\":\"map_natCast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,20]},{\"full_name\":\"map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[203,3],\"def_end_pos\":[203,14]}]"
527
+ ],
528
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : \u2191q = 0\n\u22a2 r \u2260 0",
529
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : (Ideal.Quotient.factor I M h_IM) \u2191q = (Ideal.Quotient.factor I M h_IM) 0\n\u22a2 r \u2260 0",
530
+ "tactic": "simp only [map_natCast, map_zero] at q_zero"
531
+ },
532
+ {
533
+ "annotated_tactic": [
534
+ "\"apply <a>ne_zero_of_dvd_ne_zero<\\/a> (<a>ne_of_gt<\\/a> q_pos)\"",
535
+ "[{\"full_name\":\"ne_zero_of_dvd_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,31]},{\"full_name\":\"ne_of_gt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,17]}]"
536
+ ],
537
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : \u2191q = 0\n\u22a2 r \u2223 q",
538
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : \u2191q = 0\n\u22a2 r \u2260 0",
539
+ "tactic": "apply ne_zero_of_dvd_ne_zero (ne_of_gt q_pos)"
540
+ },
541
+ {
542
+ "annotated_tactic": [
543
+ "\"exact (<a>CharP.cast_eq_zero_iff<\\/a> (R \\u29f8 M) r q).<a>mp<\\/a> q_zero\"",
544
+ "[{\"full_name\":\"CharP.cast_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[48,7],\"def_end_pos\":[48,23]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
545
+ ],
546
+ "state_after": "no goals",
547
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nq : \u2115\nq_pos : q > 0\nq_mixedChar : MixedCharZero R q\nI : Ideal R\nhI_ne_top : I \u2260 \u22a4\nright\u271d : CharP (R \u29f8 I) q\nM : Ideal R\nhM_max : M.IsMaximal\nh_IM : I \u2264 M\nr : \u2115 := ringChar (R \u29f8 M)\nq_zero : \u2191q = 0\n\u22a2 r \u2223 q",
548
+ "tactic": "exact (CharP.cast_eq_zero_iff (R \u29f8 M) r q).mp q_zero"
549
+ }
550
+ ],
551
+ "url": "https://github.com/leanprover-community/mathlib4"
552
+ },
553
+ "source_file": "Mathlib/Algebra/CharP/MixedCharZero.lean"
554
+ },
555
+ {
556
+ "error": "theorem has no selected tactic transitions",
557
+ "full_name": "CharP.quotient",
558
+ "id": "CharP.quotient",
559
+ "reason": "import_failure",
560
+ "selection": {
561
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
562
+ "end": [
563
+ 32,
564
+ 87
565
+ ],
566
+ "file_path": "Mathlib/Algebra/CharP/Quotient.lean",
567
+ "full_name": "CharP.quotient",
568
+ "start": [
569
+ 21,
570
+ 1
571
+ ],
572
+ "traced_tactics": [],
573
+ "url": "https://github.com/leanprover-community/mathlib4"
574
+ },
575
+ "source_file": "Mathlib/Algebra/CharP/Quotient.lean"
576
+ },
577
+ {
578
+ "error": "no internal trace for tactic 'simp'",
579
+ "full_name": "Ideal.Quotient.index_eq_zero",
580
+ "id": "Ideal.Quotient.index_eq_zero",
581
+ "reason": "import_failure",
582
+ "selection": {
583
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
584
+ "end": [
585
+ 66,
586
+ 38
587
+ ],
588
+ "file_path": "Mathlib/Algebra/CharP/Quotient.lean",
589
+ "full_name": "Ideal.Quotient.index_eq_zero",
590
+ "start": [
591
+ 60,
592
+ 1
593
+ ],
594
+ "traced_tactics": [
595
+ {
596
+ "annotated_tactic": [
597
+ "\"rw [<a>AddSubgroup.index<\\/a>, <a>Nat.card_eq<\\/a>]\"",
598
+ "[{\"full_name\":\"AddSubgroup.index\",\"def_path\":\"Mathlib\\/GroupTheory\\/Index.lean\",\"def_pos\":[45,3],\"def_end_pos\":[45,14]},{\"full_name\":\"Nat.card_eq\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Card.lean\",\"def_pos\":[49,9],\"def_end_pos\":[49,20]}]"
599
+ ],
600
+ "state_after": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2191(if h : Finite (R \u29f8 Submodule.toAddSubgroup I) then Fintype.card (R \u29f8 Submodule.toAddSubgroup I) else 0) = 0",
601
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2191(Submodule.toAddSubgroup I).index = 0",
602
+ "tactic": "rw [AddSubgroup.index, Nat.card_eq]"
603
+ },
604
+ {
605
+ "annotated_tactic": [
606
+ "\"split_ifs with hq\"",
607
+ "[]"
608
+ ],
609
+ "state_after": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\n\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u21910 = 0",
610
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2191(if h : Finite (R \u29f8 Submodule.toAddSubgroup I) then Fintype.card (R \u29f8 Submodule.toAddSubgroup I) else 0) = 0",
611
+ "tactic": "split_ifs with hq"
612
+ },
613
+ {
614
+ "annotated_tactic": [
615
+ "\"swap\"",
616
+ "[]"
617
+ ],
618
+ "state_after": "case neg\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u21910 = 0\n\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0",
619
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\n\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u21910 = 0",
620
+ "tactic": "swap"
621
+ },
622
+ {
623
+ "annotated_tactic": [
624
+ "\"letI : <a>Fintype<\\/a> (R \\u29f8 I) := @<a>Fintype.ofFinite<\\/a> _ hq\"",
625
+ "[{\"full_name\":\"Fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[57,7],\"def_end_pos\":[57,14]},{\"full_name\":\"Fintype.ofFinite\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[449,19],\"def_end_pos\":[449,35]}]"
626
+ ],
627
+ "state_after": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\nthis : Fintype (R \u29f8 I) := Fintype.ofFinite (R \u29f8 I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0",
628
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0",
629
+ "tactic": "letI : Fintype (R \u29f8 I) := @Fintype.ofFinite _ hq"
630
+ },
631
+ {
632
+ "annotated_tactic": [
633
+ "\"exact <a>Nat.cast_card_eq_zero<\\/a> (R \\u29f8 I)\"",
634
+ "[{\"full_name\":\"Nat.cast_card_eq_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[1315,7],\"def_end_pos\":[1315,28]}]"
635
+ ],
636
+ "state_after": "no goals",
637
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\nthis : Fintype (R \u29f8 I) := Fintype.ofFinite (R \u29f8 I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0",
638
+ "tactic": "exact Nat.cast_card_eq_zero (R \u29f8 I)"
639
+ },
640
+ {
641
+ "annotated_tactic": [
642
+ "\"simp\"",
643
+ "[]"
644
+ ],
645
+ "state_after": "no goals",
646
+ "state_before": "case neg\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u21910 = 0",
647
+ "tactic": "simp"
648
+ }
649
+ ],
650
+ "url": "https://github.com/leanprover-community/mathlib4"
651
+ },
652
+ "source_file": "Mathlib/Algebra/CharP/Quotient.lean"
653
+ }
654
+ ],
655
+ "input_proofs": 20,
656
+ "input_units": 2,
657
+ "proofs": 13,
658
+ "published_proofs": 9,
659
+ "published_transitions": 36,
660
+ "replay_failures": [],
661
+ "transitions": 57,
662
+ "validation_failures": [
663
+ {
664
+ "errors": [
665
+ "internal transition boundary mismatch",
666
+ "internal transition boundary mismatch"
667
+ ],
668
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
669
+ "proof_id": "mathlib:EqualCharZero.nonempty_algebraRat_iff",
670
+ "provenance": {
671
+ "selection": {
672
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
673
+ "end": [
674
+ 288,
675
+ 23
676
+ ],
677
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
678
+ "full_name": "EqualCharZero.nonempty_algebraRat_iff",
679
+ "start": [
680
+ 280,
681
+ 1
682
+ ],
683
+ "traced_tactics": [
684
+ {
685
+ "annotated_tactic": [
686
+ "\"constructor\"",
687
+ "[]"
688
+ ],
689
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 Nonempty (Algebra \u211a R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)\n\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)) \u2192 Nonempty (Algebra \u211a R)",
690
+ "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u22a2 Nonempty (Algebra \u211a R) \u2194 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
691
+ "tactic": "constructor"
692
+ },
693
+ {
694
+ "annotated_tactic": [
695
+ "\"intro h_alg\"",
696
+ "[]"
697
+ ],
698
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nh_alg : Nonempty (Algebra \u211a R)\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
699
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 Nonempty (Algebra \u211a R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
700
+ "tactic": "intro h_alg"
701
+ },
702
+ {
703
+ "annotated_tactic": [
704
+ "\"haveI h_alg' : <a>Algebra<\\/a> \\u211a R := h_alg.some\"",
705
+ "[{\"full_name\":\"Algebra\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[101,7],\"def_end_pos\":[101,14]}]"
706
+ ],
707
+ "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nh_alg : Nonempty (Algebra \u211a R)\nh_alg' : Algebra \u211a R\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
708
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nh_alg : Nonempty (Algebra \u211a R)\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
709
+ "tactic": "haveI h_alg' : Algebra \u211a R := h_alg.some"
710
+ },
711
+ {
712
+ "annotated_tactic": [
713
+ "\"apply <a>of_algebraRat<\\/a>\"",
714
+ "[{\"full_name\":\"EqualCharZero.of_algebraRat\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,22]}]"
715
+ ],
716
+ "state_after": "no goals",
717
+ "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nh_alg : Nonempty (Algebra \u211a R)\nh_alg' : Algebra \u211a R\n\u22a2 \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)",
718
+ "tactic": "apply of_algebraRat"
719
+ },
720
+ {
721
+ "annotated_tactic": [
722
+ "\"intro h\"",
723
+ "[]"
724
+ ],
725
+ "state_after": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nh : \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)\n\u22a2 Nonempty (Algebra \u211a R)",
726
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)) \u2192 Nonempty (Algebra \u211a R)",
727
+ "tactic": "intro h"
728
+ },
729
+ {
730
+ "annotated_tactic": [
731
+ "\"apply <a>Nonempty.intro<\\/a>\"",
732
+ "[{\"full_name\":\"Nonempty.intro\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[711,5],\"def_end_pos\":[711,10]}]"
733
+ ],
734
+ "state_after": "case mpr.val\nR : Type u_1\ninst\u271d : CommRing R\nh : \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)\n\u22a2 Algebra \u211a R",
735
+ "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nh : \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)\n\u22a2 Nonempty (Algebra \u211a R)",
736
+ "tactic": "apply Nonempty.intro"
737
+ },
738
+ {
739
+ "annotated_tactic": [
740
+ "\"exact <a>algebraRat<\\/a> h\"",
741
+ "[{\"full_name\":\"EqualCharZero.algebraRat\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[218,19],\"def_end_pos\":[218,29]}]"
742
+ ],
743
+ "state_after": "no goals",
744
+ "state_before": "case mpr.val\nR : Type u_1\ninst\u271d : CommRing R\nh : \u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I)\n\u22a2 Algebra \u211a R",
745
+ "tactic": "exact algebraRat h"
746
+ }
747
+ ],
748
+ "url": "https://github.com/leanprover-community/mathlib4"
749
+ }
750
+ },
751
+ "theorem_name": "EqualCharZero.nonempty_algebraRat_iff"
752
+ },
753
+ {
754
+ "errors": [
755
+ "internal transition boundary mismatch",
756
+ "internal transition boundary mismatch"
757
+ ],
758
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
759
+ "proof_id": "mathlib:split_by_characteristic",
760
+ "provenance": {
761
+ "selection": {
762
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
763
+ "end": [
764
+ 349,
765
+ 30
766
+ ],
767
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
768
+ "full_name": "split_by_characteristic",
769
+ "start": [
770
+ 341,
771
+ 1
772
+ ],
773
+ "traced_tactics": [
774
+ {
775
+ "annotated_tactic": [
776
+ "\"by_cases h : p = 0\"",
777
+ "[]"
778
+ ],
779
+ "state_after": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R p\nh : p = 0\n\u22a2 P\n\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R p\nh : \u00acp = 0\n\u22a2 P",
780
+ "state_before": "case intro\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R p\n\u22a2 P",
781
+ "tactic": "by_cases h : p = 0"
782
+ },
783
+ {
784
+ "annotated_tactic": [
785
+ "\"rw [h] at p_charP\"",
786
+ "[]"
787
+ ],
788
+ "state_after": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R 0\nh : p = 0\n\u22a2 P",
789
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R p\nh : p = 0\n\u22a2 P",
790
+ "tactic": "rw [h] at p_charP"
791
+ },
792
+ {
793
+ "annotated_tactic": [
794
+ "\"haveI h0 : <a>CharZero<\\/a> R := <a>CharP.charP_to_charZero<\\/a> R\"",
795
+ "[{\"full_name\":\"CharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,15]},{\"full_name\":\"CharP.charP_to_charZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,24]}]"
796
+ ],
797
+ "state_after": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R 0\nh : p = 0\nh0 : CharZero R\n\u22a2 P",
798
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R 0\nh : p = 0\n\u22a2 P",
799
+ "tactic": "haveI h0 : CharZero R := CharP.charP_to_charZero R"
800
+ },
801
+ {
802
+ "annotated_tactic": [
803
+ "\"exact <a>split_equalCharZero_mixedCharZero<\\/a> R h_equal h_mixed\"",
804
+ "[{\"full_name\":\"split_equalCharZero_mixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[321,9],\"def_end_pos\":[321,42]}]"
805
+ ],
806
+ "state_after": "no goals",
807
+ "state_before": "case pos\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R 0\nh : p = 0\nh0 : CharZero R\n\u22a2 P",
808
+ "tactic": "exact split_equalCharZero_mixedCharZero R h_equal h_mixed"
809
+ },
810
+ {
811
+ "annotated_tactic": [
812
+ "\"exact h_pos p h p_charP\"",
813
+ "[]"
814
+ ],
815
+ "state_after": "no goals",
816
+ "state_before": "case neg\nR : Type u_1\ninst\u271d : CommRing R\nP : Prop\nh_pos : \u2200 (p : \u2115), p \u2260 0 \u2192 CharP R p \u2192 P\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\np_charP : CharP R p\nh : \u00acp = 0\n\u22a2 P",
817
+ "tactic": "exact h_pos p h p_charP"
818
+ }
819
+ ],
820
+ "url": "https://github.com/leanprover-community/mathlib4"
821
+ }
822
+ },
823
+ "theorem_name": "split_by_characteristic"
824
+ },
825
+ {
826
+ "errors": [
827
+ "internal transition boundary mismatch",
828
+ "internal transition boundary mismatch"
829
+ ],
830
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
831
+ "proof_id": "mathlib:split_equalCharZero_mixedCharZero",
832
+ "provenance": {
833
+ "selection": {
834
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
835
+ "end": [
836
+ 329,
837
+ 17
838
+ ],
839
+ "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean",
840
+ "full_name": "split_equalCharZero_mixedCharZero",
841
+ "start": [
842
+ 321,
843
+ 1
844
+ ],
845
+ "traced_tactics": [
846
+ {
847
+ "annotated_tactic": [
848
+ "\"by_cases h : \\u2203 p > 0, <a>MixedCharZero<\\/a> R p\"",
849
+ "[{\"full_name\":\"MixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[74,7],\"def_end_pos\":[74,20]}]"
850
+ ],
851
+ "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u2203 p > 0, MixedCharZero R p\n\u22a2 P\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u00ac\u2203 p > 0, MixedCharZero R p\n\u22a2 P",
852
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\n\u22a2 P",
853
+ "tactic": "by_cases h : \u2203 p > 0, MixedCharZero R p"
854
+ },
855
+ {
856
+ "annotated_tactic": [
857
+ "\"rcases h with \\u27e8p, \\u27e8H, hp\\u27e9\\u27e9\"",
858
+ "[]"
859
+ ],
860
+ "state_after": "case pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\nH : p > 0\nhp : MixedCharZero R p\n\u22a2 P",
861
+ "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u2203 p > 0, MixedCharZero R p\n\u22a2 P",
862
+ "tactic": "rcases h with \u27e8p, \u27e8H, hp\u27e9\u27e9"
863
+ },
864
+ {
865
+ "annotated_tactic": [
866
+ "\"rw [\\u2190 <a>MixedCharZero.reduce_to_p_prime<\\/a>] at h_mixed\"",
867
+ "[{\"full_name\":\"MixedCharZero.reduce_to_p_prime\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,26]}]"
868
+ ],
869
+ "state_after": "case pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 p > 0, MixedCharZero R p \u2192 P\np : \u2115\nH : p > 0\nhp : MixedCharZero R p\n\u22a2 P",
870
+ "state_before": "case pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\np : \u2115\nH : p > 0\nhp : MixedCharZero R p\n\u22a2 P",
871
+ "tactic": "rw [\u2190 MixedCharZero.reduce_to_p_prime] at h_mixed"
872
+ },
873
+ {
874
+ "annotated_tactic": [
875
+ "\"exact h_mixed p H hp\"",
876
+ "[]"
877
+ ],
878
+ "state_after": "no goals",
879
+ "state_before": "case pos.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 p > 0, MixedCharZero R p \u2192 P\np : \u2115\nH : p > 0\nhp : MixedCharZero R p\n\u22a2 P",
880
+ "tactic": "exact h_mixed p H hp"
881
+ },
882
+ {
883
+ "annotated_tactic": [
884
+ "\"apply h_equal\"",
885
+ "[]"
886
+ ],
887
+ "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u00ac\u2203 p > 0, MixedCharZero R p\n\u22a2 Algebra \u211a R",
888
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u00ac\u2203 p > 0, MixedCharZero R p\n\u22a2 P",
889
+ "tactic": "apply h_equal"
890
+ },
891
+ {
892
+ "annotated_tactic": [
893
+ "\"rw [\\u2190 <a>isEmpty_algebraRat_iff_mixedCharZero<\\/a>, <a>not_isEmpty_iff<\\/a>] at h\"",
894
+ "[{\"full_name\":\"isEmpty_algebraRat_iff_mixedCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/MixedCharZero.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,45]},{\"full_name\":\"not_isEmpty_iff\",\"def_path\":\"Mathlib\\/Logic\\/IsEmpty.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,24]}]"
895
+ ],
896
+ "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : Nonempty (Algebra \u211a R)\n\u22a2 Algebra \u211a R",
897
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : \u00ac\u2203 p > 0, MixedCharZero R p\n\u22a2 Algebra \u211a R",
898
+ "tactic": "rw [\u2190 isEmpty_algebraRat_iff_mixedCharZero, not_isEmpty_iff] at h"
899
+ },
900
+ {
901
+ "annotated_tactic": [
902
+ "\"exact h.some\"",
903
+ "[]"
904
+ ],
905
+ "state_after": "no goals",
906
+ "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nP : Prop\ninst\u271d : CharZero R\nh_equal : Algebra \u211a R \u2192 P\nh_mixed : \u2200 (p : \u2115), Nat.Prime p \u2192 MixedCharZero R p \u2192 P\nh : Nonempty (Algebra \u211a R)\n\u22a2 Algebra \u211a R",
907
+ "tactic": "exact h.some"
908
+ }
909
+ ],
910
+ "url": "https://github.com/leanprover-community/mathlib4"
911
+ }
912
+ },
913
+ "theorem_name": "split_equalCharZero_mixedCharZero"
914
+ },
915
+ {
916
+ "errors": [
917
+ "internal transition boundary mismatch"
918
+ ],
919
+ "file_path": "Mathlib/Algebra/CharP/Quotient.lean",
920
+ "proof_id": "mathlib:CharP.quotient_iff",
921
+ "provenance": {
922
+ "selection": {
923
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
924
+ "end": [
925
+ 51,
926
+ 49
927
+ ],
928
+ "file_path": "Mathlib/Algebra/CharP/Quotient.lean",
929
+ "full_name": "CharP.quotient_iff",
930
+ "start": [
931
+ 47,
932
+ 1
933
+ ],
934
+ "traced_tactics": [
935
+ {
936
+ "annotated_tactic": [
937
+ "\"refine \\u27e8fun _ x hx => ?_, <a>CharP.quotient'<\\/a> n I\\u27e9\"",
938
+ "[{\"full_name\":\"CharP.quotient'\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Quotient.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,18]}]"
939
+ ],
940
+ "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nn : \u2115\ninst\u271d : CharP R n\nI : Ideal R\nx\u271d : CharP (R \u29f8 I) n\nx : \u2115\nhx : \u2191x \u2208 I\n\u22a2 \u2191x = 0",
941
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nn : \u2115\ninst\u271d : CharP R n\nI : Ideal R\n\u22a2 CharP (R \u29f8 I) n \u2194 \u2200 (x : \u2115), \u2191x \u2208 I \u2192 \u2191x = 0",
942
+ "tactic": "refine \u27e8fun _ x hx => ?_, CharP.quotient' n I\u27e9"
943
+ },
944
+ {
945
+ "annotated_tactic": [
946
+ "\"exact (<a>Submodule.Quotient.mk_eq_zero<\\/a> I).<a>mpr<\\/a> hx\"",
947
+ "[{\"full_name\":\"Submodule.Quotient.mk_eq_zero\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Quotient.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,19]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
948
+ ],
949
+ "state_after": "no goals",
950
+ "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nn : \u2115\ninst\u271d : CharP R n\nI : Ideal R\nx\u271d : CharP (R \u29f8 I) n\nx : \u2115\nhx : \u2191x \u2208 I\n\u22a2 \u2191x = 0",
951
+ "tactic": "exact (Submodule.Quotient.mk_eq_zero I).mpr hx"
952
+ }
953
+ ],
954
+ "url": "https://github.com/leanprover-community/mathlib4"
955
+ }
956
+ },
957
+ "theorem_name": "CharP.quotient_iff"
958
+ }
959
+ ],
960
+ "validation_rejected_proofs": 4,
961
+ "zero_transition_proofs": 0
962
+ },
963
+ "shards": [
964
+ {
965
+ "path": "data/train/mathlib-000082-0000-11b982170f77d618.parquet",
966
+ "sha256": "11b982170f77d618247fdd4cb830cc5c64a2132519d9c473bade893344a90a40",
967
+ "bytes": 138227,
968
+ "rows": 36,
969
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
970
+ }
971
+ ],
972
+ "proof_ids": [
973
+ "mathlib:CharP.quotient'",
974
+ "mathlib:CharP.quotient_iff_le_ker_natCast",
975
+ "mathlib:EqualCharZero.PNat.isUnit_natCast",
976
+ "mathlib:EqualCharZero.pnatCast_eq_natCast",
977
+ "mathlib:EqualCharZero.pnatCast_one",
978
+ "mathlib:EqualCharZero.to_not_mixedCharZero",
979
+ "mathlib:isEmpty_algebraRat_iff_mixedCharZero",
980
+ "mathlib:split_by_characteristic_domain",
981
+ "mathlib:split_by_characteristic_localRing"
982
+ ],
983
+ "proof_keys": [
984
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"EqualCharZero.PNat.isUnit_natCast\"]",
985
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"EqualCharZero.pnatCast_eq_natCast\"]",
986
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"EqualCharZero.pnatCast_one\"]",
987
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"EqualCharZero.to_not_mixedCharZero\"]",
988
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"isEmpty_algebraRat_iff_mixedCharZero\"]",
989
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"split_by_characteristic_domain\"]",
990
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/MixedCharZero.lean\",null,null,\"split_by_characteristic_localRing\"]",
991
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Quotient.lean\",null,null,\"CharP.quotient'\"]",
992
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Quotient.lean\",null,null,\"CharP.quotient_iff_le_ker_natCast\"]"
993
+ ]
994
+ }
manifests/batches/mathlib/000083.json ADDED
@@ -0,0 +1,295 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000083",
4
+ "batch_fingerprint": "d486418d4a613d25260fd3d4e178c31b093545df30c36123a352f990ababb76a",
5
+ "evidence": {
6
+ "batch_fingerprint": "d486418d4a613d25260fd3d4e178c31b093545df30c36123a352f990ababb76a",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "frobenius_inj",
12
+ "id": "frobenius_inj",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
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18
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19
+ ],
20
+ "file_path": "Mathlib/Algebra/CharP/Reduced.lean",
21
+ "full_name": "frobenius_inj",
22
+ "start": [
23
+ 27,
24
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25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Algebra/CharP/Reduced.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "CharTwo.add_sq",
34
+ "id": "CharTwo.add_sq",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
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+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
43
+ "full_name": "CharTwo.add_sq",
44
+ "start": [
45
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46
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47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "CharTwo.list_sum_sq",
56
+ "id": "CharTwo.list_sum_sq",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
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63
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64
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
65
+ "full_name": "CharTwo.list_sum_sq",
66
+ "start": [
67
+ 95,
68
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69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "CharTwo.multiset_sum_sq",
78
+ "id": "CharTwo.multiset_sum_sq",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
83
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85
+ ],
86
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
87
+ "full_name": "CharTwo.multiset_sum_sq",
88
+ "start": [
89
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91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
96
+ },
97
+ {
98
+ "error": "theorem has no selected tactic transitions",
99
+ "full_name": "CharTwo.neg_eq'",
100
+ "id": "CharTwo.neg_eq'",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 70,
106
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107
+ ],
108
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
109
+ "full_name": "CharTwo.neg_eq'",
110
+ "start": [
111
+ 69,
112
+ 1
113
+ ],
114
+ "traced_tactics": [],
115
+ "url": "https://github.com/leanprover-community/mathlib4"
116
+ },
117
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
118
+ },
119
+ {
120
+ "error": "theorem has no selected tactic transitions",
121
+ "full_name": "CharTwo.sub_eq_add'",
122
+ "id": "CharTwo.sub_eq_add'",
123
+ "reason": "import_failure",
124
+ "selection": {
125
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
126
+ "end": [
127
+ 78,
128
+ 49
129
+ ],
130
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
131
+ "full_name": "CharTwo.sub_eq_add'",
132
+ "start": [
133
+ 77,
134
+ 1
135
+ ],
136
+ "traced_tactics": [],
137
+ "url": "https://github.com/leanprover-community/mathlib4"
138
+ },
139
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
140
+ },
141
+ {
142
+ "error": "theorem has no selected tactic transitions",
143
+ "full_name": "CharTwo.sum_sq",
144
+ "id": "CharTwo.sum_sq",
145
+ "reason": "import_failure",
146
+ "selection": {
147
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
148
+ "end": [
149
+ 112,
150
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151
+ ],
152
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
153
+ "full_name": "CharTwo.sum_sq",
154
+ "start": [
155
+ 111,
156
+ 1
157
+ ],
158
+ "traced_tactics": [],
159
+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
162
+ },
163
+ {
164
+ "error": "no internal trace for tactic 'rw [neg_one_eq_one_iff.2 h, orderOf_one]'",
165
+ "full_name": "orderOf_neg_one",
166
+ "id": "orderOf_neg_one",
167
+ "reason": "import_failure",
168
+ "selection": {
169
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
170
+ "end": [
171
+ 139,
172
+ 37
173
+ ],
174
+ "file_path": "Mathlib/Algebra/CharP/Two.lean",
175
+ "full_name": "orderOf_neg_one",
176
+ "start": [
177
+ 134,
178
+ 1
179
+ ],
180
+ "traced_tactics": [
181
+ {
182
+ "annotated_tactic": [
183
+ "\"split_ifs with h\"",
184
+ "[]"
185
+ ],
186
+ "state_after": "case pos\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : ringChar R = 2\n\u22a2 orderOf (-1) = 1\n\ncase neg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 orderOf (-1) = 2",
187
+ "state_before": "R : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\n\u22a2 orderOf (-1) = if ringChar R = 2 then 1 else 2",
188
+ "tactic": "split_ifs with h"
189
+ },
190
+ {
191
+ "annotated_tactic": [
192
+ "\"apply <a>orderOf_eq_prime<\\/a>\"",
193
+ "[{\"full_name\":\"orderOf_eq_prime\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[529,9],\"def_end_pos\":[529,25]}]"
194
+ ],
195
+ "state_after": "case neg.hg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 (-1) ^ 2 = 1\n\ncase neg.hg1\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 -1 \u2260 1",
196
+ "state_before": "case neg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 orderOf (-1) = 2",
197
+ "tactic": "apply orderOf_eq_prime"
198
+ },
199
+ {
200
+ "annotated_tactic": [
201
+ "\"simpa [<a>neg_one_eq_one_iff<\\/a>] using h\"",
202
+ "[{\"full_name\":\"neg_one_eq_one_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Two.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,27]}]"
203
+ ],
204
+ "state_after": "no goals",
205
+ "state_before": "case neg.hg1\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 -1 \u2260 1",
206
+ "tactic": "simpa [neg_one_eq_one_iff] using h"
207
+ },
208
+ {
209
+ "annotated_tactic": [
210
+ "\"rw [<a>neg_one_eq_one_iff<\\/a>.2 h, <a>orderOf_one<\\/a>]\"",
211
+ "[{\"full_name\":\"neg_one_eq_one_iff\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Two.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,27]},{\"full_name\":\"orderOf_one\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[245,9],\"def_end_pos\":[245,20]}]"
212
+ ],
213
+ "state_after": "no goals",
214
+ "state_before": "case pos\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : ringChar R = 2\n\u22a2 orderOf (-1) = 1",
215
+ "tactic": "rw [neg_one_eq_one_iff.2 h, orderOf_one]"
216
+ },
217
+ {
218
+ "annotated_tactic": [
219
+ "\"simp\"",
220
+ "[]"
221
+ ],
222
+ "state_after": "no goals",
223
+ "state_before": "case neg.hg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 (-1) ^ 2 = 1",
224
+ "tactic": "simp"
225
+ }
226
+ ],
227
+ "url": "https://github.com/leanprover-community/mathlib4"
228
+ },
229
+ "source_file": "Mathlib/Algebra/CharP/Two.lean"
230
+ }
231
+ ],
232
+ "input_proofs": 24,
233
+ "input_units": 2,
234
+ "proofs": 16,
235
+ "published_proofs": 16,
236
+ "published_transitions": 25,
237
+ "replay_failures": [],
238
+ "transitions": 25,
239
+ "validation_failures": [],
240
+ "validation_rejected_proofs": 0,
241
+ "zero_transition_proofs": 0
242
+ },
243
+ "shards": [
244
+ {
245
+ "path": "data/train/mathlib-000083-0000-5e46854ca6b1c906.parquet",
246
+ "sha256": "5e46854ca6b1c906241ec37ff37d1de0b708eb3811adced947d86b21a3e27110",
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+ "bytes": 65232,
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+ "rows": 21,
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+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
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+ },
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+ {
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+ "path": "data/internal_test/mathlib-000083-0000-a0e6105c5d5d14e0.parquet",
253
+ "sha256": "a0e6105c5d5d14e059a1bfbe968170c8976065ab02968f0c2dce2fb544fa6cfa",
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+ "bytes": 55384,
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+ "rows": 4,
256
+ "revision": "245cb325dc449e5f52470baa7b2d9518d7860cbd"
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+ }
258
+ ],
259
+ "proof_ids": [
260
+ "mathlib:CharTwo.add_mul_self",
261
+ "mathlib:CharTwo.add_self_eq_zero",
262
+ "mathlib:CharTwo.bit0_apply_eq_zero",
263
+ "mathlib:CharTwo.bit0_eq_zero",
264
+ "mathlib:CharTwo.bit1_apply_eq_one",
265
+ "mathlib:CharTwo.bit1_eq_one",
266
+ "mathlib:CharTwo.list_sum_mul_self",
267
+ "mathlib:CharTwo.multiset_sum_mul_self",
268
+ "mathlib:CharTwo.neg_eq",
269
+ "mathlib:CharTwo.sub_eq_add",
270
+ "mathlib:CharTwo.sum_mul_self",
271
+ "mathlib:CharTwo.two_eq_zero",
272
+ "mathlib:ExpChar.pow_prime_pow_mul_eq_one_iff",
273
+ "mathlib:isSquare_of_charTwo'",
274
+ "mathlib:iterateFrobenius_inj",
275
+ "mathlib:neg_one_eq_one_iff"
276
+ ],
277
+ "proof_keys": [
278
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Reduced.lean\",null,null,\"ExpChar.pow_prime_pow_mul_eq_one_iff\"]",
279
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Reduced.lean\",null,null,\"isSquare_of_charTwo'\"]",
280
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Reduced.lean\",null,null,\"iterateFrobenius_inj\"]",
281
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.add_mul_self\"]",
282
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.add_self_eq_zero\"]",
283
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.bit0_apply_eq_zero\"]",
284
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.bit0_eq_zero\"]",
285
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.bit1_apply_eq_one\"]",
286
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.bit1_eq_one\"]",
287
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.list_sum_mul_self\"]",
288
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.multiset_sum_mul_self\"]",
289
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.neg_eq\"]",
290
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.sub_eq_add\"]",
291
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.sum_mul_self\"]",
292
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"CharTwo.two_eq_zero\"]",
293
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/CharP/Two.lean\",null,null,\"neg_one_eq_one_iff\"]"
294
+ ]
295
+ }
manifests/batches/numina/000836.json ADDED
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19
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+ "numina:eb9105cc-0256-54a1-8772-d2bfec808af2"
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+ ],
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_004182.lean\",\"2561d15905d5974b41576c1084faa84fddc3ea3dae254928daa593a0de2fe1d8\",0,\"algrebra_608132\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_004183.lean\",\"63977402659c7835c3ce08bcd38d48781d22bcf4b8fae8d3c1a7ea6bc8569c63\",0,\"number_theory_23059\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_004184.lean\",\"a91d2f9f2154eb54774441aee56441dfc6d92efd140b7c4b8900f790d914c642\",0,\"algebra_96240\"]"
43
+ ]
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+ }
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_004187.lean\",\"d0a3f480e32393a29e22e0317858a12adddb5454c1f78c4dc99d2b2579f76bb9\",0,\"algebra_1148\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_004188.lean\",\"121a5a5a6db5d5b7e53dc5e6275edb6c6aa7cb5a99520b4a0c75c6c0dd8b7a05\",0,\"number_theory_23070\"]",
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50
+ ]
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+ }
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+ ]
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