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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
4b36a61
·
verified ·
1 Parent(s): 61f21f7

Record 11 verified HPC batch manifests

Browse files
Files changed (11) hide show
  1. manifests/batches/deepseek/000052.json +148 -0
  2. manifests/batches/deepseek/000055.json +166 -0
  3. manifests/batches/deepseek/000058.json +141 -0
  4. manifests/batches/deepseek/000059.json +148 -0
  5. manifests/batches/mathlib/000037.json +1411 -0
  6. manifests/batches/mathlib/000038.json +0 -0
  7. manifests/batches/mathlib/000039.json +0 -0
  8. manifests/batches/mathlib/000040.json +0 -0
  9. manifests/batches/mathlib/000041.json +0 -0
  10. manifests/batches/mathlib/000042.json +839 -0
  11. manifests/batches/mathlib/000045.json +0 -0
manifests/batches/deepseek/000052.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000052",
4
+ "batch_fingerprint": "a332a3fb239adc354d18201fcd7bded13913dec48d08033e124fb5353c37054d",
5
+ "evidence": {
6
+ "batch_fingerprint": "a332a3fb239adc354d18201fcd7bded13913dec48d08033e124fb5353c37054d",
7
+ "contamination_matches": [],
8
+ "failures": [],
9
+ "input_proofs": 50,
10
+ "input_units": 50,
11
+ "proofs": 50,
12
+ "published_proofs": 50,
13
+ "published_transitions": 124,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 124,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
20
+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000052-0000-01a0f1a359121b73.parquet",
24
+ "sha256": "01a0f1a359121b73b82be0717f0621c9f0096b91a9af17b448413420c5f3d8ff",
25
+ "bytes": 350994,
26
+ "rows": 118,
27
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
28
+ },
29
+ {
30
+ "path": "data/dev/deepseek-000052-0000-8f208c1308a65158.parquet",
31
+ "sha256": "8f208c1308a65158e819e278582fd2d68998554c4b3b7d3bfe62a081d3a0eb11",
32
+ "bytes": 36702,
33
+ "rows": 3,
34
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
35
+ },
36
+ {
37
+ "path": "data/internal_test/deepseek-000052-0000-df071c64fc732688.parquet",
38
+ "sha256": "df071c64fc73268857ea44532dd5f02a272f34df4a01904e202ce2f9cbe1b7c3",
39
+ "bytes": 50922,
40
+ "rows": 3,
41
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
42
+ }
43
+ ],
44
+ "proof_ids": [
45
+ "deepseek:thm_2619",
46
+ "deepseek:thm_2620",
47
+ "deepseek:thm_2621",
48
+ "deepseek:thm_2622",
49
+ "deepseek:thm_2623",
50
+ "deepseek:thm_2624",
51
+ "deepseek:thm_2625",
52
+ "deepseek:thm_2626",
53
+ "deepseek:thm_2627",
54
+ "deepseek:thm_2628",
55
+ "deepseek:thm_2629",
56
+ "deepseek:thm_2630",
57
+ "deepseek:thm_2631",
58
+ "deepseek:thm_2632",
59
+ "deepseek:thm_2633",
60
+ "deepseek:thm_2634",
61
+ "deepseek:thm_2635",
62
+ "deepseek:thm_2636",
63
+ "deepseek:thm_2637",
64
+ "deepseek:thm_2638",
65
+ "deepseek:thm_2639",
66
+ "deepseek:thm_2640",
67
+ "deepseek:thm_2641",
68
+ "deepseek:thm_2642",
69
+ "deepseek:thm_2643",
70
+ "deepseek:thm_2644",
71
+ "deepseek:thm_2645",
72
+ "deepseek:thm_2646",
73
+ "deepseek:thm_2647",
74
+ "deepseek:thm_2648",
75
+ "deepseek:thm_2649",
76
+ "deepseek:thm_2650",
77
+ "deepseek:thm_2651",
78
+ "deepseek:thm_2652",
79
+ "deepseek:thm_2653",
80
+ "deepseek:thm_2654",
81
+ "deepseek:thm_2655",
82
+ "deepseek:thm_2656",
83
+ "deepseek:thm_2657",
84
+ "deepseek:thm_2658",
85
+ "deepseek:thm_2659",
86
+ "deepseek:thm_2660",
87
+ "deepseek:thm_2661",
88
+ "deepseek:thm_2662",
89
+ "deepseek:thm_2663",
90
+ "deepseek:thm_2664",
91
+ "deepseek:thm_2665",
92
+ "deepseek:thm_2666",
93
+ "deepseek:thm_2667",
94
+ "deepseek:thm_2668"
95
+ ],
96
+ "proof_keys": [
97
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002600.lean\",\"ec476e0192b75fe50402654315251fce627d3d83774ecc75878d6ddded0277bf\",0,\"thm_2619\"]",
98
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002601.lean\",\"937347046258872eac02a65eb2cfc9e8c8869771f9fe0154f7ace14785836b45\",0,\"thm_2620\"]",
99
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002602.lean\",\"10ce19f12787aa4bdc9eefe6234c091b41f9c7266e6bee276f65cba510e49fde\",0,\"thm_2621\"]",
100
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002603.lean\",\"d8a44ecc406585af4888ac0316a17fd9582676ce63c4ce3a70f684ebfcc3cd28\",0,\"thm_2622\"]",
101
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002604.lean\",\"ac65da13693231abc0c081362204186d94a8abfa41e2ab36b5cbd603ded6e4ad\",0,\"thm_2623\"]",
102
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002605.lean\",\"dfd75d9f82f905440af8b81b183a37a9bc4415cabb2575bb1b7c9e0400939b0c\",0,\"thm_2624\"]",
103
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002606.lean\",\"d1f276adf307df6d5c46ac4a8f86e9204a5b8909819b504f9a88c6c7795fde84\",0,\"thm_2625\"]",
104
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002607.lean\",\"1a0d1932d7122da753aa4c6dd687c54e2336a088e7916d153144b40fdb16a1b8\",0,\"thm_2626\"]",
105
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002608.lean\",\"ab5e66dd538afeefab97b4ce750b9e9aee64daa8a77e8a15e14c3837c826866f\",0,\"thm_2627\"]",
106
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002609.lean\",\"d8f3a6d12f1e7d132a551ef833b048a30461f55421854438975e3ffd5d9d4488\",0,\"thm_2628\"]",
107
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002610.lean\",\"6b0292b7ad417dfe4de22ca82e75b2ca47e189f668c655b208aac448f82a17c0\",0,\"thm_2629\"]",
108
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002611.lean\",\"661b9c15c182ecaeac40534cc1fdbfecce840ef8454a799effc6cc3b5085bf54\",0,\"thm_2630\"]",
109
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002612.lean\",\"88d6545212938828ede7dbea560d60f37fd38f8a35a20aed1bdb1803f9c37693\",0,\"thm_2631\"]",
110
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002613.lean\",\"f77a3a5cac9f84110602787958a2a274fb9917dd8cc2f17919c9e8fbb834d00b\",0,\"thm_2632\"]",
111
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002614.lean\",\"d88d199d6b8e0413adc064b47c271286d7d5da87e8d06fa11c4f4a33baf9617f\",0,\"thm_2633\"]",
112
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002615.lean\",\"41aa5ca8e950a00610fa6f075361c348c61e138f64f227506c1b76873cc0409a\",0,\"thm_2634\"]",
113
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002616.lean\",\"fe9165ae8cb372897336f67262d4182db95b85c6d08eb42a33de5214c861937e\",0,\"thm_2635\"]",
114
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002617.lean\",\"86ebc795887b1c95b034f36ccdb5041589382cd3a5785bd2622f8016875ede7c\",0,\"thm_2636\"]",
115
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002618.lean\",\"228080224de7233ae450651488dbac15f34fedde46602f26f164195507d72103\",0,\"thm_2637\"]",
116
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002619.lean\",\"ec7df5faa29f946b1a762bf740b28212391b1a9ad102497dde61ed71178d49f9\",0,\"thm_2638\"]",
117
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002620.lean\",\"13b06f0467263b2c304ae2095db90e38aa0ddf0c20f7166e4448f7c163cb4277\",0,\"thm_2639\"]",
118
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002621.lean\",\"3663eb46fb28593a255e7e8891001160617d6595f05a0fde8470981c6e6f2cdb\",0,\"thm_2640\"]",
119
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002622.lean\",\"686fa7a83d3adc94376f459720518e465de24d8a0d78ee8fc622f265d4c8cd49\",0,\"thm_2641\"]",
120
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002623.lean\",\"f10b2f785c6979be86ba77586bc3f7654631be6dbd4b033183d26a5cdd777a43\",0,\"thm_2642\"]",
121
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002624.lean\",\"c9e2f938c298b510829f9fbcc6b422963030b9492615e6c20b62c7a10d6d7bdd\",0,\"thm_2643\"]",
122
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002625.lean\",\"903d27c338e5161f9edff1d604f89eadcce948ad3bb54421cf67ef91bdd4fa88\",0,\"thm_2644\"]",
123
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002626.lean\",\"167d652625c5ef048fdfe44acce306ab9816d910564fa9ec27da53cf81475700\",0,\"thm_2645\"]",
124
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002627.lean\",\"daba8de50a3405cd8b022dedd931a1532c2755a14dd7c9dc05f91ec8c25d6617\",0,\"thm_2646\"]",
125
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002628.lean\",\"350ea787481c2c5fc6149526880623332c8ca2831232fd01c0f44a2aa370d6b1\",0,\"thm_2647\"]",
126
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002629.lean\",\"64ca7d8cdb07bb46981cc3b9163f20bd32c72264ebba15eafdc23150b2418345\",0,\"thm_2648\"]",
127
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002630.lean\",\"66d8756a2103e9a0307e1744b04cb50270200dcc79a436ddc7fd8d386f3022b2\",0,\"thm_2649\"]",
128
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002631.lean\",\"3c971187a9732aaca5a41a67e1e6b051248889fc756b402340d6aabcbefdf53a\",0,\"thm_2650\"]",
129
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002632.lean\",\"0a53dbfe3de33f79a20cad8f3e5f0dc663d20f1d1e24e8cf1b752c499ecf7d45\",0,\"thm_2651\"]",
130
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002633.lean\",\"bcc6783d3ee03ceca139d7a26f15facd1ad7a91c85912aece0b045894bd7dc7e\",0,\"thm_2652\"]",
131
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002634.lean\",\"af2ccf5e0152c5e3e9b19b656b094adfd70d6a4e45fd8f1af1c17765d4a04f4e\",0,\"thm_2653\"]",
132
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002635.lean\",\"856747f554a6e63d16769f55e9395d5555cab447cccdff7ab6257eac76d9165b\",0,\"thm_2654\"]",
133
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002636.lean\",\"e8e5efcc7178237eebf079de2082dd296f222745087a764d4d5da3ba67132541\",0,\"thm_2655\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002637.lean\",\"638e474bc0a81fa36f8876509ab6736755a77eb37a2fcbbffe13cee90694ee2b\",0,\"thm_2656\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002638.lean\",\"79ab6ec32908a9b78bbd548bf14a4d5abcb496be58582806c937bebe30b3737e\",0,\"thm_2657\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002639.lean\",\"46892cbee199d36c7178c7284f445013bdbbd1635870f1594410c165245f2c24\",0,\"thm_2658\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002640.lean\",\"889b1041105222b5f77396e4a2dde40036ce565c9ff1d1db7f40db08e07d894d\",0,\"thm_2659\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002641.lean\",\"b6cb84e620c71e4e928edd897d4d7cc35e2f0eaee77aae3df485e65d04011d13\",0,\"thm_2660\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002642.lean\",\"f7e7f6c31fdc219a66f0886b419aacfd7523df46c25fa3b9cdfa618babc8e1ab\",0,\"thm_2661\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002643.lean\",\"89130d2702441b962e332b8d9db24cb1cb68e280828327beceba8d74b8aa51f1\",0,\"thm_2662\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002644.lean\",\"daee5e65a8b00997884a339c0243b19c4ac8bb9fbc7dc2def37fcd5bdbb65286\",0,\"thm_2663\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002645.lean\",\"5793e0e46c4054406ad67a2e8ef12c37282eda112cdf24b927c76fcd60b1b7cd\",0,\"thm_2664\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002646.lean\",\"4d657a7b82cfdb7ef9367568ce29eb736f9f550341e4c6bde7126f1a9700e9d4\",0,\"thm_2665\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002647.lean\",\"795188927ee23417ab25fc170c7ea70d0c57e774f8ec48ee5a5b2662b8cec332\",0,\"thm_2666\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002648.lean\",\"582399162eb419e68c182c7149667ac78e6d72b5f34b7f5bff840f565dcdb08f\",0,\"thm_2667\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002649.lean\",\"3f6bcfde5839719b7e035d577b3d079acd5ca0f8e22c543787b8f31451610434\",0,\"thm_2668\"]"
147
+ ]
148
+ }
manifests/batches/deepseek/000055.json ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000055",
4
+ "batch_fingerprint": "4e63f95fc829489f549ca0c4a40d82dc061e7a1981c824948d3d7d50a91813e7",
5
+ "evidence": {
6
+ "batch_fingerprint": "4e63f95fc829489f549ca0c4a40d82dc061e7a1981c824948d3d7d50a91813e7",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "final replayable tactic does not close all goals",
11
+ "id": "thm_2777",
12
+ "index": 2758,
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "file_path": "Full/Proof_002758.lean",
16
+ "id": "thm_2777",
17
+ "index": 2758,
18
+ "metadata": {
19
+ "declaration_kind": "theorem",
20
+ "source_dataset": "deepseek",
21
+ "theorem_name": "thm_2777",
22
+ "theorem_ordinal": 0
23
+ },
24
+ "source_sha256": "71132b4dd9f647a4db98be82ce85af9633d771a7241e100a6ce8039ad75972c1"
25
+ },
26
+ "source_file": "Full/Proof_002758.lean"
27
+ }
28
+ ],
29
+ "input_proofs": 50,
30
+ "input_units": 50,
31
+ "proofs": 49,
32
+ "published_proofs": 49,
33
+ "published_transitions": 115,
34
+ "replay_failures": [],
35
+ "trace_mode": "individual",
36
+ "transitions": 115,
37
+ "validation_failures": [],
38
+ "validation_rejected_proofs": 0,
39
+ "zero_transition_proofs": 0
40
+ },
41
+ "shards": [
42
+ {
43
+ "path": "data/train/deepseek-000055-0000-b7898f17a139eb23.parquet",
44
+ "sha256": "b7898f17a139eb23407600b17b38259b94d8ff92f47b6b779afa440f19dc77ec",
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+ "bytes": 321499,
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+ "rows": 96,
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+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
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+ },
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+ {
50
+ "path": "data/dev/deepseek-000055-0000-89cd3da23a01c031.parquet",
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+ "sha256": "89cd3da23a01c031d174ce140efa41f9e5ac73cd9a34a290239d5785c1d588e8",
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+ "bytes": 52287,
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+ "rows": 7,
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+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
55
+ },
56
+ {
57
+ "path": "data/internal_test/deepseek-000055-0000-ba0d7511f1758238.parquet",
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manifests/batches/deepseek/000058.json ADDED
@@ -0,0 +1,141 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ ]
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+ }
manifests/batches/deepseek/000059.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ },
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+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
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+ },
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+ "start": [
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+ {
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+ "state_after": "case h\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring S\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : Module R A\ninst\u271d\u2078 : SMulCommClass R A A\ninst\u271d\u2077 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : Algebra S B\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : DistribMulAction S A\ninst\u271d\u00b2 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nx\u271d : Unitization R A\n\u22a2 (NonUnitalAlgHom.toAlgHom 0) x\u271d = x\u271d.fst",
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+ "state_before": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring S\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : Module R A\ninst\u271d\u2078 : SMulCommClass R A A\ninst\u271d\u2077 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : Algebra S B\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : DistribMulAction S A\ninst\u271d\u00b2 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u22a2 \u21d1(NonUnitalAlgHom.toAlgHom 0) = fst",
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+ },
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+ {
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+ "\"simp\"",
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+ "[]"
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63
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64
+ }
65
+ ],
66
+ "url": "https://github.com/leanprover-community/mathlib4"
67
+ },
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+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
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+ },
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+ {
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+ "id": "Unitization.algHom_ext",
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+ "start": [
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+ 684,
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+ 1
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+ ],
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+ "traced_tactics": [
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+ {
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+ "annotated_tactic": [
89
+ "\"refine <a>DFunLike.ext<\\/a> \\u03c6 \\u03c8 (fun x \\u21a6 ?_)\"",
90
+ "[{\"full_name\":\"DFunLike.ext\",\"def_path\":\"Mathlib\\/Data\\/FunLike\\/Basic.lean\",\"def_pos\":[201,9],\"def_end_pos\":[201,12]}]"
91
+ ],
92
+ "state_after": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring S\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : NonUnitalSemiring A\ninst\u271d\u00b9\u00b9 : Module R A\ninst\u271d\u00b9\u2070 : SMulCommClass R A A\ninst\u271d\u2079 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2078 : Semiring B\ninst\u271d\u2077 : Algebra S B\ninst\u271d\u2076 : Algebra S R\ninst\u271d\u2075 : DistribMulAction S A\ninst\u271d\u2074 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : Algebra R C\nF : Type u_6\ninst\u271d\u00b9 : FunLike F (Unitization R A) B\ninst\u271d : AlgHomClass F S (Unitization R A) B\n\u03c6 \u03c8 : F\nh : \u2200 (a : A), \u03c6 \u2191a = \u03c8 \u2191a\nh' : \u2200 (r : R), \u03c6 ((algebraMap R (Unitization R A)) r) = \u03c8 ((algebraMap R (Unitization R A)) r)\nx : Unitization R A\n\u22a2 \u03c6 x = \u03c8 x",
93
+ "state_before": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring S\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : NonUnitalSemiring A\ninst\u271d\u00b9\u00b9 : Module R A\ninst\u271d\u00b9\u2070 : SMulCommClass R A A\ninst\u271d\u2079 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2078 : Semiring B\ninst\u271d\u2077 : Algebra S B\ninst\u271d\u2076 : Algebra S R\ninst\u271d\u2075 : DistribMulAction S A\ninst\u271d\u2074 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : Algebra R C\nF : Type u_6\ninst\u271d\u00b9 : FunLike F (Unitization R A) B\ninst\u271d : AlgHomClass F S (Unitization R A) B\n\u03c6 \u03c8 : F\nh : \u2200 (a : A), \u03c6 \u2191a = \u03c8 \u2191a\nh' : \u2200 (r : R), \u03c6 ((algebraMap R (Unitization R A)) r) = \u03c8 ((algebraMap R (Unitization R A)) r)\n\u22a2 \u03c6 = \u03c8",
94
+ "tactic": "refine DFunLike.ext \u03c6 \u03c8 (fun x \u21a6 ?_)"
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+ },
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+ {
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+ "annotated_tactic": [
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+ "\"induction x\"",
99
+ "[]"
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+ ],
101
+ "state_after": "case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring S\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : NonUnitalSemiring A\ninst\u271d\u00b9\u00b9 : Module R A\ninst\u271d\u00b9\u2070 : SMulCommClass R A A\ninst\u271d\u2079 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2078 : Semiring B\ninst\u271d\u2077 : Algebra S B\ninst\u271d\u2076 : Algebra S R\ninst\u271d\u2075 : DistribMulAction S A\ninst\u271d\u2074 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : Algebra R C\nF : Type u_6\ninst\u271d\u00b9 : FunLike F (Unitization R A) B\ninst\u271d : AlgHomClass F S (Unitization R A) B\n\u03c6 \u03c8 : F\nh : \u2200 (a : A), \u03c6 \u2191a = \u03c8 \u2191a\nh' : \u2200 (r : R), \u03c6 ((algebraMap R (Unitization R A)) r) = \u03c8 ((algebraMap R (Unitization R A)) r)\nr\u271d : R\na\u271d : A\n\u22a2 \u03c6 (inl r\u271d + \u2191a\u271d) = \u03c8 (inl r\u271d + \u2191a\u271d)",
102
+ "state_before": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u00b9\u2074 : CommSemiring S\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : NonUnitalSemiring A\ninst\u271d\u00b9\u00b9 : Module R A\ninst\u271d\u00b9\u2070 : SMulCommClass R A A\ninst\u271d\u2079 : IsScalarTower R A A\nB : Type u_4\ninst\u271d\u2078 : Semiring B\ninst\u271d\u2077 : Algebra S B\ninst\u271d\u2076 : Algebra S R\ninst\u271d\u2075 : DistribMulAction S A\ninst\u271d\u2074 : IsScalarTower S R A\nC : Type u_5\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : Algebra R C\nF : Type u_6\ninst\u271d\u00b9 : FunLike F (Unitization R A) B\ninst\u271d : AlgHomClass F S (Unitization R A) B\n\u03c6 \u03c8 : F\nh : \u2200 (a : A), \u03c6 \u2191a = \u03c8 \u2191a\nh' : \u2200 (r : R), \u03c6 ((algebraMap R (Unitization R A)) r) = \u03c8 ((algebraMap R (Unitization R A)) r)\nx : Unitization R A\n\u22a2 \u03c6 x = \u03c8 x",
103
+ "tactic": "induction x"
104
+ },
105
+ {
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997
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998
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1000
+ "tactic": "simp only [isStarNormal_iff, commute_iff_eq, \u2190 inr_star, \u2190 inr_mul, inr_injective.eq_iff]"
1001
+ }
1002
+ ],
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ },
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+ },
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+ {
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+ "full_name": "Unitization.snd_neg",
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+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
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+ },
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+ {
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+ "full_name": "Unitization.snd_one",
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+ "id": "Unitization.snd_one",
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+ "reason": "missing_trace",
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+ },
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+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
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+ },
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+ {
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+ "full_name": "Unitization.snd_star",
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+ "reason": "missing_trace",
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
1194
+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
1195
+ },
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+ {
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+ "full_name": "Unitization.snd_zero",
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+ "reason": "missing_trace",
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+ "file_path": "Mathlib/Algebra/Algebra/Unitization.lean",
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+ },
1215
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1216
+ },
1217
+ {
1218
+ "full_name": "Unitization.starAlgHom_ext",
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+ "reason": "missing_trace",
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+ "file_path": "Mathlib/Algebra/Algebra/Unitization.lean",
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+ "full_name": "Unitization.starAlgHom_ext",
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+ ],
1233
+ "traced_tactics": [],
1234
+ "url": "https://github.com/leanprover-community/mathlib4"
1235
+ },
1236
+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
1237
+ },
1238
+ {
1239
+ "full_name": "Unitization.starLift_symm_apply_apply",
1240
+ "id": "Unitization.starLift_symm_apply_apply",
1241
+ "reason": "missing_trace",
1242
+ "selection": {
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+ "file_path": "Mathlib/Algebra/Algebra/Unitization.lean",
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+ "full_name": "Unitization.starLift_symm_apply_apply",
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+ ],
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
1256
+ },
1257
+ "source_file": "Mathlib/Algebra/Algebra/Unitization.lean"
1258
+ },
1259
+ {
1260
+ "error": "theorem has no selected tactic transitions",
1261
+ "full_name": "Algebraic.aleph0_le_cardinal_mk_of_charZero",
1262
+ "id": "Algebraic.aleph0_le_cardinal_mk_of_charZero",
1263
+ "reason": "import_failure",
1264
+ "selection": {
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+ "end": [
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+ ],
1270
+ "file_path": "Mathlib/Algebra/AlgebraicCard.lean",
1271
+ "full_name": "Algebraic.aleph0_le_cardinal_mk_of_charZero",
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+ "start": [
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+ ],
1276
+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
1278
+ },
1279
+ "source_file": "Mathlib/Algebra/AlgebraicCard.lean"
1280
+ },
1281
+ {
1282
+ "error": "theorem has no selected tactic transitions",
1283
+ "full_name": "Algebraic.cardinal_mk_lift_of_infinite",
1284
+ "id": "Algebraic.cardinal_mk_lift_of_infinite",
1285
+ "reason": "import_failure",
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1292
+ "file_path": "Mathlib/Algebra/AlgebraicCard.lean",
1293
+ "full_name": "Algebraic.cardinal_mk_lift_of_infinite",
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+ "start": [
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+ ],
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+ "url": "https://github.com/leanprover-community/mathlib4"
1300
+ },
1301
+ "source_file": "Mathlib/Algebra/AlgebraicCard.lean"
1302
+ },
1303
+ {
1304
+ "error": "theorem has no selected tactic transitions",
1305
+ "full_name": "Algebraic.cardinal_mk_of_countable_of_charZero",
1306
+ "id": "Algebraic.cardinal_mk_of_countable_of_charZero",
1307
+ "reason": "import_failure",
1308
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+ "file_path": "Mathlib/Algebra/AlgebraicCard.lean",
1315
+ "full_name": "Algebraic.cardinal_mk_of_countable_of_charZero",
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1322
+ },
1323
+ "source_file": "Mathlib/Algebra/AlgebraicCard.lean"
1324
+ },
1325
+ {
1326
+ "error": "theorem has no selected tactic transitions",
1327
+ "full_name": "Algebraic.cardinal_mk_of_infinite",
1328
+ "id": "Algebraic.cardinal_mk_of_infinite",
1329
+ "reason": "import_failure",
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1336
+ "file_path": "Mathlib/Algebra/AlgebraicCard.lean",
1337
+ "full_name": "Algebraic.cardinal_mk_of_infinite",
1338
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1342
+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
1344
+ },
1345
+ "source_file": "Mathlib/Algebra/AlgebraicCard.lean"
1346
+ },
1347
+ {
1348
+ "error": "theorem has no selected tactic transitions",
1349
+ "full_name": "Algebraic.infinite_of_charZero",
1350
+ "id": "Algebraic.infinite_of_charZero",
1351
+ "reason": "import_failure",
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+ "selection": {
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1358
+ "file_path": "Mathlib/Algebra/AlgebraicCard.lean",
1359
+ "full_name": "Algebraic.infinite_of_charZero",
1360
+ "start": [
1361
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1362
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1363
+ ],
1364
+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/Algebra/AlgebraicCard.lean"
1368
+ }
1369
+ ],
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+ "input_proofs": 63,
1371
+ "input_units": 2,
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+ "proofs": 5,
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+ "published_proofs": 5,
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+ "published_transitions": 15,
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+ "replay_failures": [],
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+ "validation_failures": [],
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+ "validation_rejected_proofs": 0,
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+ "zero_transition_proofs": 0
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+ },
1381
+ "shards": [
1382
+ {
1383
+ "path": "data/train/mathlib-000037-0000-2b584582add5ad1d.parquet",
1384
+ "sha256": "2b584582add5ad1d4b477890c01b588fe6f5c41728f8437396e7a98a66aceb22",
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+ "bytes": 51143,
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+ "rows": 8,
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+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
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+ },
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+ {
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+ "path": "data/dev/mathlib-000037-0000-2ce359679e2d283e.parquet",
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+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
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+ }
1396
+ ],
1397
+ "proof_ids": [
1398
+ "mathlib:Algebraic.cardinal_mk_le_max",
1399
+ "mathlib:Algebraic.cardinal_mk_le_mul",
1400
+ "mathlib:Algebraic.cardinal_mk_lift_le_max",
1401
+ "mathlib:Algebraic.cardinal_mk_lift_le_mul",
1402
+ "mathlib:Algebraic.countable"
1403
+ ],
1404
+ "proof_keys": [
1405
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/AlgebraicCard.lean\",null,null,\"Algebraic.cardinal_mk_le_max\"]",
1406
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/AlgebraicCard.lean\",null,null,\"Algebraic.cardinal_mk_le_mul\"]",
1407
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/AlgebraicCard.lean\",null,null,\"Algebraic.cardinal_mk_lift_le_max\"]",
1408
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/AlgebraicCard.lean\",null,null,\"Algebraic.cardinal_mk_lift_le_mul\"]",
1409
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/AlgebraicCard.lean\",null,null,\"Algebraic.countable\"]"
1410
+ ]
1411
+ }
manifests/batches/mathlib/000038.json ADDED
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manifests/batches/mathlib/000039.json ADDED
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manifests/batches/mathlib/000040.json ADDED
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manifests/batches/mathlib/000041.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/000042.json ADDED
@@ -0,0 +1,839 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000042",
4
+ "batch_fingerprint": "1a34cf82a033355341eed1f370d442956f411d5bf6d87f5dc64761721ac70999",
5
+ "evidence": {
6
+ "batch_fingerprint": "1a34cf82a033355341eed1f370d442956f411d5bf6d87f5dc64761721ac70999",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic 'rw [\u2190 prod_erase_mul _ _ hi, h, mul_zero]'",
11
+ "full_name": "Finset.prod_eq_zero",
12
+ "id": "Finset.prod_eq_zero",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 26,
18
+ 54
19
+ ],
20
+ "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean",
21
+ "full_name": "Finset.prod_eq_zero",
22
+ "start": [
23
+ 25,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"classical rw [\\u2190 <a>prod_erase_mul<\\/a> _ _ hi, h, <a>mul_zero<\\/a>]\"",
30
+ "[{\"full_name\":\"Finset.prod_erase_mul\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1969,9],\"def_end_pos\":[1969,23]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
31
+ ],
32
+ "state_after": "no goals",
33
+ "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh : f i = 0\n\u22a2 \u220f j \u2208 s, f j = 0",
34
+ "tactic": "classical rw [\u2190 prod_erase_mul _ _ hi, h, mul_zero]"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"rw [\\u2190 <a>prod_erase_mul<\\/a> _ _ hi, h, <a>mul_zero<\\/a>]\"",
39
+ "[{\"full_name\":\"Finset.prod_erase_mul\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1969,9],\"def_end_pos\":[1969,23]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
40
+ ],
41
+ "state_after": "no goals",
42
+ "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh : f i = 0\n\u22a2 \u220f j \u2208 s, f j = 0",
43
+ "tactic": "rw [\u2190 prod_erase_mul _ _ hi, h, mul_zero]"
44
+ }
45
+ ],
46
+ "url": "https://github.com/leanprover-community/mathlib4"
47
+ },
48
+ "source_file": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean"
49
+ },
50
+ {
51
+ "error": "no internal trace for tactic 'simp at H'",
52
+ "full_name": "Finset.prod_eq_zero_iff",
53
+ "id": "Finset.prod_eq_zero_iff",
54
+ "reason": "import_failure",
55
+ "selection": {
56
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
57
+ "end": [
58
+ 51,
59
+ 62
60
+ ],
61
+ "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean",
62
+ "full_name": "Finset.prod_eq_zero_iff",
63
+ "start": [
64
+ 47,
65
+ 1
66
+ ],
67
+ "traced_tactics": [
68
+ {
69
+ "annotated_tactic": [
70
+ "\"induction' s using <a>Finset.induction_on<\\/a> with a s ha ih\"",
71
+ "[{\"full_name\":\"Finset.induction_on\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1278,19],\"def_end_pos\":[1278,31]}]"
72
+ ],
73
+ "state_after": "case empty\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 \u220f x \u2208 \u2205, f x = 0 \u2194 \u2203 a \u2208 \u2205, f a = 0\n\ncase insert\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns\u271d : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nih : \u220f x \u2208 s, f x = 0 \u2194 \u2203 a \u2208 s, f a = 0\n\u22a2 \u220f x \u2208 insert a s, f x = 0 \u2194 \u2203 a_1 \u2208 insert a s, f a_1 = 0",
74
+ "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 \u220f x \u2208 s, f x = 0 \u2194 \u2203 a \u2208 s, f a = 0",
75
+ "tactic": "induction' s using Finset.induction_on with a s ha ih"
76
+ },
77
+ {
78
+ "annotated_tactic": [
79
+ "\"exact \\u27e8<a>Not.elim<\\/a> <a>one_ne_zero<\\/a>, fun \\u27e8_, H, _\\u27e9 => by simp at H\\u27e9\"",
80
+ "[{\"full_name\":\"Not.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1333,5],\"def_end_pos\":[1333,13]},{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
81
+ ],
82
+ "state_after": "no goals",
83
+ "state_before": "case empty\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 \u220f x \u2208 \u2205, f x = 0 \u2194 \u2203 a \u2208 \u2205, f a = 0",
84
+ "tactic": "exact \u27e8Not.elim one_ne_zero, fun \u27e8_, H, _\u27e9 => by simp at H\u27e9"
85
+ },
86
+ {
87
+ "annotated_tactic": [
88
+ "\"simp at H\"",
89
+ "[]"
90
+ ],
91
+ "state_after": "no goals",
92
+ "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\nx\u271d : \u2203 a \u2208 \u2205, f a = 0\nw\u271d : \u03b9\nH : w\u271d \u2208 \u2205\nright\u271d : f w\u271d = 0\n\u22a2 \u220f x \u2208 \u2205, f x = 0",
93
+ "tactic": "simp at H"
94
+ },
95
+ {
96
+ "annotated_tactic": [
97
+ "\"rw [<a>prod_insert<\\/a> ha, <a>mul_eq_zero<\\/a>, <a>exists_mem_insert<\\/a>, ih]\"",
98
+ "[{\"full_name\":\"Finset.prod_insert\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[351,9],\"def_end_pos\":[351,20]},{\"full_name\":\"mul_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]},{\"full_name\":\"Finset.exists_mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[3028,9],\"def_end_pos\":[3028,26]}]"
99
+ ],
100
+ "state_after": "no goals",
101
+ "state_before": "case insert\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b3 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns\u271d : Finset \u03b9\ni : \u03b9\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nih : \u220f x \u2208 s, f x = 0 \u2194 \u2203 a \u2208 s, f a = 0\n\u22a2 \u220f x \u2208 insert a s, f x = 0 \u2194 \u2203 a_1 \u2208 insert a s, f a_1 = 0",
102
+ "tactic": "rw [prod_insert ha, mul_eq_zero, exists_mem_insert, ih]"
103
+ }
104
+ ],
105
+ "url": "https://github.com/leanprover-community/mathlib4"
106
+ },
107
+ "source_file": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean"
108
+ },
109
+ {
110
+ "error": "theorem has no selected tactic transitions",
111
+ "full_name": "Finset.support_prod_subset",
112
+ "id": "Finset.support_prod_subset",
113
+ "reason": "import_failure",
114
+ "selection": {
115
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
116
+ "end": [
117
+ 42,
118
+ 68
119
+ ],
120
+ "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean",
121
+ "full_name": "Finset.support_prod_subset",
122
+ "start": [
123
+ 40,
124
+ 1
125
+ ],
126
+ "traced_tactics": [],
127
+ "url": "https://github.com/leanprover-community/mathlib4"
128
+ },
129
+ "source_file": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean"
130
+ },
131
+ {
132
+ "error": "theorem has no selected tactic transitions",
133
+ "full_name": "Finset.prod_Ico_consecutive",
134
+ "id": "Finset.prod_Ico_consecutive",
135
+ "reason": "import_failure",
136
+ "selection": {
137
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
138
+ "end": [
139
+ 128,
140
+ 91
141
+ ],
142
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
143
+ "full_name": "Finset.prod_Ico_consecutive",
144
+ "start": [
145
+ 126,
146
+ 1
147
+ ],
148
+ "traced_tactics": [],
149
+ "url": "https://github.com/leanprover-community/mathlib4"
150
+ },
151
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
152
+ },
153
+ {
154
+ "error": "theorem has no selected tactic transitions",
155
+ "full_name": "Finset.prod_Ico_div_bot",
156
+ "id": "Finset.prod_Ico_div_bot",
157
+ "reason": "import_failure",
158
+ "selection": {
159
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
160
+ "end": [
161
+ 329,
162
+ 60
163
+ ],
164
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
165
+ "full_name": "Finset.prod_Ico_div_bot",
166
+ "start": [
167
+ 328,
168
+ 1
169
+ ],
170
+ "traced_tactics": [],
171
+ "url": "https://github.com/leanprover-community/mathlib4"
172
+ },
173
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
174
+ },
175
+ {
176
+ "error": "no internal trace for tactic 'rw [mul_comm]'",
177
+ "full_name": "Finset.prod_Ico_eq_mul_inv",
178
+ "id": "Finset.prod_Ico_eq_mul_inv",
179
+ "reason": "import_failure",
180
+ "selection": {
181
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
182
+ "end": [
183
+ 164,
184
+ 83
185
+ ],
186
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
187
+ "full_name": "Finset.prod_Ico_eq_mul_inv",
188
+ "start": [
189
+ 162,
190
+ 1
191
+ ],
192
+ "traced_tactics": [
193
+ {
194
+ "annotated_tactic": [
195
+ "\"(rw [<a>mul_comm<\\/a>]; exact <a>prod_range_mul_prod_Ico<\\/a> f h)\"",
196
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Finset.prod_range_mul_prod_Ico\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,32]}]"
197
+ ],
198
+ "state_after": "no goals",
199
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k \u2208 Ico m n, f k) * \u220f k \u2208 range m, f k = \u220f k \u2208 range n, f k",
200
+ "tactic": "(rw [mul_comm]; exact prod_range_mul_prod_Ico f h)"
201
+ },
202
+ {
203
+ "annotated_tactic": [
204
+ "\"rw [<a>mul_comm<\\/a>]\"",
205
+ "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
206
+ ],
207
+ "state_after": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k \u2208 range m, f k) * \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range n, f k",
208
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k \u2208 Ico m n, f k) * \u220f k \u2208 range m, f k = \u220f k \u2208 range n, f k",
209
+ "tactic": "rw [mul_comm]"
210
+ },
211
+ {
212
+ "annotated_tactic": [
213
+ "\"exact <a>prod_range_mul_prod_Ico<\\/a> f h\"",
214
+ "[{\"full_name\":\"Finset.prod_range_mul_prod_Ico\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,32]}]"
215
+ ],
216
+ "state_after": "no goals",
217
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k \u2208 range m, f k) * \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range n, f k",
218
+ "tactic": "exact prod_range_mul_prod_Ico f h"
219
+ }
220
+ ],
221
+ "url": "https://github.com/leanprover-community/mathlib4"
222
+ },
223
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
224
+ },
225
+ {
226
+ "error": "no internal trace for tactic 'intro i hi'",
227
+ "full_name": "Finset.prod_Ico_reflect",
228
+ "id": "Finset.prod_Ico_reflect",
229
+ "reason": "import_failure",
230
+ "selection": {
231
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
232
+ "end": [
233
+ 235,
234
+ 12
235
+ ],
236
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
237
+ "full_name": "Finset.prod_Ico_reflect",
238
+ "start": [
239
+ 220,
240
+ 1
241
+ ],
242
+ "traced_tactics": [
243
+ {
244
+ "annotated_tactic": [
245
+ "\"have : \\u2200 i < m, i \\u2264 n := by\\n intro i hi\\n exact (<a>add_le_add_iff_right<\\/a> 1).1 (<a>le_trans<\\/a> (<a>Nat.lt_iff_add_one_le<\\/a>.1 hi) h)\"",
246
+ "[{\"full_name\":\"add_le_add_iff_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[90,3],\"def_end_pos\":[90,14]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"Nat.lt_iff_add_one_le\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Defs.lean\",\"def_pos\":[152,7],\"def_end_pos\":[152,24]}]"
247
+ ],
248
+ "state_after": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
249
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
250
+ "tactic": "have : \u2200 i < m, i \u2264 n := by\n intro i hi\n exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)"
251
+ },
252
+ {
253
+ "annotated_tactic": [
254
+ "\"cases' <a>lt_or_le<\\/a> k m with hkm hkm\"",
255
+ "[{\"full_name\":\"lt_or_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,17]}]"
256
+ ],
257
+ "state_after": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j\n\ncase inr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
258
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
259
+ "tactic": "cases' lt_or_le k m with hkm hkm"
260
+ },
261
+ {
262
+ "annotated_tactic": [
263
+ "\"intro i hi\"",
264
+ "[]"
265
+ ],
266
+ "state_after": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\ni : \u2115\nhi : i < m\n\u22a2 i \u2264 n",
267
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\n\u22a2 \u2200 i < m, i \u2264 n",
268
+ "tactic": "intro i hi"
269
+ },
270
+ {
271
+ "annotated_tactic": [
272
+ "\"exact (<a>add_le_add_iff_right<\\/a> 1).1 (<a>le_trans<\\/a> (<a>Nat.lt_iff_add_one_le<\\/a>.1 hi) h)\"",
273
+ "[{\"full_name\":\"add_le_add_iff_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[90,3],\"def_end_pos\":[90,14]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"Nat.lt_iff_add_one_le\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Defs.lean\",\"def_pos\":[152,7],\"def_end_pos\":[152,24]}]"
274
+ ],
275
+ "state_after": "no goals",
276
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\ni : \u2115\nhi : i < m\n\u22a2 i \u2264 n",
277
+ "tactic": "exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)"
278
+ },
279
+ {
280
+ "annotated_tactic": [
281
+ "\"rw [\\u2190 <a>Nat.Ico_image_const_sub_eq_Ico<\\/a> (this _ hkm)]\"",
282
+ "[{\"full_name\":\"Nat.Ico_image_const_sub_eq_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Nat.lean\",\"def_pos\":[205,9],\"def_end_pos\":[205,35]}]"
283
+ ],
284
+ "state_after": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 image (fun x => n - x) (Ico k m), f j",
285
+ "state_before": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
286
+ "tactic": "rw [\u2190 Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]"
287
+ },
288
+ {
289
+ "annotated_tactic": [
290
+ "\"refine (<a>prod_image<\\/a> ?_).<a>symm<\\/a>\"",
291
+ "[{\"full_name\":\"Finset.prod_image\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[401,9],\"def_end_pos\":[401,19]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
292
+ ],
293
+ "state_after": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u2200 x \u2208 Ico k m, \u2200 y \u2208 Ico k m, n - x = n - y \u2192 x = y",
294
+ "state_before": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 image (fun x => n - x) (Ico k m), f j",
295
+ "tactic": "refine (prod_image ?_).symm"
296
+ },
297
+ {
298
+ "annotated_tactic": [
299
+ "\"simp only [<a>mem_Ico<\\/a>]\"",
300
+ "[{\"full_name\":\"Finset.mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]}]"
301
+ ],
302
+ "state_after": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u2200 (x : \u2115), k \u2264 x \u2227 x < m \u2192 \u2200 (y : \u2115), k \u2264 y \u2227 y < m \u2192 n - x = n - y \u2192 x = y",
303
+ "state_before": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u2200 x \u2208 Ico k m, \u2200 y \u2208 Ico k m, n - x = n - y \u2192 x = y",
304
+ "tactic": "simp only [mem_Ico]"
305
+ },
306
+ {
307
+ "annotated_tactic": [
308
+ "\"rintro i \\u27e8_, im\\u27e9 j \\u27e8_, jm\\u27e9 Hij\"",
309
+ "[]"
310
+ ],
311
+ "state_after": "case inl.intro.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\ni : \u2115\nleft\u271d\u00b9 : k \u2264 i\nim : i < m\nj : \u2115\nleft\u271d : k \u2264 j\njm : j < m\nHij : n - i = n - j\n\u22a2 i = j",
312
+ "state_before": "case inl\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\n\u22a2 \u2200 (x : \u2115), k \u2264 x \u2227 x < m \u2192 \u2200 (y : \u2115), k \u2264 y \u2227 y < m \u2192 n - x = n - y \u2192 x = y",
313
+ "tactic": "rintro i \u27e8_, im\u27e9 j \u27e8_, jm\u27e9 Hij"
314
+ },
315
+ {
316
+ "annotated_tactic": [
317
+ "\"rw [\\u2190 <a>tsub_tsub_cancel_of_le<\\/a> (this _ im), Hij, <a>tsub_tsub_cancel_of_le<\\/a> (this _ jm)]\"",
318
+ "[{\"full_name\":\"tsub_tsub_cancel_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[296,9],\"def_end_pos\":[296,31]},{\"full_name\":\"tsub_tsub_cancel_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[296,9],\"def_end_pos\":[296,31]}]"
319
+ ],
320
+ "state_after": "no goals",
321
+ "state_before": "case inl.intro.intro\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : k < m\ni : \u2115\nleft\u271d\u00b9 : k \u2264 i\nim : i < m\nj : \u2115\nleft\u271d : k \u2264 j\njm : j < m\nHij : n - i = n - j\n\u22a2 i = j",
322
+ "tactic": "rw [\u2190 tsub_tsub_cancel_of_le (this _ im), Hij, tsub_tsub_cancel_of_le (this _ jm)]"
323
+ },
324
+ {
325
+ "annotated_tactic": [
326
+ "\"have : n + 1 - k \\u2264 n + 1 - m := by\\n rw [<a>tsub_le_tsub_iff_left<\\/a> h]\\n exact hkm\"",
327
+ "[{\"full_name\":\"tsub_le_tsub_iff_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[395,9],\"def_end_pos\":[395,30]}]"
328
+ ],
329
+ "state_after": "case inr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis\u271d : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\nthis : n + 1 - k \u2264 n + 1 - m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
330
+ "state_before": "case inr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
331
+ "tactic": "have : n + 1 - k \u2264 n + 1 - m := by\n rw [tsub_le_tsub_iff_left h]\n exact hkm"
332
+ },
333
+ {
334
+ "annotated_tactic": [
335
+ "\"simp only [<a>ge_iff_le<\\/a>, hkm, <a>Ico_eq_empty_of_le<\\/a>, <a>prod_empty<\\/a>, <a>tsub_le_iff_right<\\/a>, <a>Ico_eq_empty_of_le<\\/a>\\n this]\"",
336
+ "[{\"full_name\":\"ge_iff_le\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1947,17],\"def_end_pos\":[1947,26]},{\"full_name\":\"Finset.Ico_eq_empty_of_le\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,27]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"tsub_le_iff_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Defs.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,26]},{\"full_name\":\"Finset.Ico_eq_empty_of_le\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,27]}]"
337
+ ],
338
+ "state_after": "no goals",
339
+ "state_before": "case inr\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis\u271d : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\nthis : n + 1 - k \u2264 n + 1 - m\n\u22a2 \u220f j \u2208 Ico k m, f (n - j) = \u220f j \u2208 Ico (n + 1 - m) (n + 1 - k), f j",
340
+ "tactic": "simp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le\n this]"
341
+ },
342
+ {
343
+ "annotated_tactic": [
344
+ "\"rw [<a>tsub_le_tsub_iff_left<\\/a> h]\"",
345
+ "[{\"full_name\":\"tsub_le_tsub_iff_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[395,9],\"def_end_pos\":[395,30]}]"
346
+ ],
347
+ "state_after": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\n\u22a2 m \u2264 k",
348
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\n\u22a2 n + 1 - k \u2264 n + 1 - m",
349
+ "tactic": "rw [tsub_le_tsub_iff_left h]"
350
+ },
351
+ {
352
+ "annotated_tactic": [
353
+ "\"exact hkm\"",
354
+ "[]"
355
+ ],
356
+ "state_after": "no goals",
357
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 i < m, i \u2264 n\nhkm : m \u2264 k\n\u22a2 m \u2264 k",
358
+ "tactic": "exact hkm"
359
+ }
360
+ ],
361
+ "url": "https://github.com/leanprover-community/mathlib4"
362
+ },
363
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
364
+ },
365
+ {
366
+ "error": "theorem has no selected tactic transitions",
367
+ "full_name": "Finset.prod_Ico_succ_div_top",
368
+ "id": "Finset.prod_Ico_succ_div_top",
369
+ "reason": "import_failure",
370
+ "selection": {
371
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
372
+ "end": [
373
+ 336,
374
+ 51
375
+ ],
376
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
377
+ "full_name": "Finset.prod_Ico_succ_div_top",
378
+ "start": [
379
+ 334,
380
+ 1
381
+ ],
382
+ "traced_tactics": [],
383
+ "url": "https://github.com/leanprover-community/mathlib4"
384
+ },
385
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
386
+ },
387
+ {
388
+ "error": "no internal trace for tactic 'simp only [\u2190 Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]'",
389
+ "full_name": "Finset.prod_range_reflect",
390
+ "id": "Finset.prod_range_reflect",
391
+ "reason": "import_failure",
392
+ "selection": {
393
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
394
+ "end": [
395
+ 249,
396
+ 9
397
+ ],
398
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
399
+ "full_name": "Finset.prod_range_reflect",
400
+ "start": [
401
+ 243,
402
+ 1
403
+ ],
404
+ "traced_tactics": [
405
+ {
406
+ "annotated_tactic": [
407
+ "\"cases n\"",
408
+ "[]"
409
+ ],
410
+ "state_after": "case zero\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\n\u22a2 \u220f j \u2208 range 0, f (0 - 1 - j) = \u220f j \u2208 range 0, f j\n\ncase succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f j \u2208 range (n\u271d + 1), f (n\u271d + 1 - 1 - j) = \u220f j \u2208 range (n\u271d + 1), f j",
411
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f j \u2208 range n, f (n - 1 - j) = \u220f j \u2208 range n, f j",
412
+ "tactic": "cases n"
413
+ },
414
+ {
415
+ "annotated_tactic": [
416
+ "\"simp\"",
417
+ "[]"
418
+ ],
419
+ "state_after": "no goals",
420
+ "state_before": "case zero\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\n\u22a2 \u220f j \u2208 range 0, f (0 - 1 - j) = \u220f j \u2208 range 0, f j",
421
+ "tactic": "simp"
422
+ },
423
+ {
424
+ "annotated_tactic": [
425
+ "\"simp only [\\u2190 <a>Nat.Ico_zero_eq_range<\\/a>, <a>Nat.succ_sub_succ_eq_sub<\\/a>, <a>tsub_zero<\\/a>]\"",
426
+ "[{\"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.succ_sub_succ_eq_sub\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[276,9],\"def_end_pos\":[276,29]},{\"full_name\":\"tsub_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Defs.lean\",\"def_pos\":[448,9],\"def_end_pos\":[448,18]}]"
427
+ ],
428
+ "state_after": "case succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f x \u2208 Ico 0 (n\u271d + 1), f (n\u271d - x) = \u220f x \u2208 Ico 0 (n\u271d + 1), f x",
429
+ "state_before": "case succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f j \u2208 range (n\u271d + 1), f (n\u271d + 1 - 1 - j) = \u220f j \u2208 range (n\u271d + 1), f j",
430
+ "tactic": "simp only [\u2190 Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]"
431
+ },
432
+ {
433
+ "annotated_tactic": [
434
+ "\"rw [<a>prod_Ico_reflect<\\/a> _ _ <a>le_rfl<\\/a>]\"",
435
+ "[{\"full_name\":\"Finset.prod_Ico_reflect\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,25]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
436
+ ],
437
+ "state_after": "case succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f j \u2208 Ico (n\u271d + 1 - (n\u271d + 1)) (n\u271d + 1 - 0), f j = \u220f x \u2208 Ico 0 (n\u271d + 1), f x",
438
+ "state_before": "case succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f x \u2208 Ico 0 (n\u271d + 1), f (n\u271d - x) = \u220f x \u2208 Ico 0 (n\u271d + 1), f x",
439
+ "tactic": "rw [prod_Ico_reflect _ _ le_rfl]"
440
+ },
441
+ {
442
+ "annotated_tactic": [
443
+ "\"simp\"",
444
+ "[]"
445
+ ],
446
+ "state_after": "no goals",
447
+ "state_before": "case succ\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nn\u271d : \u2115\n\u22a2 \u220f j \u2208 Ico (n\u271d + 1 - (n\u271d + 1)) (n\u271d + 1 - 0), f j = \u220f x \u2208 Ico 0 (n\u271d + 1), f x",
448
+ "tactic": "simp"
449
+ }
450
+ ],
451
+ "url": "https://github.com/leanprover-community/mathlib4"
452
+ },
453
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
454
+ },
455
+ {
456
+ "error": "theorem has no selected tactic transitions",
457
+ "full_name": "Finset.prod_range_succ_div_prod",
458
+ "id": "Finset.prod_range_succ_div_prod",
459
+ "reason": "import_failure",
460
+ "selection": {
461
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
462
+ "end": [
463
+ 317,
464
+ 48
465
+ ],
466
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
467
+ "full_name": "Finset.prod_range_succ_div_prod",
468
+ "start": [
469
+ 316,
470
+ 1
471
+ ],
472
+ "traced_tactics": [],
473
+ "url": "https://github.com/leanprover-community/mathlib4"
474
+ },
475
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
476
+ },
477
+ {
478
+ "error": "theorem has no selected tactic transitions",
479
+ "full_name": "Finset.prod_range_succ_div_top",
480
+ "id": "Finset.prod_range_succ_div_top",
481
+ "reason": "import_failure",
482
+ "selection": {
483
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
484
+ "end": [
485
+ 323,
486
+ 47
487
+ ],
488
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
489
+ "full_name": "Finset.prod_range_succ_div_top",
490
+ "start": [
491
+ 322,
492
+ 1
493
+ ],
494
+ "traced_tactics": [],
495
+ "url": "https://github.com/leanprover-community/mathlib4"
496
+ },
497
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
498
+ },
499
+ {
500
+ "error": "theorem has no selected tactic transitions",
501
+ "full_name": "Finset.sum_Ico_reflect",
502
+ "id": "Finset.sum_Ico_reflect",
503
+ "reason": "import_failure",
504
+ "selection": {
505
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
506
+ "end": [
507
+ 240,
508
+ 51
509
+ ],
510
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
511
+ "full_name": "Finset.sum_Ico_reflect",
512
+ "start": [
513
+ 238,
514
+ 1
515
+ ],
516
+ "traced_tactics": [],
517
+ "url": "https://github.com/leanprover-community/mathlib4"
518
+ },
519
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
520
+ },
521
+ {
522
+ "error": "theorem has no selected tactic transitions",
523
+ "full_name": "Finset.sum_range_reflect",
524
+ "id": "Finset.sum_range_reflect",
525
+ "reason": "import_failure",
526
+ "selection": {
527
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
528
+ "end": [
529
+ 254,
530
+ 47
531
+ ],
532
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
533
+ "full_name": "Finset.sum_range_reflect",
534
+ "start": [
535
+ 252,
536
+ 1
537
+ ],
538
+ "traced_tactics": [],
539
+ "url": "https://github.com/leanprover-community/mathlib4"
540
+ },
541
+ "source_file": "Mathlib/Algebra/BigOperators/Intervals.lean"
542
+ }
543
+ ],
544
+ "input_proofs": 41,
545
+ "input_units": 2,
546
+ "proofs": 28,
547
+ "published_proofs": 25,
548
+ "published_transitions": 43,
549
+ "replay_failures": [],
550
+ "transitions": 57,
551
+ "validation_failures": [
552
+ {
553
+ "errors": [
554
+ "internal transition boundary mismatch",
555
+ "internal transition boundary mismatch"
556
+ ],
557
+ "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean",
558
+ "proof_id": "mathlib:Finset.prod_boole",
559
+ "provenance": {
560
+ "selection": {
561
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
562
+ "end": [
563
+ 37,
564
+ 19
565
+ ],
566
+ "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean",
567
+ "full_name": "Finset.prod_boole",
568
+ "start": [
569
+ 29,
570
+ 1
571
+ ],
572
+ "traced_tactics": [
573
+ {
574
+ "annotated_tactic": [
575
+ "\"split_ifs with h\"",
576
+ "[]"
577
+ ],
578
+ "state_after": "case pos\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2200 i \u2208 s, p i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 1\n\ncase neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u00ac\u2200 i \u2208 s, p i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
579
+ "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = if \u2200 i \u2208 s, p i then 1 else 0",
580
+ "tactic": "split_ifs with h"
581
+ },
582
+ {
583
+ "annotated_tactic": [
584
+ "\"apply <a>prod_eq_one<\\/a>\"",
585
+ "[{\"full_name\":\"Finset.prod_eq_one\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[427,9],\"def_end_pos\":[427,20]}]"
586
+ ],
587
+ "state_after": "case pos.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2200 i \u2208 s, p i\n\u22a2 \u2200 x \u2208 s, (if p x then 1 else 0) = 1",
588
+ "state_before": "case pos\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2200 i \u2208 s, p i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 1",
589
+ "tactic": "apply prod_eq_one"
590
+ },
591
+ {
592
+ "annotated_tactic": [
593
+ "\"intro i hi\"",
594
+ "[]"
595
+ ],
596
+ "state_after": "case pos.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d : \u03b9\nh : \u2200 i \u2208 s, p i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (if p i then 1 else 0) = 1",
597
+ "state_before": "case pos.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2200 i \u2208 s, p i\n\u22a2 \u2200 x \u2208 s, (if p x then 1 else 0) = 1",
598
+ "tactic": "intro i hi"
599
+ },
600
+ {
601
+ "annotated_tactic": [
602
+ "\"rw [<a>if_pos<\\/a> (h i hi)]\"",
603
+ "[{\"full_name\":\"if_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[932,9],\"def_end_pos\":[932,15]}]"
604
+ ],
605
+ "state_after": "no goals",
606
+ "state_before": "case pos.h\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d : \u03b9\nh : \u2200 i \u2208 s, p i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (if p i then 1 else 0) = 1",
607
+ "tactic": "rw [if_pos (h i hi)]"
608
+ },
609
+ {
610
+ "annotated_tactic": [
611
+ "\"push_neg at h\"",
612
+ "[]"
613
+ ],
614
+ "state_after": "case neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2203 i \u2208 s, \u00acp i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
615
+ "state_before": "case neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u00ac\u2200 i \u2208 s, p i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
616
+ "tactic": "push_neg at h"
617
+ },
618
+ {
619
+ "annotated_tactic": [
620
+ "\"rcases h with \\u27e8i, hi, hq\\u27e9\"",
621
+ "[]"
622
+ ],
623
+ "state_after": "case neg.intro.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d i : \u03b9\nhi : i \u2208 s\nhq : \u00acp i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
624
+ "state_before": "case neg\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni : \u03b9\nh : \u2203 i \u2208 s, \u00acp i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
625
+ "tactic": "rcases h with \u27e8i, hi, hq\u27e9"
626
+ },
627
+ {
628
+ "annotated_tactic": [
629
+ "\"apply <a>prod_eq_zero<\\/a> hi\"",
630
+ "[{\"full_name\":\"Finset.prod_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/GroupWithZero\\/Finset.lean\",\"def_pos\":[25,7],\"def_end_pos\":[25,19]}]"
631
+ ],
632
+ "state_after": "case neg.intro.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d i : \u03b9\nhi : i \u2208 s\nhq : \u00acp i\n\u22a2 (if p i then 1 else 0) = 0",
633
+ "state_before": "case neg.intro.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d i : \u03b9\nhi : i \u2208 s\nhq : \u00acp i\n\u22a2 (\u220f i \u2208 s, if p i then 1 else 0) = 0",
634
+ "tactic": "apply prod_eq_zero hi"
635
+ },
636
+ {
637
+ "annotated_tactic": [
638
+ "\"rw [<a>if_neg<\\/a> hq]\"",
639
+ "[{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]}]"
640
+ ],
641
+ "state_after": "no goals",
642
+ "state_before": "case neg.intro.intro\n\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d\u00b9 : CommMonoidWithZero M\u2080\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b9 \u2192 M\u2080\ns : Finset \u03b9\ni\u271d i : \u03b9\nhi : i \u2208 s\nhq : \u00acp i\n\u22a2 (if p i then 1 else 0) = 0",
643
+ "tactic": "rw [if_neg hq]"
644
+ }
645
+ ],
646
+ "url": "https://github.com/leanprover-community/mathlib4"
647
+ }
648
+ },
649
+ "theorem_name": "Finset.prod_boole"
650
+ },
651
+ {
652
+ "errors": [
653
+ "internal transition boundary mismatch",
654
+ "internal transition boundary mismatch"
655
+ ],
656
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
657
+ "proof_id": "mathlib:Finset.prod_Ico_eq_prod_range",
658
+ "provenance": {
659
+ "selection": {
660
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
662
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665
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
666
+ "full_name": "Finset.prod_Ico_eq_prod_range",
667
+ "start": [
668
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670
+ ],
671
+ "traced_tactics": [
672
+ {
673
+ "annotated_tactic": [
674
+ "\"by_cases h : m \\u2264 n\"",
675
+ "[]"
676
+ ],
677
+ "state_after": "case pos\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)\n\ncase neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : \u00acm \u2264 n\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
678
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
679
+ "tactic": "by_cases h : m \u2264 n"
680
+ },
681
+ {
682
+ "annotated_tactic": [
683
+ "\"rw [\\u2190 <a>Nat.Ico_zero_eq_range<\\/a>, <a>prod_Ico_add<\\/a>, <a>zero_add<\\/a>, <a>tsub_add_cancel_of_le<\\/a> h]\"",
684
+ "[{\"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\":\"Finset.prod_Ico_add\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,21]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"tsub_add_cancel_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[31,9],\"def_end_pos\":[31,30]}]"
685
+ ],
686
+ "state_after": "no goals",
687
+ "state_before": "case pos\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
688
+ "tactic": "rw [\u2190 Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]"
689
+ },
690
+ {
691
+ "annotated_tactic": [
692
+ "\"replace h : n \\u2264 m := <a>le_of_not_ge<\\/a> h\"",
693
+ "[{\"full_name\":\"le_of_not_ge\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[301,9],\"def_end_pos\":[301,21]}]"
694
+ ],
695
+ "state_after": "case neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
696
+ "state_before": "case neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : \u00acm \u2264 n\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
697
+ "tactic": "replace h : n \u2264 m := le_of_not_ge h"
698
+ },
699
+ {
700
+ "annotated_tactic": [
701
+ "\"rw [<a>Ico_eq_empty_of_le<\\/a> h, tsub_eq_zero_iff_le.mpr h, <a>range_zero<\\/a>, <a>prod_empty<\\/a>, <a>prod_empty<\\/a>]\"",
702
+ "[{\"full_name\":\"Finset.Ico_eq_empty_of_le\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,27]},{\"full_name\":\"Finset.range_zero\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2945,9],\"def_end_pos\":[2945,19]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]}]"
703
+ ],
704
+ "state_after": "no goals",
705
+ "state_before": "case neg\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 M\nf : \u2115 \u2192 M\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u220f k \u2208 Ico m n, f k = \u220f k \u2208 range (n - m), f (m + k)",
706
+ "tactic": "rw [Ico_eq_empty_of_le h, tsub_eq_zero_iff_le.mpr h, range_zero, prod_empty, prod_empty]"
707
+ }
708
+ ],
709
+ "url": "https://github.com/leanprover-community/mathlib4"
710
+ }
711
+ },
712
+ "theorem_name": "Finset.prod_Ico_eq_prod_range"
713
+ },
714
+ {
715
+ "errors": [
716
+ "internal transition boundary mismatch"
717
+ ],
718
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
719
+ "proof_id": "mathlib:Finset.sum_range_id_mul_two",
720
+ "provenance": {
721
+ "selection": {
722
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
723
+ "end": [
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+ 281,
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+ 71
726
+ ],
727
+ "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean",
728
+ "full_name": "Finset.sum_range_id_mul_two",
729
+ "start": [
730
+ 274,
731
+ 1
732
+ ],
733
+ "traced_tactics": [
734
+ {
735
+ "annotated_tactic": [
736
+ "\"rw [<a>sum_range_reflect<\\/a> (fun i => i) n, <a>mul_two<\\/a>]\"",
737
+ "[{\"full_name\":\"Finset.sum_range_reflect\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[252,9],\"def_end_pos\":[252,26]},{\"full_name\":\"mul_two\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,16]}]"
738
+ ],
739
+ "state_after": "no goals",
740
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 M\nn : \u2115\n\u22a2 (\u2211 i \u2208 range n, i) * 2 = \u2211 i \u2208 range n, i + \u2211 i \u2208 range n, (n - 1 - i)",
741
+ "tactic": "rw [sum_range_reflect (fun i => i) n, mul_two]"
742
+ },
743
+ {
744
+ "annotated_tactic": [
745
+ "\"rw [<a>sum_const<\\/a>, <a>card_range<\\/a>, <a>Nat.nsmul_eq_mul<\\/a>]\"",
746
+ "[{\"full_name\":\"Finset.sum_const\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1736,3],\"def_end_pos\":[1736,14]},{\"full_name\":\"Finset.card_range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[206,9],\"def_end_pos\":[206,19]},{\"full_name\":\"Nat.nsmul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Nat.lean\",\"def_pos\":[69,40],\"def_end_pos\":[69,52]}]"
747
+ ],
748
+ "state_after": "no goals",
749
+ "state_before": "\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 M\nn : \u2115\n\u22a2 \u2211 i \u2208 range n, (n - 1) = n * (n - 1)",
750
+ "tactic": "rw [sum_const, card_range, Nat.nsmul_eq_mul]"
751
+ }
752
+ ],
753
+ "url": "https://github.com/leanprover-community/mathlib4"
754
+ }
755
+ },
756
+ "theorem_name": "Finset.sum_range_id_mul_two"
757
+ }
758
+ ],
759
+ "validation_rejected_proofs": 3,
760
+ "zero_transition_proofs": 0
761
+ },
762
+ "shards": [
763
+ {
764
+ "path": "data/train/mathlib-000042-0000-876a6a5d84e394b3.parquet",
765
+ "sha256": "876a6a5d84e394b35165dc0c761cd05908e26f314e6ae84f6da1318c28dc3d8c",
766
+ "bytes": 158906,
767
+ "rows": 35,
768
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
769
+ },
770
+ {
771
+ "path": "data/dev/mathlib-000042-0000-58a1f25c7ce0f974.parquet",
772
+ "sha256": "58a1f25c7ce0f974906234d896076246bafe300b80b9fa2c976f1758eb81bde4",
773
+ "bytes": 41208,
774
+ "rows": 2,
775
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
776
+ },
777
+ {
778
+ "path": "data/internal_test/mathlib-000042-0000-3217ef06fd60ce61.parquet",
779
+ "sha256": "3217ef06fd60ce61be91c8491a682d1a467641839920eabb28dffc39e0df734f",
780
+ "bytes": 57967,
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+ "rows": 6,
782
+ "revision": "61f21f76cbf338a5d96f72def0b41e8b4955531a"
783
+ }
784
+ ],
785
+ "proof_ids": [
786
+ "mathlib:Finset.mul_prod_Ico_eq_prod_Icc",
787
+ "mathlib:Finset.mul_prod_Iio_eq_prod_Iic",
788
+ "mathlib:Finset.mul_prod_Ioc_eq_prod_Icc",
789
+ "mathlib:Finset.mul_prod_Ioi_eq_prod_Ici",
790
+ "mathlib:Finset.prod_Icc_succ_top",
791
+ "mathlib:Finset.prod_Ico_add",
792
+ "mathlib:Finset.prod_Ico_add'",
793
+ "mathlib:Finset.prod_Ico_eq_div",
794
+ "mathlib:Finset.prod_Ico_id_eq_factorial",
795
+ "mathlib:Finset.prod_Ico_mul_eq_prod_Icc",
796
+ "mathlib:Finset.prod_Ico_succ_top",
797
+ "mathlib:Finset.prod_Iio_mul_eq_prod_Iic",
798
+ "mathlib:Finset.prod_Ioc_consecutive",
799
+ "mathlib:Finset.prod_Ioc_mul_eq_prod_Icc",
800
+ "mathlib:Finset.prod_Ioc_succ_top",
801
+ "mathlib:Finset.prod_Ioi_mul_eq_prod_Ici",
802
+ "mathlib:Finset.prod_ne_zero_iff",
803
+ "mathlib:Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag",
804
+ "mathlib:Finset.prod_range_add_one_eq_factorial",
805
+ "mathlib:Finset.prod_range_diag_flip",
806
+ "mathlib:Finset.prod_range_div_prod_range",
807
+ "mathlib:Finset.sum_Ico_Ico_comm",
808
+ "mathlib:Finset.sum_Ico_Ico_comm'",
809
+ "mathlib:Finset.sum_range_id",
810
+ "mathlib:Finset.support_prod"
811
+ ],
812
+ "proof_keys": [
813
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean\",null,null,\"Finset.prod_ne_zero_iff\"]",
814
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean\",null,null,\"Finset.support_prod\"]",
815
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.mul_prod_Ico_eq_prod_Icc\"]",
816
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.mul_prod_Iio_eq_prod_Iic\"]",
817
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.mul_prod_Ioc_eq_prod_Icc\"]",
818
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.mul_prod_Ioi_eq_prod_Ici\"]",
819
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Icc_succ_top\"]",
820
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_add\"]",
821
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_add'\"]",
822
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_eq_div\"]",
823
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_id_eq_factorial\"]",
824
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_mul_eq_prod_Icc\"]",
825
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ico_succ_top\"]",
826
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Iio_mul_eq_prod_Iic\"]",
827
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ioc_consecutive\"]",
828
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ioc_mul_eq_prod_Icc\"]",
829
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ioc_succ_top\"]",
830
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_Ioi_mul_eq_prod_Ici\"]",
831
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_prod_Ioi_mul_eq_prod_prod_off_diag\"]",
832
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_range_add_one_eq_factorial\"]",
833
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_range_diag_flip\"]",
834
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.prod_range_div_prod_range\"]",
835
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.sum_Ico_Ico_comm\"]",
836
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.sum_Ico_Ico_comm'\"]",
837
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/BigOperators/Intervals.lean\",null,null,\"Finset.sum_range_id\"]"
838
+ ]
839
+ }
manifests/batches/mathlib/000045.json ADDED
The diff for this file is too large to render. See raw diff