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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
22d62b6
·
verified ·
1 Parent(s): 2034373

Record 11 verified HPC batch manifests

Browse files
Files changed (11) hide show
  1. manifests/batches/deepseek/000045.json +148 -0
  2. manifests/batches/deepseek/000047.json +148 -0
  3. manifests/batches/deepseek/000048.json +148 -0
  4. manifests/batches/deepseek/000050.json +148 -0
  5. manifests/batches/mathlib/000030.json +0 -0
  6. manifests/batches/mathlib/000031.json +0 -0
  7. manifests/batches/mathlib/000032.json +0 -0
  8. manifests/batches/mathlib/000033.json +937 -0
  9. manifests/batches/mathlib/000034.json +471 -0
  10. manifests/batches/mathlib/000035.json +366 -0
  11. manifests/batches/mathlib/000036.json +1259 -0
manifests/batches/deepseek/000045.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000045",
4
+ "batch_fingerprint": "06d04991b86c0bde8f87c8e32cb504d025e308e66a004edd4e742d88949b8b49",
5
+ "evidence": {
6
+ "batch_fingerprint": "06d04991b86c0bde8f87c8e32cb504d025e308e66a004edd4e742d88949b8b49",
7
+ "contamination_matches": [],
8
+ "failures": [],
9
+ "input_proofs": 50,
10
+ "input_units": 50,
11
+ "proofs": 50,
12
+ "published_proofs": 50,
13
+ "published_transitions": 111,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 111,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
20
+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000045-0000-9f28fc18060d7fee.parquet",
24
+ "sha256": "9f28fc18060d7fee73a898f954188f84d22207e25f2cdbce815e41ca73a851cd",
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+ "bytes": 314861,
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+ "rows": 88,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
28
+ },
29
+ {
30
+ "path": "data/dev/deepseek-000045-0000-906b6f596ed0e9b0.parquet",
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+ "sha256": "906b6f596ed0e9b04df2b6eddd827cedf81a450a8511c4ca3a6b779d9e95be02",
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+ "bytes": 40614,
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+ "rows": 7,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
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+ },
36
+ {
37
+ "path": "data/internal_test/deepseek-000045-0000-05e11af6e8d07887.parquet",
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+ "sha256": "05e11af6e8d0788792cbc8a117c26cf193e6a1010e0ee5dbe3f8eef124c551f2",
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+ "bytes": 59713,
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+ "rows": 16,
41
+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
42
+ }
43
+ ],
44
+ "proof_ids": [
45
+ "deepseek:thm_2266",
46
+ "deepseek:thm_2267",
47
+ "deepseek:thm_2268",
48
+ "deepseek:thm_2269",
49
+ "deepseek:thm_2270",
50
+ "deepseek:thm_2271",
51
+ "deepseek:thm_2272",
52
+ "deepseek:thm_2273",
53
+ "deepseek:thm_2274",
54
+ "deepseek:thm_2275",
55
+ "deepseek:thm_2276",
56
+ "deepseek:thm_2277",
57
+ "deepseek:thm_2278",
58
+ "deepseek:thm_2279",
59
+ "deepseek:thm_2280",
60
+ "deepseek:thm_2281",
61
+ "deepseek:thm_2282",
62
+ "deepseek:thm_2283",
63
+ "deepseek:thm_2284",
64
+ "deepseek:thm_2285",
65
+ "deepseek:thm_2286",
66
+ "deepseek:thm_2287",
67
+ "deepseek:thm_2288",
68
+ "deepseek:thm_2289",
69
+ "deepseek:thm_2290",
70
+ "deepseek:thm_2291",
71
+ "deepseek:thm_2292",
72
+ "deepseek:thm_2293",
73
+ "deepseek:thm_2294",
74
+ "deepseek:thm_2295",
75
+ "deepseek:thm_2296",
76
+ "deepseek:thm_2297",
77
+ "deepseek:thm_2298",
78
+ "deepseek:thm_2299",
79
+ "deepseek:thm_2300",
80
+ "deepseek:thm_2301",
81
+ "deepseek:thm_2302",
82
+ "deepseek:thm_2303",
83
+ "deepseek:thm_2304",
84
+ "deepseek:thm_2305",
85
+ "deepseek:thm_2306",
86
+ "deepseek:thm_2307",
87
+ "deepseek:thm_2308",
88
+ "deepseek:thm_2309",
89
+ "deepseek:thm_2310",
90
+ "deepseek:thm_2311",
91
+ "deepseek:thm_2312",
92
+ "deepseek:thm_2313",
93
+ "deepseek:thm_2314",
94
+ "deepseek:thm_2315"
95
+ ],
96
+ "proof_keys": [
97
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002250.lean\",\"ae8deefebc1f882427426cca9fab12da86cb0e2e0fc120e8f97a6911679f82a1\",0,\"thm_2266\"]",
98
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002251.lean\",\"048b2a3a3baaf65d3d9b555c3749e4d5be372cddf7156b8dd69a8c6418ea3ff3\",0,\"thm_2267\"]",
99
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002252.lean\",\"ff95ab54c422e293604175d7b6b636a625a8fcc70d2ef6b60395e91b48b92b76\",0,\"thm_2268\"]",
100
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002253.lean\",\"468440dd5c0eb5ba319e26c8821d3b5977eb0ec9f97329fc5ce9b846d4f0c303\",0,\"thm_2269\"]",
101
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002254.lean\",\"fb09c1ee54178022abc2abf2019c5e71f44de9bee82862e8b30a5cad62ef03fe\",0,\"thm_2270\"]",
102
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002255.lean\",\"b3a09dfa5ffe0c9b1ad32b3fec70f82d7a1cf01bf4710b9a0beeab3aeb3faac9\",0,\"thm_2271\"]",
103
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002256.lean\",\"8bcff9647839b7e255754524e02fc2701e92441ae21dabd1d027ea616dd2b717\",0,\"thm_2272\"]",
104
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002257.lean\",\"0ad13acd141196e6dfb4e96a5cbf7cdb84707d177bd3db6e5938b1da5dfeb026\",0,\"thm_2273\"]",
105
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002258.lean\",\"e27005b702bc706e1065919086c1c3303b22201a491c6fb7e0c01e1135cbc12f\",0,\"thm_2274\"]",
106
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002259.lean\",\"e4fa9eea5f20bf205dd1c741afc755a87d551699117d4ae39850be2fe90fb22d\",0,\"thm_2275\"]",
107
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002260.lean\",\"6eab420182a09c09ef28642a4d8fc5b09ca6a57be38870efb8ecf6867b092405\",0,\"thm_2276\"]",
108
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002261.lean\",\"b4a0a983c14b9fb8e91e11e2b1f83c39bf9949fc63c6e4d1ccf5b1d066480246\",0,\"thm_2277\"]",
109
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002262.lean\",\"98b1ce26d73082659789849fe6b0d09b5b2dc37038599e1caf1d744cd1f9d1cf\",0,\"thm_2278\"]",
110
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002263.lean\",\"abd86ea9e9c6e68e1d3b4032e0b8bfb1239d55d468451bc7f4f1622b1410da95\",0,\"thm_2279\"]",
111
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002264.lean\",\"80b99d33d0966059dbde38bd85d5eca7f6a1e87f833457fc783cbb3968c06709\",0,\"thm_2280\"]",
112
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002265.lean\",\"132f3bd7dda929ebe6c18ac6793d1ca2d7a4a8d3b0bfe69c9e1dde452291a7d1\",0,\"thm_2281\"]",
113
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002266.lean\",\"b07ceff3197671749238c7d70d86f1984a96d89750a728c6ba7c91a0edb6dd87\",0,\"thm_2282\"]",
114
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002267.lean\",\"9c060d8d4bfa0c434a8d35b9fa648a998e23e0472e9c32be6250e19172de7c29\",0,\"thm_2283\"]",
115
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002268.lean\",\"a252fb8a822d84351b4247e74f1027f6b5c43f766388988c37607661d56647d2\",0,\"thm_2284\"]",
116
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002269.lean\",\"862f6e5704fadf0cfb5a9c3c3aa0be0f97602a6aaec53fa8c09865df902f4b30\",0,\"thm_2285\"]",
117
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002270.lean\",\"9bbddc5a3f89e4562d87df69c675b581c09f3ae898ff1adafa4b8d27a5bfd54a\",0,\"thm_2286\"]",
118
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002271.lean\",\"eaca7d5ecb7f96aad50069c71d683e414723cf56ee8b6d73db8db8a5a97f0179\",0,\"thm_2287\"]",
119
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002272.lean\",\"9787811202356b03c06dde229d046b8e6680fa9180cccbac61a376208b4e16ca\",0,\"thm_2288\"]",
120
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002273.lean\",\"1851c3d2c7cda5a49db7772e7401d41d6392f8d7568230ae3d85d2fe19f759d2\",0,\"thm_2289\"]",
121
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002274.lean\",\"3f45bb3f097bcbac57673fa13f7b8843929e97662f45215e47534b63999febec\",0,\"thm_2290\"]",
122
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002275.lean\",\"54de2b1e412d4c9563212386b13d07b209a836068aa51ae2738d6f2d3d0cfd3e\",0,\"thm_2291\"]",
123
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002276.lean\",\"8b96ca706061c881654209bb6de476145a8910bb8c829570907c488cbf7f0024\",0,\"thm_2292\"]",
124
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002277.lean\",\"3e8dae4626d17c7a36c44b0ef857f0b38a6b423af42fc0c81e0eeec9da1026d9\",0,\"thm_2293\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002278.lean\",\"a40e53f54d7f1be75af17a70f44175193ca90579bdb362ffca4ff5137dd7967f\",0,\"thm_2294\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002279.lean\",\"024afaf88c6b245a5c350436844aa0f324646b56fb9b3635d6d2a9e36074f26c\",0,\"thm_2295\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002280.lean\",\"11920834a220ff3706b60632ff15a671d68b0487f8057e397c9eb3baf959e615\",0,\"thm_2296\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002281.lean\",\"a29f9473bda3cffd5ed6f6f3e04b9221deae191dfcc0cb4f18007b4c02d037e6\",0,\"thm_2297\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002282.lean\",\"f2de9b3a6a6ce7809f94f45c47052dffe575a947ec60facd3ad30dc21c871ce0\",0,\"thm_2298\"]",
130
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002283.lean\",\"c3356beb1216fa09337ed384dff158730e49eb478f77e2a6d2cab205c28901ca\",0,\"thm_2299\"]",
131
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002284.lean\",\"18b9ab3a31ce5b634953aa9b8f6a6b520ccf83f41f571aef755ed445bed316ae\",0,\"thm_2300\"]",
132
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002285.lean\",\"2e843acc4264a3c26d2ab0b350cdc36cdfa0ae7e762d648a72f2b87e4c4f5f87\",0,\"thm_2301\"]",
133
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002286.lean\",\"02a1f2bbdbd8669a5dd4b5f72213f99a5dbeefc35085e4f91d3cdcc86b7dc55e\",0,\"thm_2302\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002287.lean\",\"066ff4f4fe2ca55f07739565c34f35315a6419d2ffa8aa9750b8bab22e4ef2d8\",0,\"thm_2303\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002288.lean\",\"d7e36cae75600665f7e9925ffad5c14162583ef68ea10e02e88f4f83e55602c0\",0,\"thm_2304\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002289.lean\",\"e469f5976bcc40412ac3ca93a9319e329509eb469e5cff663b091700f0e1dfdf\",0,\"thm_2305\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002290.lean\",\"31b422a41b3362bfd23b8927535a4c6b5192c9c66a30a06f50f82fac81992fa4\",0,\"thm_2306\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002291.lean\",\"1fab3536eb10895563d3d205dcf7758cabc30b40bbbd9297fc5c5483d86b9252\",0,\"thm_2307\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002292.lean\",\"243c82505570068e4e9c70a1319fbf6336585798a8212ab53b6724242e5e09f0\",0,\"thm_2308\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002293.lean\",\"f8c96f16713ff6be378d8764a5343d1790b57c0db3e4305a30072426da86f717\",0,\"thm_2309\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002294.lean\",\"4a7443df0ae1522e54bba37a57db4465aa88cf5552a59b0630c5afe604489719\",0,\"thm_2310\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002295.lean\",\"91e8e91ac07dededa8a38a9a1eb87205229123f1794b6560c7009f564fbe5f3b\",0,\"thm_2311\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002296.lean\",\"32e9bc7a33775df3cf29b4891b28109b875447aaf1cca64d798ee60a2dc97741\",0,\"thm_2312\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002297.lean\",\"6e284b33ed455bb614a214a30281c01cb5ba61cd97a28e0a37cfd00ff471d217\",0,\"thm_2313\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002298.lean\",\"4582f8e51a39c39183d3772e2b166173a48cf184872c703573bb572d50914e54\",0,\"thm_2314\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002299.lean\",\"034370867fc3e3d1dc252cd6eee50ad7bf44e1939d69535bfcc65d5d0d721bd9\",0,\"thm_2315\"]"
147
+ ]
148
+ }
manifests/batches/deepseek/000047.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000047",
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+ "batch_fingerprint": "d44acf6e2f312b2789c1e51d2fba536a62c788c0087c37f20d7b69d845ae7b97",
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+ "evidence": {
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+ "batch_fingerprint": "d44acf6e2f312b2789c1e51d2fba536a62c788c0087c37f20d7b69d845ae7b97",
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+ "contamination_matches": [],
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+ "failures": [],
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+ "input_proofs": 50,
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+ "input_units": 50,
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+ "proofs": 50,
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+ "published_proofs": 50,
13
+ "published_transitions": 109,
14
+ "replay_failures": [],
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+ "trace_mode": "individual",
16
+ "transitions": 109,
17
+ "validation_failures": [],
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+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
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+ },
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+ "shards": [
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+ {
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+ "path": "data/train/deepseek-000047-0000-996b8aac5e98130d.parquet",
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+ "sha256": "996b8aac5e98130d0a6f783f5d0cb18e4435def7ad8c65a21dd211b27d61ad09",
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+ "bytes": 350404,
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+ "rows": 98,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
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+ },
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+ {
30
+ "path": "data/dev/deepseek-000047-0000-185441a9b7c92a16.parquet",
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+ "sha256": "185441a9b7c92a165f1acc42c584132b96467aabe6b5166d1083f7b0aef3e1e2",
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+ "bytes": 30774,
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+ "rows": 2,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
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+ },
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+ {
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+ "path": "data/internal_test/deepseek-000047-0000-a5aba7c1b98bc21c.parquet",
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+ "sha256": "a5aba7c1b98bc21c88912d76ff6bd2a386899d44e02adccc415a91344593073f",
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+ "bytes": 51808,
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+ "rows": 9,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
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+ }
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+ ],
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+ "proof_ids": [
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+ "deepseek:thm_2366",
46
+ "deepseek:thm_2367",
47
+ "deepseek:thm_2368",
48
+ "deepseek:thm_2369",
49
+ "deepseek:thm_2370",
50
+ "deepseek:thm_2371",
51
+ "deepseek:thm_2372",
52
+ "deepseek:thm_2373",
53
+ "deepseek:thm_2374",
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+ "deepseek:thm_2377",
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+ "deepseek:thm_2378",
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+ "deepseek:thm_2379",
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+ "deepseek:thm_2380",
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+ "deepseek:thm_2382",
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+ "deepseek:thm_2383",
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+ "deepseek:thm_2384",
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+ "deepseek:thm_2385",
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+ "deepseek:thm_2386",
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+ "deepseek:thm_2387",
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+ "deepseek:thm_2388",
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+ "deepseek:thm_2389",
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+ "deepseek:thm_2390",
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+ "deepseek:thm_2391",
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+ "deepseek:thm_2392",
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+ "deepseek:thm_2393",
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+ "deepseek:thm_2394",
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+ "deepseek:thm_2395",
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+ "deepseek:thm_2396",
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+ "deepseek:thm_2397",
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+ "deepseek:thm_2398",
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+ "deepseek:thm_2399",
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+ "deepseek:thm_2400",
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+ "deepseek:thm_2401",
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+ "deepseek:thm_2402",
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+ "deepseek:thm_2403",
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+ "deepseek:thm_2404",
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+ "deepseek:thm_2405",
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+ "deepseek:thm_2406",
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+ "deepseek:thm_2407",
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+ "deepseek:thm_2408",
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+ "deepseek:thm_2409",
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+ "deepseek:thm_2410",
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+ "deepseek:thm_2411",
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+ "deepseek:thm_2412",
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+ "deepseek:thm_2413",
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+ "deepseek:thm_2414",
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+ "deepseek:thm_2415"
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002393.lean\",\"a80e39b8a80d50d81f88df955ff315b6154411b9f57ca1e014f603940f798a30\",0,\"thm_2409\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_002395.lean\",\"7bfbb8809b0568f0d9a9fbe88c8c89a8b47bb40c2d2f49322c3b3cfbba54e081\",0,\"thm_2411\"]",
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147
+ ]
148
+ }
manifests/batches/deepseek/000048.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "batch": "deepseek/000048",
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+ "batch_fingerprint": "c9d2b639230b826a1760a5a57551b3684314c0123ef5137577a7370a460a0733",
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+ "evidence": {
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+ "batch_fingerprint": "c9d2b639230b826a1760a5a57551b3684314c0123ef5137577a7370a460a0733",
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+ "contamination_matches": [],
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+ "failures": [],
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+ "input_proofs": 50,
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+ "input_units": 50,
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+ "proofs": 50,
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+ "published_proofs": 50,
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+ "published_transitions": 118,
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+ "replay_failures": [],
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+ "trace_mode": "individual",
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+ "transitions": 118,
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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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+ "shards": [
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+ {
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+ "path": "data/train/deepseek-000048-0000-1a763629c2292ebb.parquet",
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+ "path": "data/internal_test/deepseek-000048-0000-7a5ce4b894f76683.parquet",
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+ "proof_ids": [
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+ "deepseek:thm_2416",
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+ "deepseek:thm_2418",
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+ "deepseek:thm_2419",
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+ "deepseek:thm_2422",
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+ "deepseek:thm_2423",
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+ "deepseek:thm_2424",
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+ "deepseek:thm_2438",
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+ "deepseek:thm_2440",
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+ "deepseek:thm_2462",
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+ "deepseek:thm_2463",
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+ "deepseek:thm_2464",
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+ ],
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+ "proof_keys": [
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+ ]
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+ }
manifests/batches/deepseek/000050.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ "contamination_matches": [],
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+ "published_transitions": 110,
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+ "replay_failures": [],
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+ "trace_mode": "individual",
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+ "transitions": 110,
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+ "validation_failures": [],
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+ "traced_tactics": [],
313
+ "url": "https://github.com/leanprover-community/mathlib4"
314
+ },
315
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
316
+ },
317
+ {
318
+ "error": "theorem has no selected tactic transitions",
319
+ "full_name": "Subalgebra.op_top",
320
+ "id": "Subalgebra.op_top",
321
+ "reason": "import_failure",
322
+ "selection": {
323
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324
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325
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328
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
329
+ "full_name": "Subalgebra.op_top",
330
+ "start": [
331
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334
+ "traced_tactics": [],
335
+ "url": "https://github.com/leanprover-community/mathlib4"
336
+ },
337
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
338
+ },
339
+ {
340
+ "error": "theorem has no selected tactic transitions",
341
+ "full_name": "Subalgebra.op_unop",
342
+ "id": "Subalgebra.op_unop",
343
+ "reason": "import_failure",
344
+ "selection": {
345
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
346
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347
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348
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350
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
351
+ "full_name": "Subalgebra.op_unop",
352
+ "start": [
353
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354
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355
+ ],
356
+ "traced_tactics": [],
357
+ "url": "https://github.com/leanprover-community/mathlib4"
358
+ },
359
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
360
+ },
361
+ {
362
+ "error": "theorem has no selected tactic transitions",
363
+ "full_name": "Subalgebra.unop_bot",
364
+ "id": "Subalgebra.unop_bot",
365
+ "reason": "import_failure",
366
+ "selection": {
367
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368
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369
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370
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371
+ ],
372
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
373
+ "full_name": "Subalgebra.unop_bot",
374
+ "start": [
375
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376
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377
+ ],
378
+ "traced_tactics": [],
379
+ "url": "https://github.com/leanprover-community/mathlib4"
380
+ },
381
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
382
+ },
383
+ {
384
+ "error": "theorem has no selected tactic transitions",
385
+ "full_name": "Subalgebra.unop_coe",
386
+ "id": "Subalgebra.unop_coe",
387
+ "reason": "import_failure",
388
+ "selection": {
389
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
390
+ "end": [
391
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392
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393
+ ],
394
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
395
+ "full_name": "Subalgebra.unop_coe",
396
+ "start": [
397
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398
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399
+ ],
400
+ "traced_tactics": [],
401
+ "url": "https://github.com/leanprover-community/mathlib4"
402
+ },
403
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
404
+ },
405
+ {
406
+ "error": "theorem has no selected tactic transitions",
407
+ "full_name": "Subalgebra.unop_iInf",
408
+ "id": "Subalgebra.unop_iInf",
409
+ "reason": "import_failure",
410
+ "selection": {
411
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
412
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413
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414
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415
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416
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
417
+ "full_name": "Subalgebra.unop_iInf",
418
+ "start": [
419
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420
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421
+ ],
422
+ "traced_tactics": [],
423
+ "url": "https://github.com/leanprover-community/mathlib4"
424
+ },
425
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
426
+ },
427
+ {
428
+ "error": "theorem has no selected tactic transitions",
429
+ "full_name": "Subalgebra.unop_iSup",
430
+ "id": "Subalgebra.unop_iSup",
431
+ "reason": "import_failure",
432
+ "selection": {
433
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
434
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435
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436
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437
+ ],
438
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
439
+ "full_name": "Subalgebra.unop_iSup",
440
+ "start": [
441
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442
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443
+ ],
444
+ "traced_tactics": [],
445
+ "url": "https://github.com/leanprover-community/mathlib4"
446
+ },
447
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
448
+ },
449
+ {
450
+ "error": "theorem has no selected tactic transitions",
451
+ "full_name": "Subalgebra.unop_inf",
452
+ "id": "Subalgebra.unop_inf",
453
+ "reason": "import_failure",
454
+ "selection": {
455
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
456
+ "end": [
457
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458
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459
+ ],
460
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
461
+ "full_name": "Subalgebra.unop_inf",
462
+ "start": [
463
+ 99,
464
+ 1
465
+ ],
466
+ "traced_tactics": [],
467
+ "url": "https://github.com/leanprover-community/mathlib4"
468
+ },
469
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
470
+ },
471
+ {
472
+ "error": "theorem has no selected tactic transitions",
473
+ "full_name": "Subalgebra.unop_le_unop_iff",
474
+ "id": "Subalgebra.unop_le_unop_iff",
475
+ "reason": "import_failure",
476
+ "selection": {
477
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
478
+ "end": [
479
+ 68,
480
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481
+ ],
482
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
483
+ "full_name": "Subalgebra.unop_le_unop_iff",
484
+ "start": [
485
+ 67,
486
+ 1
487
+ ],
488
+ "traced_tactics": [],
489
+ "url": "https://github.com/leanprover-community/mathlib4"
490
+ },
491
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
492
+ },
493
+ {
494
+ "error": "theorem has no selected tactic transitions",
495
+ "full_name": "Subalgebra.unop_op",
496
+ "id": "Subalgebra.unop_op",
497
+ "reason": "import_failure",
498
+ "selection": {
499
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
500
+ "end": [
501
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502
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503
+ ],
504
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
505
+ "full_name": "Subalgebra.unop_op",
506
+ "start": [
507
+ 49,
508
+ 1
509
+ ],
510
+ "traced_tactics": [],
511
+ "url": "https://github.com/leanprover-community/mathlib4"
512
+ },
513
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
514
+ },
515
+ {
516
+ "error": "theorem has no selected tactic transitions",
517
+ "full_name": "Subalgebra.unop_sInf",
518
+ "id": "Subalgebra.unop_sInf",
519
+ "reason": "import_failure",
520
+ "selection": {
521
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
522
+ "end": [
523
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524
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525
+ ],
526
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
527
+ "full_name": "Subalgebra.unop_sInf",
528
+ "start": [
529
+ 111,
530
+ 1
531
+ ],
532
+ "traced_tactics": [],
533
+ "url": "https://github.com/leanprover-community/mathlib4"
534
+ },
535
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
536
+ },
537
+ {
538
+ "error": "theorem has no selected tactic transitions",
539
+ "full_name": "Subalgebra.unop_sSup",
540
+ "id": "Subalgebra.unop_sSup",
541
+ "reason": "import_failure",
542
+ "selection": {
543
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
544
+ "end": [
545
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546
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547
+ ],
548
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
549
+ "full_name": "Subalgebra.unop_sSup",
550
+ "start": [
551
+ 105,
552
+ 1
553
+ ],
554
+ "traced_tactics": [],
555
+ "url": "https://github.com/leanprover-community/mathlib4"
556
+ },
557
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
558
+ },
559
+ {
560
+ "error": "theorem has no selected tactic transitions",
561
+ "full_name": "Subalgebra.unop_sup",
562
+ "id": "Subalgebra.unop_sup",
563
+ "reason": "import_failure",
564
+ "selection": {
565
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
566
+ "end": [
567
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568
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569
+ ],
570
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
571
+ "full_name": "Subalgebra.unop_sup",
572
+ "start": [
573
+ 94,
574
+ 1
575
+ ],
576
+ "traced_tactics": [],
577
+ "url": "https://github.com/leanprover-community/mathlib4"
578
+ },
579
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
580
+ },
581
+ {
582
+ "error": "theorem has no selected tactic transitions",
583
+ "full_name": "Subalgebra.unop_toSubring",
584
+ "id": "Subalgebra.unop_toSubring",
585
+ "reason": "import_failure",
586
+ "selection": {
587
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
588
+ "end": [
589
+ 168,
590
+ 92
591
+ ],
592
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
593
+ "full_name": "Subalgebra.unop_toSubring",
594
+ "start": [
595
+ 168,
596
+ 1
597
+ ],
598
+ "traced_tactics": [],
599
+ "url": "https://github.com/leanprover-community/mathlib4"
600
+ },
601
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
602
+ },
603
+ {
604
+ "error": "theorem has no selected tactic transitions",
605
+ "full_name": "Subalgebra.unop_top",
606
+ "id": "Subalgebra.unop_top",
607
+ "reason": "import_failure",
608
+ "selection": {
609
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
610
+ "end": [
611
+ 89,
612
+ 76
613
+ ],
614
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean",
615
+ "full_name": "Subalgebra.unop_top",
616
+ "start": [
617
+ 89,
618
+ 1
619
+ ],
620
+ "traced_tactics": [],
621
+ "url": "https://github.com/leanprover-community/mathlib4"
622
+ },
623
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean"
624
+ },
625
+ {
626
+ "error": "theorem has no selected tactic transitions",
627
+ "full_name": "AlgHom.ker_rangeRestrict",
628
+ "id": "AlgHom.ker_rangeRestrict",
629
+ "reason": "import_failure",
630
+ "selection": {
631
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
632
+ "end": [
633
+ 26,
634
+ 36
635
+ ],
636
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean",
637
+ "full_name": "AlgHom.ker_rangeRestrict",
638
+ "start": [
639
+ 25,
640
+ 1
641
+ ],
642
+ "traced_tactics": [],
643
+ "url": "https://github.com/leanprover-community/mathlib4"
644
+ },
645
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean"
646
+ },
647
+ {
648
+ "error": "no internal trace for tactic 'obtain \u27e8x, rfl\u27e9 := this'",
649
+ "full_name": "Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem",
650
+ "id": "Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem",
651
+ "reason": "import_failure",
652
+ "selection": {
653
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
654
+ "end": [
655
+ 68,
656
+ 25
657
+ ],
658
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean",
659
+ "full_name": "Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem",
660
+ "start": [
661
+ 40,
662
+ 1
663
+ ],
664
+ "traced_tactics": [
665
+ {
666
+ "annotated_tactic": [
667
+ "\"let _i : <a>Algebra<\\/a> { x \\/\\/ x \\u2208 S' } { x \\/\\/ x \\u2208 S' } := <a>Algebra.id<\\/a> _\"",
668
+ "[{\"full_name\":\"Algebra\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[101,7],\"def_end_pos\":[101,14]},{\"full_name\":\"Algebra.id\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[426,29],\"def_end_pos\":[426,31]}]"
669
+ ],
670
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\n\u22a2 x \u2208 S'",
671
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n\u22a2 x \u2208 S'",
672
+ "tactic": "let _i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id _"
673
+ },
674
+ {
675
+ "annotated_tactic": [
676
+ "\"suffices x \\u2208 <a>Subalgebra.toSubmodule<\\/a> (<a>Algebra.ofId<\\/a> S' S).<a>range<\\/a> by\\n obtain \\u27e8x, rfl\\u27e9 := this\\n exact x.2\"",
677
+ "[{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]},{\"full_name\":\"Algebra.ofId\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[542,5],\"def_end_pos\":[542,9]},{\"full_name\":\"AlgHom.range\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[592,15],\"def_end_pos\":[592,20]}]"
678
+ ],
679
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
680
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\n\u22a2 x \u2208 S'",
681
+ "tactic": "suffices x \u2208 Subalgebra.toSubmodule (Algebra.ofId S' S).range by\n obtain \u27e8x, rfl\u27e9 := this\n exact x.2"
682
+ },
683
+ {
684
+ "annotated_tactic": [
685
+ "\"choose n hn using H\"",
686
+ "[]"
687
+ ],
688
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
689
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
690
+ "tactic": "choose n hn using H"
691
+ },
692
+ {
693
+ "annotated_tactic": [
694
+ "\"let s' : \\u03b9 \\u2192 S' := fun x => \\u27e8s x, hs x\\u27e9\"",
695
+ "[]"
696
+ ],
697
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
698
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
699
+ "tactic": "let s' : \u03b9 \u2192 S' := fun x => \u27e8s x, hs x\u27e9"
700
+ },
701
+ {
702
+ "annotated_tactic": [
703
+ "\"let l' : \\u03b9 \\u2192 S' := fun x => \\u27e8l x, hl x\\u27e9\"",
704
+ "[]"
705
+ ],
706
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
707
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
708
+ "tactic": "let l' : \u03b9 \u2192 S' := fun x => \u27e8l x, hl x\u27e9"
709
+ },
710
+ {
711
+ "annotated_tactic": [
712
+ "\"have e' : \\u2211 i \\u2208 \\u03b9', l' i * s' i = 1 := by\\n ext\\n show S'.subtype (\\u2211 i \\u2208 \\u03b9', l' i * s' i) = 1\\n simpa only [<a>map_sum<\\/a>, <a>map_mul<\\/a>] using e\"",
713
+ "[{\"full_name\":\"map_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[295,3],\"def_end_pos\":[295,14]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]}]"
714
+ ],
715
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
716
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
717
+ "tactic": "have e' : \u2211 i \u2208 \u03b9', l' i * s' i = 1 := by\n ext\n show S'.subtype (\u2211 i \u2208 \u03b9', l' i * s' i) = 1\n simpa only [map_sum, map_mul] using e"
718
+ },
719
+ {
720
+ "annotated_tactic": [
721
+ "\"have : <a>Ideal.span<\\/a> (s' '' \\u03b9') = \\u22a4 := by\\n rw [<a>Ideal.eq_top_iff_one<\\/a>, \\u2190 e']\\n apply <a>sum_mem<\\/a>\\n intros i hi\\n exact <a>Ideal.mul_mem_left<\\/a> _ _ <| <a>Ideal.subset_span<\\/a> <| <a>Set.mem_image_of_mem<\\/a> s' hi\"",
722
+ "[{\"full_name\":\"Ideal.span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[115,5],\"def_end_pos\":[115,9]},{\"full_name\":\"Ideal.eq_top_iff_one\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,23]},{\"full_name\":\"sum_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Membership.lean\",\"def_pos\":[94,3],\"def_end_pos\":[94,14]},{\"full_name\":\"Ideal.mul_mem_left\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[69,9],\"def_end_pos\":[69,21]},{\"full_name\":\"Ideal.subset_span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[146,9],\"def_end_pos\":[146,20]},{\"full_name\":\"Set.mem_image_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[132,9],\"def_end_pos\":[132,25]}]"
723
+ ],
724
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
725
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
726
+ "tactic": "have : Ideal.span (s' '' \u03b9') = \u22a4 := by\n rw [Ideal.eq_top_iff_one, \u2190 e']\n apply sum_mem\n intros i hi\n exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi"
727
+ },
728
+ {
729
+ "annotated_tactic": [
730
+ "\"let N := \\u03b9'.sup n\"",
731
+ "[]"
732
+ ],
733
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
734
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
735
+ "tactic": "let N := \u03b9'.sup n"
736
+ },
737
+ {
738
+ "annotated_tactic": [
739
+ "\"have hN := <a>Ideal.span_pow_eq_top<\\/a> _ this N\"",
740
+ "[{\"full_name\":\"Ideal.span_pow_eq_top\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[654,9],\"def_end_pos\":[654,24]}]"
741
+ ],
742
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
743
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
744
+ "tactic": "have hN := Ideal.span_pow_eq_top _ this N"
745
+ },
746
+ {
747
+ "annotated_tactic": [
748
+ "\"apply (<a>Algebra.ofId<\\/a> S' S).range.toSubmodule.mem_of_span_top_of_smul_mem _ hN\"",
749
+ "[{\"full_name\":\"Algebra.ofId\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[542,5],\"def_end_pos\":[542,9]}]"
750
+ ],
751
+ "state_after": "case H\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 \u2200 (r : \u2191((fun x => x ^ N) '' (s' '' \u2191\u03b9'))), \u2191r \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
752
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
753
+ "tactic": "apply (Algebra.ofId S' S).range.toSubmodule.mem_of_span_top_of_smul_mem _ hN"
754
+ },
755
+ {
756
+ "annotated_tactic": [
757
+ "\"rintro \\u27e8_, _, \\u27e8i, hi, rfl\\u27e9, rfl\\u27e9\"",
758
+ "[]"
759
+ ],
760
+ "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 \u2191\u27e8(fun x => x ^ N) (s' i), \u22ef\u27e9 \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
761
+ "state_before": "case H\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 \u2200 (r : \u2191((fun x => x ^ N) '' (s' '' \u2191\u03b9'))), \u2191r \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
762
+ "tactic": "rintro \u27e8_, _, \u27e8i, hi, rfl\u27e9, rfl\u27e9"
763
+ },
764
+ {
765
+ "annotated_tactic": [
766
+ "\"change s' i ^ N \\u2022 x \\u2208 _\"",
767
+ "[]"
768
+ ],
769
+ "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ N \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
770
+ "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 \u2191\u27e8(fun x => x ^ N) (s' i), \u22ef\u27e9 \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
771
+ "tactic": "change s' i ^ N \u2022 x \u2208 _"
772
+ },
773
+ {
774
+ "annotated_tactic": [
775
+ "\"rw [\\u2190 <a>tsub_add_cancel_of_le<\\/a> (show n i \\u2264 N from <a>Finset.le_sup<\\/a> hi), <a>pow_add<\\/a>, <a>mul_smul<\\/a>]\"",
776
+ "[{\"full_name\":\"tsub_add_cancel_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[31,9],\"def_end_pos\":[31,30]},{\"full_name\":\"Finset.le_sup\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Lattice.lean\",\"def_pos\":[119,9],\"def_end_pos\":[119,15]},{\"full_name\":\"pow_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[703,7],\"def_end_pos\":[703,14]},{\"full_name\":\"MulAction.mul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[114,3],\"def_end_pos\":[114,11]}]"
777
+ ],
778
+ "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ (N - n i) \u2022 s' i ^ n i \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
779
+ "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ N \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
780
+ "tactic": "rw [\u2190 tsub_add_cancel_of_le (show n i \u2264 N from Finset.le_sup hi), pow_add, mul_smul]"
781
+ },
782
+ {
783
+ "annotated_tactic": [
784
+ "\"refine <a>Submodule.smul_mem<\\/a> _ (\\u27e8_, <a>pow_mem<\\/a> (hs i) _\\u27e9 : S') ?_\"",
785
+ "[{\"full_name\":\"Submodule.smul_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,17]},{\"full_name\":\"pow_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,16]}]"
786
+ ],
787
+ "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ n i \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
788
+ "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ (N - n i) \u2022 s' i ^ n i \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
789
+ "tactic": "refine Submodule.smul_mem _ (\u27e8_, pow_mem (hs i) _\u27e9 : S') ?_"
790
+ },
791
+ {
792
+ "annotated_tactic": [
793
+ "\"exact \\u27e8\\u27e8_, hn i\\u27e9, <a>rfl<\\/a>\\u27e9\"",
794
+ "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
795
+ ],
796
+ "state_after": "no goals",
797
+ "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := \u03b9'.sup n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ n i \u2022 x \u2208 toSubmodule (ofId (\u21a5S') S).range",
798
+ "tactic": "exact \u27e8\u27e8_, hn i\u27e9, rfl\u27e9"
799
+ },
800
+ {
801
+ "annotated_tactic": [
802
+ "\"obtain \\u27e8x, rfl\\u27e9 := this\"",
803
+ "[]"
804
+ ],
805
+ "state_after": "case intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nx : \u21a5S'\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 (ofId (\u21a5S') S).toRingHom x \u2208 S'\n\u22a2 (ofId (\u21a5S') S).toRingHom x \u2208 S'",
806
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nthis : x \u2208 toSubmodule (ofId (\u21a5S') S).range\n\u22a2 x \u2208 S'",
807
+ "tactic": "obtain \u27e8x, rfl\u27e9 := this"
808
+ },
809
+ {
810
+ "annotated_tactic": [
811
+ "\"exact x.2\"",
812
+ "[]"
813
+ ],
814
+ "state_after": "no goals",
815
+ "state_before": "case intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nx : \u21a5S'\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 (ofId (\u21a5S') S).toRingHom x \u2208 S'\n\u22a2 (ofId (\u21a5S') S).toRingHom x \u2208 S'",
816
+ "tactic": "exact x.2"
817
+ },
818
+ {
819
+ "annotated_tactic": [
820
+ "\"ext\"",
821
+ "[]"
822
+ ],
823
+ "state_after": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 \u2191(\u2211 i \u2208 \u03b9', l' i * s' i) = \u21911",
824
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 \u2211 i \u2208 \u03b9', l' i * s' i = 1",
825
+ "tactic": "ext"
826
+ },
827
+ {
828
+ "annotated_tactic": [
829
+ "\"show S'.subtype (\\u2211 i \\u2208 \\u03b9', l' i * s' i) = 1\"",
830
+ "[]"
831
+ ],
832
+ "state_after": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 S'.subtype (\u2211 i \u2208 \u03b9', l' i * s' i) = 1",
833
+ "state_before": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 \u2191(\u2211 i \u2208 \u03b9', l' i * s' i) = \u21911",
834
+ "tactic": "show S'.subtype (\u2211 i \u2208 \u03b9', l' i * s' i) = 1"
835
+ },
836
+ {
837
+ "annotated_tactic": [
838
+ "\"simpa only [<a>map_sum<\\/a>, <a>map_mul<\\/a>] using e\"",
839
+ "[{\"full_name\":\"map_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[295,3],\"def_end_pos\":[295,14]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]}]"
840
+ ],
841
+ "state_after": "no goals",
842
+ "state_before": "case a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\n\u22a2 S'.subtype (\u2211 i \u2208 \u03b9', l' i * s' i) = 1",
843
+ "tactic": "simpa only [map_sum, map_mul] using e"
844
+ },
845
+ {
846
+ "annotated_tactic": [
847
+ "\"rw [<a>Ideal.eq_top_iff_one<\\/a>, \\u2190 e']\"",
848
+ "[{\"full_name\":\"Ideal.eq_top_iff_one\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,23]}]"
849
+ ],
850
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 \u2211 i \u2208 \u03b9', l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')",
851
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 Ideal.span (s' '' \u2191\u03b9') = \u22a4",
852
+ "tactic": "rw [Ideal.eq_top_iff_one, \u2190 e']"
853
+ },
854
+ {
855
+ "annotated_tactic": [
856
+ "\"apply <a>sum_mem<\\/a>\"",
857
+ "[{\"full_name\":\"sum_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Membership.lean\",\"def_pos\":[94,3],\"def_end_pos\":[94,14]}]"
858
+ ],
859
+ "state_after": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 \u2200 c \u2208 \u03b9', l' c * s' c \u2208 Ideal.span (s' '' \u2191\u03b9')",
860
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 \u2211 i \u2208 \u03b9', l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')",
861
+ "tactic": "apply sum_mem"
862
+ },
863
+ {
864
+ "annotated_tactic": [
865
+ "\"intros i hi\"",
866
+ "[]"
867
+ ],
868
+ "state_after": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\ni : \u03b9\nhi : i \u2208 \u03b9'\n\u22a2 l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')",
869
+ "state_before": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\n\u22a2 \u2200 c \u2208 \u03b9', l' c * s' c \u2208 Ideal.span (s' '' \u2191\u03b9')",
870
+ "tactic": "intros i hi"
871
+ },
872
+ {
873
+ "annotated_tactic": [
874
+ "\"exact <a>Ideal.mul_mem_left<\\/a> _ _ <| <a>Ideal.subset_span<\\/a> <| <a>Set.mem_image_of_mem<\\/a> s' hi\"",
875
+ "[{\"full_name\":\"Ideal.mul_mem_left\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[69,9],\"def_end_pos\":[69,21]},{\"full_name\":\"Ideal.subset_span\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[146,9],\"def_end_pos\":[146,20]},{\"full_name\":\"Set.mem_image_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[132,9],\"def_end_pos\":[132,25]}]"
876
+ ],
877
+ "state_after": "no goals",
878
+ "state_before": "case h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_3\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i \u2208 \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra \u21a5S' \u21a5S' := Algebra.id \u21a5S'\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8s x, \u22ef\u27e9\nl' : \u03b9 \u2192 \u21a5S' := fun x => \u27e8l x, \u22ef\u27e9\ne' : \u2211 i \u2208 \u03b9', l' i * s' i = 1\ni : \u03b9\nhi : i \u2208 \u03b9'\n\u22a2 l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')",
879
+ "tactic": "exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi"
880
+ }
881
+ ],
882
+ "url": "https://github.com/leanprover-community/mathlib4"
883
+ },
884
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean"
885
+ },
886
+ {
887
+ "error": "theorem has no selected tactic transitions",
888
+ "full_name": "Subalgebra.mem_of_span_eq_top_of_smul_pow_mem",
889
+ "id": "Subalgebra.mem_of_span_eq_top_of_smul_pow_mem",
890
+ "reason": "import_failure",
891
+ "selection": {
892
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
893
+ "end": [
894
+ 75,
895
+ 91
896
+ ],
897
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean",
898
+ "full_name": "Subalgebra.mem_of_span_eq_top_of_smul_pow_mem",
899
+ "start": [
900
+ 71,
901
+ 1
902
+ ],
903
+ "traced_tactics": [],
904
+ "url": "https://github.com/leanprover-community/mathlib4"
905
+ },
906
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Operations.lean"
907
+ }
908
+ ],
909
+ "input_proofs": 33,
910
+ "input_units": 2,
911
+ "proofs": 2,
912
+ "published_proofs": 2,
913
+ "published_transitions": 10,
914
+ "replay_failures": [],
915
+ "transitions": 10,
916
+ "validation_failures": [],
917
+ "validation_rejected_proofs": 0,
918
+ "zero_transition_proofs": 0
919
+ },
920
+ "shards": [
921
+ {
922
+ "path": "data/train/mathlib-000033-0000-bfb25f3e8866e986.parquet",
923
+ "sha256": "bfb25f3e8866e986df4899d93899cb3bc7f9eceee93a00eedd9b56a7e83b9123",
924
+ "bytes": 60799,
925
+ "rows": 10,
926
+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
927
+ }
928
+ ],
929
+ "proof_ids": [
930
+ "mathlib:Subalgebra.op_adjoin",
931
+ "mathlib:Subalgebra.unop_adjoin"
932
+ ],
933
+ "proof_keys": [
934
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean\",null,null,\"Subalgebra.op_adjoin\"]",
935
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean\",null,null,\"Subalgebra.unop_adjoin\"]"
936
+ ]
937
+ }
manifests/batches/mathlib/000034.json ADDED
@@ -0,0 +1,471 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000034",
4
+ "batch_fingerprint": "5243f48a89d26b5eafe5a4b0100d7eddc425d2b9775617e085fec58bdfc073ff",
5
+ "evidence": {
6
+ "batch_fingerprint": "5243f48a89d26b5eafe5a4b0100d7eddc425d2b9775617e085fec58bdfc073ff",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "Subalgebra.coe_pointwise_smul",
12
+ "id": "Subalgebra.coe_pointwise_smul",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 86,
18
+ 6
19
+ ],
20
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
21
+ "full_name": "Subalgebra.coe_pointwise_smul",
22
+ "start": [
23
+ 85,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
30
+ },
31
+ {
32
+ "error": "no internal trace for tactic \"cases' hx with hxS hxT\"",
33
+ "full_name": "Subalgebra.mul_toSubmodule",
34
+ "id": "Subalgebra.mul_toSubmodule",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 65,
40
+ 82
41
+ ],
42
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
43
+ "full_name": "Subalgebra.mul_toSubmodule",
44
+ "start": [
45
+ 48,
46
+ 1
47
+ ],
48
+ "traced_tactics": [
49
+ {
50
+ "annotated_tactic": [
51
+ "\"refine <a>le_antisymm<\\/a> (<a>mul_toSubmodule_le<\\/a> _ _) ?_\"",
52
+ "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Subalgebra.mul_toSubmodule_le\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Pointwise.lean\",\"def_pos\":[27,9],\"def_end_pos\":[27,27]}]"
53
+ ],
54
+ "state_after": "R\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 toSubmodule (S \u2294 T) \u2264 toSubmodule S * toSubmodule T",
55
+ "state_before": "R\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 toSubmodule S * toSubmodule T = toSubmodule (S \u2294 T)",
56
+ "tactic": "refine le_antisymm (mul_toSubmodule_le _ _) ?_"
57
+ },
58
+ {
59
+ "annotated_tactic": [
60
+ "\"rintro x (hx : x \\u2208 <a>Algebra.adjoin<\\/a> R (S \\u222a T : <a>Set<\\/a> A))\"",
61
+ "[{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
62
+ ],
63
+ "state_after": "R\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
64
+ "state_before": "R\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\n\u22a2 toSubmodule (S \u2294 T) \u2264 toSubmodule S * toSubmodule T",
65
+ "tactic": "rintro x (hx : x \u2208 Algebra.adjoin R (S \u222a T : Set A))"
66
+ },
67
+ {
68
+ "annotated_tactic": [
69
+ "\"refine\\n <a>Algebra.adjoin_induction<\\/a> hx (fun x hx => ?_) (fun r => ?_) (fun _ _ => <a>Submodule.add_mem<\\/a> _)\\n fun x y hx hy => ?_\"",
70
+ "[{\"full_name\":\"Algebra.adjoin_induction\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,25]},{\"full_name\":\"Submodule.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[220,19],\"def_end_pos\":[220,26]}]"
71
+ ],
72
+ "state_after": "case refine_1\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhx : x \u2208 \u2191S \u222a \u2191T\n\u22a2 x \u2208 toSubmodule S * toSubmodule T\n\ncase refine_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 (algebraMap R A) r \u2208 toSubmodule S * toSubmodule T\n\ncase refine_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 toSubmodule S * toSubmodule T\nhy : y \u2208 toSubmodule S * toSubmodule T\n\u22a2 x * y \u2208 toSubmodule S * toSubmodule T",
73
+ "state_before": "R\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
74
+ "tactic": "refine\n Algebra.adjoin_induction hx (fun x hx => ?_) (fun r => ?_) (fun _ _ => Submodule.add_mem _)\n fun x y hx hy => ?_"
75
+ },
76
+ {
77
+ "annotated_tactic": [
78
+ "\"have := <a>Submodule.mul_mem_mul<\\/a> hx hy\"",
79
+ "[{\"full_name\":\"Submodule.mul_mem_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Operations.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,20]}]"
80
+ ],
81
+ "state_after": "case refine_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 toSubmodule S * toSubmodule T\nhy : y \u2208 toSubmodule S * toSubmodule T\nthis : x * y \u2208 toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)\n\u22a2 x * y \u2208 toSubmodule S * toSubmodule T",
82
+ "state_before": "case refine_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 toSubmodule S * toSubmodule T\nhy : y \u2208 toSubmodule S * toSubmodule T\n\u22a2 x * y \u2208 toSubmodule S * toSubmodule T",
83
+ "tactic": "have := Submodule.mul_mem_mul hx hy"
84
+ },
85
+ {
86
+ "annotated_tactic": [
87
+ "\"rwa [<a>mul_assoc<\\/a>, <a>mul_comm<\\/a> _ (<a>Subalgebra.toSubmodule<\\/a> T), \\u2190 <a>mul_assoc<\\/a> _ _ (<a>Subalgebra.toSubmodule<\\/a> S),\\n <a>mul_self<\\/a>, <a>mul_comm<\\/a> (<a>Subalgebra.toSubmodule<\\/a> T), \\u2190 <a>mul_assoc<\\/a>, <a>mul_self<\\/a>] at this\"",
88
+ "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]},{\"full_name\":\"Subalgebra.mul_self\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Pointwise.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,17]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Subalgebra.toSubmodule\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[280,5],\"def_end_pos\":[280,16]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Subalgebra.mul_self\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Pointwise.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,17]}]"
89
+ ],
90
+ "state_after": "no goals",
91
+ "state_before": "case refine_3\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx y : A\nhx : x \u2208 toSubmodule S * toSubmodule T\nhy : y \u2208 toSubmodule S * toSubmodule T\nthis : x * y \u2208 toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)\n\u22a2 x * y \u2208 toSubmodule S * toSubmodule T",
92
+ "tactic": "rwa [mul_assoc, mul_comm _ (Subalgebra.toSubmodule T), \u2190 mul_assoc _ _ (Subalgebra.toSubmodule S),\n mul_self, mul_comm (Subalgebra.toSubmodule T), \u2190 mul_assoc, mul_self] at this"
93
+ },
94
+ {
95
+ "annotated_tactic": [
96
+ "\"cases' hx with hxS hxT\"",
97
+ "[]"
98
+ ],
99
+ "state_after": "case refine_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x \u2208 toSubmodule S * toSubmodule T\n\ncase refine_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
100
+ "state_before": "case refine_1\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx\u271d : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhx : x \u2208 \u2191S \u222a \u2191T\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
101
+ "tactic": "cases' hx with hxS hxT"
102
+ },
103
+ {
104
+ "annotated_tactic": [
105
+ "\"rw [\\u2190 <a>mul_one<\\/a> x]\"",
106
+ "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
107
+ ],
108
+ "state_after": "case refine_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x * 1 \u2208 toSubmodule S * toSubmodule T",
109
+ "state_before": "case refine_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
110
+ "tactic": "rw [\u2190 mul_one x]"
111
+ },
112
+ {
113
+ "annotated_tactic": [
114
+ "\"exact <a>Submodule.mul_mem_mul<\\/a> hxS (show (1 : A) \\u2208 T from <a>one_mem<\\/a> T)\"",
115
+ "[{\"full_name\":\"Submodule.mul_mem_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Operations.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,20]},{\"full_name\":\"OneMemClass.one_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]}]"
116
+ ],
117
+ "state_after": "no goals",
118
+ "state_before": "case refine_1.inl\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxS : x \u2208 \u2191S\n\u22a2 x * 1 \u2208 toSubmodule S * toSubmodule T",
119
+ "tactic": "exact Submodule.mul_mem_mul hxS (show (1 : A) \u2208 T from one_mem T)"
120
+ },
121
+ {
122
+ "annotated_tactic": [
123
+ "\"rw [\\u2190 <a>one_mul<\\/a> x]\"",
124
+ "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
125
+ ],
126
+ "state_after": "case refine_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 1 * x \u2208 toSubmodule S * toSubmodule T",
127
+ "state_before": "case refine_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 x \u2208 toSubmodule S * toSubmodule T",
128
+ "tactic": "rw [\u2190 one_mul x]"
129
+ },
130
+ {
131
+ "annotated_tactic": [
132
+ "\"exact <a>Submodule.mul_mem_mul<\\/a> (show (1 : A) \\u2208 S from <a>one_mem<\\/a> S) hxT\"",
133
+ "[{\"full_name\":\"Submodule.mul_mem_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Operations.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,20]},{\"full_name\":\"OneMemClass.one_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]}]"
134
+ ],
135
+ "state_after": "no goals",
136
+ "state_before": "case refine_1.inr\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx\u271d : A\nhx : x\u271d \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nx : A\nhxT : x \u2208 \u2191T\n\u22a2 1 * x \u2208 toSubmodule S * toSubmodule T",
137
+ "tactic": "exact Submodule.mul_mem_mul (show (1 : A) \u2208 S from one_mem S) hxT"
138
+ },
139
+ {
140
+ "annotated_tactic": [
141
+ "\"rw [\\u2190 <a>one_mul<\\/a> (<a>algebraMap<\\/a> _ _ _)]\"",
142
+ "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
143
+ ],
144
+ "state_after": "case refine_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 1 * (algebraMap R A) r \u2208 toSubmodule S * toSubmodule T",
145
+ "state_before": "case refine_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 (algebraMap R A) r \u2208 toSubmodule S * toSubmodule T",
146
+ "tactic": "rw [\u2190 one_mul (algebraMap _ _ _)]"
147
+ },
148
+ {
149
+ "annotated_tactic": [
150
+ "\"exact <a>Submodule.mul_mem_mul<\\/a> (show (1 : A) \\u2208 S from <a>one_mem<\\/a> S) (<a>algebraMap_mem<\\/a> T _)\"",
151
+ "[{\"full_name\":\"Submodule.mul_mem_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Operations.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,20]},{\"full_name\":\"OneMemClass.one_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]},{\"full_name\":\"algebraMap_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,30]}]"
152
+ ],
153
+ "state_after": "no goals",
154
+ "state_before": "case refine_2\nR\u271d : Type u_1\nA\u271d : Type u_2\ninst\u271d\u2075 : CommSemiring R\u271d\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS T : Subalgebra R A\nx : A\nhx : x \u2208 Algebra.adjoin R (\u2191S \u222a \u2191T)\nr : R\n\u22a2 1 * (algebraMap R A) r \u2208 toSubmodule S * toSubmodule T",
155
+ "tactic": "exact Submodule.mul_mem_mul (show (1 : A) \u2208 S from one_mem S) (algebraMap_mem T _)"
156
+ }
157
+ ],
158
+ "url": "https://github.com/leanprover-community/mathlib4"
159
+ },
160
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
161
+ },
162
+ {
163
+ "error": "theorem has no selected tactic transitions",
164
+ "full_name": "Subalgebra.pointwise_smul_toSubmodule",
165
+ "id": "Subalgebra.pointwise_smul_toSubmodule",
166
+ "reason": "import_failure",
167
+ "selection": {
168
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
169
+ "end": [
170
+ 98,
171
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172
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173
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
174
+ "full_name": "Subalgebra.pointwise_smul_toSubmodule",
175
+ "start": [
176
+ 96,
177
+ 1
178
+ ],
179
+ "traced_tactics": [],
180
+ "url": "https://github.com/leanprover-community/mathlib4"
181
+ },
182
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
183
+ },
184
+ {
185
+ "error": "theorem has no selected tactic transitions",
186
+ "full_name": "Subalgebra.pointwise_smul_toSubring",
187
+ "id": "Subalgebra.pointwise_smul_toSubring",
188
+ "reason": "import_failure",
189
+ "selection": {
190
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
191
+ "end": [
192
+ 105,
193
+ 6
194
+ ],
195
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
196
+ "full_name": "Subalgebra.pointwise_smul_toSubring",
197
+ "start": [
198
+ 102,
199
+ 1
200
+ ],
201
+ "traced_tactics": [],
202
+ "url": "https://github.com/leanprover-community/mathlib4"
203
+ },
204
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
205
+ },
206
+ {
207
+ "error": "theorem has no selected tactic transitions",
208
+ "full_name": "Subalgebra.pointwise_smul_toSubsemiring",
209
+ "id": "Subalgebra.pointwise_smul_toSubsemiring",
210
+ "reason": "import_failure",
211
+ "selection": {
212
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
213
+ "end": [
214
+ 92,
215
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216
+ ],
217
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
218
+ "full_name": "Subalgebra.pointwise_smul_toSubsemiring",
219
+ "start": [
220
+ 90,
221
+ 1
222
+ ],
223
+ "traced_tactics": [],
224
+ "url": "https://github.com/leanprover-community/mathlib4"
225
+ },
226
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
227
+ },
228
+ {
229
+ "error": "theorem has no selected tactic transitions",
230
+ "full_name": "Subalgebra.smul_mem_pointwise_smul",
231
+ "id": "Subalgebra.smul_mem_pointwise_smul",
232
+ "reason": "import_failure",
233
+ "selection": {
234
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
235
+ "end": [
236
+ 109,
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238
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239
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
240
+ "full_name": "Subalgebra.smul_mem_pointwise_smul",
241
+ "start": [
242
+ 108,
243
+ 1
244
+ ],
245
+ "traced_tactics": [],
246
+ "url": "https://github.com/leanprover-community/mathlib4"
247
+ },
248
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean"
249
+ },
250
+ {
251
+ "error": "theorem has no selected tactic transitions",
252
+ "full_name": "Subalgebra.coe_prod",
253
+ "id": "Subalgebra.coe_prod",
254
+ "reason": "import_failure",
255
+ "selection": {
256
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
257
+ "end": [
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+ 38,
259
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260
+ ],
261
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean",
262
+ "full_name": "Subalgebra.coe_prod",
263
+ "start": [
264
+ 37,
265
+ 1
266
+ ],
267
+ "traced_tactics": [],
268
+ "url": "https://github.com/leanprover-community/mathlib4"
269
+ },
270
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean"
271
+ },
272
+ {
273
+ "error": "theorem has no selected tactic transitions",
274
+ "full_name": "Subalgebra.mem_prod",
275
+ "id": "Subalgebra.mem_prod",
276
+ "reason": "import_failure",
277
+ "selection": {
278
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
279
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282
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283
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean",
284
+ "full_name": "Subalgebra.mem_prod",
285
+ "start": [
286
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287
+ 1
288
+ ],
289
+ "traced_tactics": [],
290
+ "url": "https://github.com/leanprover-community/mathlib4"
291
+ },
292
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean"
293
+ },
294
+ {
295
+ "error": "theorem has no selected tactic transitions",
296
+ "full_name": "Subalgebra.prod_inf_prod",
297
+ "id": "Subalgebra.prod_inf_prod",
298
+ "reason": "import_failure",
299
+ "selection": {
300
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
301
+ "end": [
302
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304
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305
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean",
306
+ "full_name": "Subalgebra.prod_inf_prod",
307
+ "start": [
308
+ 60,
309
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310
+ ],
311
+ "traced_tactics": [],
312
+ "url": "https://github.com/leanprover-community/mathlib4"
313
+ },
314
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean"
315
+ },
316
+ {
317
+ "error": "theorem has no selected tactic transitions",
318
+ "full_name": "Subalgebra.prod_mono",
319
+ "id": "Subalgebra.prod_mono",
320
+ "reason": "import_failure",
321
+ "selection": {
322
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
323
+ "end": [
324
+ 56,
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326
+ ],
327
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean",
328
+ "full_name": "Subalgebra.prod_mono",
329
+ "start": [
330
+ 54,
331
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332
+ ],
333
+ "traced_tactics": [],
334
+ "url": "https://github.com/leanprover-community/mathlib4"
335
+ },
336
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean"
337
+ },
338
+ {
339
+ "error": "theorem has no selected tactic transitions",
340
+ "full_name": "Subalgebra.prod_toSubmodule",
341
+ "id": "Subalgebra.prod_toSubmodule",
342
+ "reason": "import_failure",
343
+ "selection": {
344
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
345
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346
+ 42,
347
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348
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349
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350
+ "full_name": "Subalgebra.prod_toSubmodule",
351
+ "start": [
352
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353
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354
+ ],
355
+ "traced_tactics": [],
356
+ "url": "https://github.com/leanprover-community/mathlib4"
357
+ },
358
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Prod.lean"
359
+ }
360
+ ],
361
+ "input_proofs": 14,
362
+ "input_units": 2,
363
+ "proofs": 3,
364
+ "published_proofs": 2,
365
+ "published_transitions": 6,
366
+ "replay_failures": [],
367
+ "transitions": 12,
368
+ "validation_failures": [
369
+ {
370
+ "errors": [
371
+ "internal transition boundary mismatch",
372
+ "internal transition boundary mismatch"
373
+ ],
374
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
375
+ "proof_id": "mathlib:Subalgebra.mul_self",
376
+ "provenance": {
377
+ "selection": {
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+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
379
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382
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383
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean",
384
+ "full_name": "Subalgebra.mul_self",
385
+ "start": [
386
+ 37,
387
+ 1
388
+ ],
389
+ "traced_tactics": [
390
+ {
391
+ "annotated_tactic": [
392
+ "\"apply <a>le_antisymm<\\/a>\"",
393
+ "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]}]"
394
+ ],
395
+ "state_after": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule S * toSubmodule S \u2264 toSubmodule S\n\ncase a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule S \u2264 toSubmodule S * toSubmodule S",
396
+ "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule S * toSubmodule S = toSubmodule S",
397
+ "tactic": "apply le_antisymm"
398
+ },
399
+ {
400
+ "annotated_tactic": [
401
+ "\"refine (<a>mul_toSubmodule_le<\\/a> _ _).<a>trans_eq<\\/a> ?_\"",
402
+ "[{\"full_name\":\"Subalgebra.mul_toSubmodule_le\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Pointwise.lean\",\"def_pos\":[27,9],\"def_end_pos\":[27,27]},{\"full_name\":\"LE.le.trans_eq\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[190,7],\"def_end_pos\":[190,21]}]"
403
+ ],
404
+ "state_after": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule (S \u2294 S) = toSubmodule S",
405
+ "state_before": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule S * toSubmodule S \u2264 toSubmodule S",
406
+ "tactic": "refine (mul_toSubmodule_le _ _).trans_eq ?_"
407
+ },
408
+ {
409
+ "annotated_tactic": [
410
+ "\"rw [<a>sup_idem<\\/a>]\"",
411
+ "[{\"full_name\":\"sup_idem\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[219,9],\"def_end_pos\":[219,17]}]"
412
+ ],
413
+ "state_after": "no goals",
414
+ "state_before": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule (S \u2294 S) = toSubmodule S",
415
+ "tactic": "rw [sup_idem]"
416
+ },
417
+ {
418
+ "annotated_tactic": [
419
+ "\"intro x hx1\"",
420
+ "[]"
421
+ ],
422
+ "state_after": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x \u2208 toSubmodule S\n\u22a2 x \u2208 toSubmodule S * toSubmodule S",
423
+ "state_before": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\n\u22a2 toSubmodule S \u2264 toSubmodule S * toSubmodule S",
424
+ "tactic": "intro x hx1"
425
+ },
426
+ {
427
+ "annotated_tactic": [
428
+ "\"rw [\\u2190 <a>mul_one<\\/a> x]\"",
429
+ "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
430
+ ],
431
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432
+ "state_before": "case a\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\nx : A\nhx1 : x \u2208 toSubmodule S\n\u22a2 x \u2208 toSubmodule S * toSubmodule S",
433
+ "tactic": "rw [\u2190 mul_one x]"
434
+ },
435
+ {
436
+ "annotated_tactic": [
437
+ "\"exact <a>Submodule.mul_mem_mul<\\/a> hx1 (show (1 : A) \\u2208 S from <a>one_mem<\\/a> S)\"",
438
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439
+ ],
440
+ "state_after": "no goals",
441
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442
+ "tactic": "exact Submodule.mul_mem_mul hx1 (show (1 : A) \u2208 S from one_mem S)"
443
+ }
444
+ ],
445
+ "url": "https://github.com/leanprover-community/mathlib4"
446
+ }
447
+ },
448
+ "theorem_name": "Subalgebra.mul_self"
449
+ }
450
+ ],
451
+ "validation_rejected_proofs": 1,
452
+ "zero_transition_proofs": 0
453
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454
+ "shards": [
455
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456
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457
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+ "rows": 6,
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+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
461
+ }
462
+ ],
463
+ "proof_ids": [
464
+ "mathlib:Subalgebra.mul_toSubmodule_le",
465
+ "mathlib:Subalgebra.prod_top"
466
+ ],
467
+ "proof_keys": [
468
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean\",null,null,\"Subalgebra.mul_toSubmodule_le\"]",
469
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Prod.lean\",null,null,\"Subalgebra.prod_top\"]"
470
+ ]
471
+ }
manifests/batches/mathlib/000035.json ADDED
@@ -0,0 +1,366 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
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+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "Subalgebra.finrank_left_dvd_finrank_sup_of_free",
12
+ "id": "Subalgebra.finrank_left_dvd_finrank_sup_of_free",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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20
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean",
21
+ "full_name": "Subalgebra.finrank_left_dvd_finrank_sup_of_free",
22
+ "start": [
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+ 55,
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+ "traced_tactics": [],
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+ },
29
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "Subalgebra.finrank_right_dvd_finrank_sup_of_free",
34
+ "id": "Subalgebra.finrank_right_dvd_finrank_sup_of_free",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 59,
40
+ 98
41
+ ],
42
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean",
43
+ "full_name": "Subalgebra.finrank_right_dvd_finrank_sup_of_free",
44
+ "start": [
45
+ 58,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean"
52
+ },
53
+ {
54
+ "error": "no internal trace for tactic 'haveI := Module.subsingleton R S'",
55
+ "full_name": "Subalgebra.rank_sup_eq_rank_left_mul_rank_of_free",
56
+ "id": "Subalgebra.rank_sup_eq_rank_left_mul_rank_of_free",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 41,
62
+ 43
63
+ ],
64
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean",
65
+ "full_name": "Subalgebra.rank_sup_eq_rank_left_mul_rank_of_free",
66
+ "start": [
67
+ 30,
68
+ 1
69
+ ],
70
+ "traced_tactics": [
71
+ {
72
+ "annotated_tactic": [
73
+ "\"rcases <a>subsingleton_or_nontrivial<\\/a> R with _ | _\"",
74
+ "[{\"full_name\":\"subsingleton_or_nontrivial\",\"def_path\":\"Mathlib\\/Logic\\/Nontrivial\\/Defs.lean\",\"def_pos\":[99,9],\"def_end_pos\":[99,35]}]"
75
+ ],
76
+ "state_after": "case inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Subsingleton R\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
77
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
78
+ "tactic": "rcases subsingleton_or_nontrivial R with _ | _"
79
+ },
80
+ {
81
+ "annotated_tactic": [
82
+ "\"nontriviality S using <a>rank_subsingleton'<\\/a>\"",
83
+ "[{\"full_name\":\"rank_subsingleton'\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Finite.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,27]}]"
84
+ ],
85
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
86
+ "state_before": "case inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
87
+ "tactic": "nontriviality S using rank_subsingleton'"
88
+ },
89
+ {
90
+ "annotated_tactic": [
91
+ "\"letI : <a>Algebra<\\/a> A (<a>Algebra.adjoin<\\/a> A (B : <a>Set<\\/a> S)) := <a>Subalgebra.algebra<\\/a> _\"",
92
+ "[{\"full_name\":\"Algebra\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[101,7],\"def_end_pos\":[101,14]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Subalgebra.algebra\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[338,10],\"def_end_pos\":[338,17]}]"
93
+ ],
94
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
95
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
96
+ "tactic": "letI : Algebra A (Algebra.adjoin A (B : Set S)) := Subalgebra.algebra _"
97
+ },
98
+ {
99
+ "annotated_tactic": [
100
+ "\"letI : <a>SMul<\\/a> A (<a>Algebra.adjoin<\\/a> A (B : <a>Set<\\/a> S)) := Algebra.toSMul\"",
101
+ "[{\"full_name\":\"SMul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[102,7],\"def_end_pos\":[102,11]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
102
+ ],
103
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
104
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
105
+ "tactic": "letI : SMul A (Algebra.adjoin A (B : Set S)) := Algebra.toSMul"
106
+ },
107
+ {
108
+ "annotated_tactic": [
109
+ "\"haveI : <a>IsScalarTower<\\/a> R A (<a>Algebra.adjoin<\\/a> A (B : <a>Set<\\/a> S)) :=\\n <a>IsScalarTower.of_algebraMap_eq<\\/a> (<a>congrFun<\\/a> <a>rfl<\\/a>)\"",
110
+ "[{\"full_name\":\"IsScalarTower\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[249,7],\"def_end_pos\":[249,20]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"IsScalarTower.of_algebraMap_eq\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Tower.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,25]},{\"full_name\":\"congrFun\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[376,9],\"def_end_pos\":[376,17]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
111
+ ],
112
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
113
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
114
+ "tactic": "haveI : IsScalarTower R A (Algebra.adjoin A (B : Set S)) :=\n IsScalarTower.of_algebraMap_eq (congrFun rfl)"
115
+ },
116
+ {
117
+ "annotated_tactic": [
118
+ "\"rw [<a>rank_mul_rank<\\/a> R A (<a>Algebra.adjoin<\\/a> A (B : <a>Set<\\/a> S))]\"",
119
+ "[{\"full_name\":\"rank_mul_rank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Free.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,22]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
120
+ ],
121
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(Algebra.adjoin \u21a5A \u2191B)",
122
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
123
+ "tactic": "rw [rank_mul_rank R A (Algebra.adjoin A (B : Set S))]"
124
+ },
125
+ {
126
+ "annotated_tactic": [
127
+ "\"change _ = <a>Module.rank<\\/a> R ((<a>Algebra.adjoin<\\/a> A (B : <a>Set<\\/a> S)).<a>restrictScalars<\\/a> R)\"",
128
+ "[{\"full_name\":\"Module.rank\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Basic.lean\",\"def_pos\":[49,1],\"def_end_pos\":[61,67]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Subalgebra.restrictScalars\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Tower.lean\",\"def_pos\":[84,5],\"def_end_pos\":[84,20]}]"
129
+ ],
130
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(restrictScalars R (Algebra.adjoin \u21a5A \u2191B))",
131
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(Algebra.adjoin \u21a5A \u2191B)",
132
+ "tactic": "change _ = Module.rank R ((Algebra.adjoin A (B : Set S)).restrictScalars R)"
133
+ },
134
+ {
135
+ "annotated_tactic": [
136
+ "\"rw [<a>Algebra.restrictScalars_adjoin<\\/a>]\"",
137
+ "[{\"full_name\":\"Algebra.restrictScalars_adjoin\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[538,9],\"def_end_pos\":[538,39]}]"
138
+ ],
139
+ "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(Algebra.adjoin R (\u2191A \u222a \u2191B))",
140
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(restrictScalars R (Algebra.adjoin \u21a5A \u2191B))",
141
+ "tactic": "rw [Algebra.restrictScalars_adjoin]"
142
+ },
143
+ {
144
+ "annotated_tactic": [
145
+ "\"rfl\"",
146
+ "[]"
147
+ ],
148
+ "state_after": "no goals",
149
+ "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Nontrivial R\na\u271d : Nontrivial S\nthis\u271d\u00b9 : Algebra \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := (Algebra.adjoin \u21a5A \u2191B).algebra\nthis\u271d : SMul \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B) := Algebra.toSMul\nthis : IsScalarTower R \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5(Algebra.adjoin R (\u2191A \u222a \u2191B))",
150
+ "tactic": "rfl"
151
+ },
152
+ {
153
+ "annotated_tactic": [
154
+ "\"haveI := <a>Module.subsingleton<\\/a> R S\"",
155
+ "[{\"full_name\":\"Module.subsingleton\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[303,19],\"def_end_pos\":[303,38]}]"
156
+ ],
157
+ "state_after": "case inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Subsingleton R\nthis : Subsingleton S\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
158
+ "state_before": "case inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Subsingleton R\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
159
+ "tactic": "haveI := Module.subsingleton R S"
160
+ },
161
+ {
162
+ "annotated_tactic": [
163
+ "\"simp\"",
164
+ "[]"
165
+ ],
166
+ "state_after": "no goals",
167
+ "state_before": "case inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nA B : Subalgebra R S\ninst\u271d\u00b3 : Module.Free R \u21a5A\ninst\u271d\u00b2 : Module.Free R \u21a5B\ninst\u271d\u00b9 : Module.Free \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)\ninst\u271d : Module.Free \u21a5B \u21a5(Algebra.adjoin \u21a5B \u2191A)\nh\u271d : Subsingleton R\nthis : Subsingleton S\n\u22a2 Module.rank R \u21a5(A \u2294 B) = Module.rank R \u21a5A * Module.rank \u21a5A \u21a5(Algebra.adjoin \u21a5A \u2191B)",
168
+ "tactic": "simp"
169
+ }
170
+ ],
171
+ "url": "https://github.com/leanprover-community/mathlib4"
172
+ },
173
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Rank.lean"
174
+ },
175
+ {
176
+ "error": "theorem has no selected tactic transitions",
177
+ "full_name": "Algebra.lmul_algebraMap",
178
+ "id": "Algebra.lmul_algebraMap",
179
+ "reason": "import_failure",
180
+ "selection": {
181
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
182
+ "end": [
183
+ 45,
184
+ 41
185
+ ],
186
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean",
187
+ "full_name": "Algebra.lmul_algebraMap",
188
+ "start": [
189
+ 44,
190
+ 1
191
+ ],
192
+ "traced_tactics": [],
193
+ "url": "https://github.com/leanprover-community/mathlib4"
194
+ },
195
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean"
196
+ },
197
+ {
198
+ "error": "no internal trace for tactic 'rw [this]'",
199
+ "full_name": "IsScalarTower.adjoin_range_toAlgHom",
200
+ "id": "IsScalarTower.adjoin_range_toAlgHom",
201
+ "reason": "import_failure",
202
+ "selection": {
203
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
204
+ "end": [
205
+ 158,
206
+ 83
207
+ ],
208
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean",
209
+ "full_name": "IsScalarTower.adjoin_range_toAlgHom",
210
+ "start": [
211
+ 149,
212
+ 1
213
+ ],
214
+ "traced_tactics": [
215
+ {
216
+ "annotated_tactic": [
217
+ "\"suffices <a>Set.range<\\/a> (<a>algebraMap<\\/a> (<a>toAlgHom<\\/a> R S A).<a>range<\\/a> A) = <a>Set.range<\\/a> (<a>algebraMap<\\/a> S A) by\\n rw [this]\"",
218
+ "[{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"IsScalarTower.toAlgHom\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Tower.lean\",\"def_pos\":[143,5],\"def_end_pos\":[143,13]},{\"full_name\":\"AlgHom.range\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[592,15],\"def_end_pos\":[592,20]},{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
219
+ ],
220
+ "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz : A\n\u22a2 Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) = Set.range \u21d1(algebraMap S A)",
221
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz : A\n\u22a2 z \u2208 Subsemiring.closure (Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) \u222a t) \u2194\n z \u2208 Subsemiring.closure (Set.range \u21d1(algebraMap S A) \u222a t)",
222
+ "tactic": "suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n rw [this]"
223
+ },
224
+ {
225
+ "annotated_tactic": [
226
+ "\"ext z\"",
227
+ "[]"
228
+ ],
229
+ "state_after": "case h\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz\u271d z : A\n\u22a2 z \u2208 Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) \u2194 z \u2208 Set.range \u21d1(algebraMap S A)",
230
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz : A\n\u22a2 Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) = Set.range \u21d1(algebraMap S A)",
231
+ "tactic": "ext z"
232
+ },
233
+ {
234
+ "annotated_tactic": [
235
+ "\"exact \\u27e8fun \\u27e8\\u27e8_, y, h1\\u27e9, h2\\u27e9 \\u21a6 \\u27e8y, h2 \\u25b8 h1\\u27e9, fun \\u27e8y, hy\\u27e9 \\u21a6 \\u27e8\\u27e8z, y, hy\\u27e9, <a>rfl<\\/a>\\u27e9\\u27e9\"",
236
+ "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
237
+ ],
238
+ "state_after": "no goals",
239
+ "state_before": "case h\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz\u271d z : A\n\u22a2 z \u2208 Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) \u2194 z \u2208 Set.range \u21d1(algebraMap S A)",
240
+ "tactic": "exact \u27e8fun \u27e8\u27e8_, y, h1\u27e9, h2\u27e9 \u21a6 \u27e8y, h2 \u25b8 h1\u27e9, fun \u27e8y, hy\u27e9 \u21a6 \u27e8\u27e8z, y, hy\u27e9, rfl\u27e9\u27e9"
241
+ },
242
+ {
243
+ "annotated_tactic": [
244
+ "\"rw [this]\"",
245
+ "[]"
246
+ ],
247
+ "state_after": "no goals",
248
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nt : Set A\nz : A\nthis : Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) = Set.range \u21d1(algebraMap S A)\n\u22a2 z \u2208 Subsemiring.closure (Set.range \u21d1(algebraMap (\u21a5(toAlgHom R S A).range) A) \u222a t) \u2194\n z \u2208 Subsemiring.closure (Set.range \u21d1(algebraMap S A) \u222a t)",
249
+ "tactic": "rw [this]"
250
+ }
251
+ ],
252
+ "url": "https://github.com/leanprover-community/mathlib4"
253
+ },
254
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean"
255
+ },
256
+ {
257
+ "error": "theorem has no selected tactic transitions",
258
+ "full_name": "Subalgebra.coe_restrictScalars",
259
+ "id": "Subalgebra.coe_restrictScalars",
260
+ "reason": "import_failure",
261
+ "selection": {
262
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
263
+ "end": [
264
+ 93,
265
+ 6
266
+ ],
267
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean",
268
+ "full_name": "Subalgebra.coe_restrictScalars",
269
+ "start": [
270
+ 92,
271
+ 1
272
+ ],
273
+ "traced_tactics": [],
274
+ "url": "https://github.com/leanprover-community/mathlib4"
275
+ },
276
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean"
277
+ },
278
+ {
279
+ "error": "theorem has no selected tactic transitions",
280
+ "full_name": "Subalgebra.mem_restrictScalars",
281
+ "id": "Subalgebra.mem_restrictScalars",
282
+ "reason": "import_failure",
283
+ "selection": {
284
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
285
+ "end": [
286
+ 109,
287
+ 10
288
+ ],
289
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean",
290
+ "full_name": "Subalgebra.mem_restrictScalars",
291
+ "start": [
292
+ 108,
293
+ 1
294
+ ],
295
+ "traced_tactics": [],
296
+ "url": "https://github.com/leanprover-community/mathlib4"
297
+ },
298
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean"
299
+ },
300
+ {
301
+ "error": "theorem has no selected tactic transitions",
302
+ "full_name": "Subalgebra.restrictScalars_toSubmodule",
303
+ "id": "Subalgebra.restrictScalars_toSubmodule",
304
+ "reason": "import_failure",
305
+ "selection": {
306
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
307
+ "end": [
308
+ 104,
309
+ 28
310
+ ],
311
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean",
312
+ "full_name": "Subalgebra.restrictScalars_toSubmodule",
313
+ "start": [
314
+ 102,
315
+ 1
316
+ ],
317
+ "traced_tactics": [],
318
+ "url": "https://github.com/leanprover-community/mathlib4"
319
+ },
320
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean"
321
+ }
322
+ ],
323
+ "input_proofs": 14,
324
+ "input_units": 2,
325
+ "proofs": 6,
326
+ "published_proofs": 6,
327
+ "published_transitions": 8,
328
+ "replay_failures": [],
329
+ "transitions": 8,
330
+ "validation_failures": [],
331
+ "validation_rejected_proofs": 0,
332
+ "zero_transition_proofs": 0
333
+ },
334
+ "shards": [
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+ {
336
+ "path": "data/train/mathlib-000035-0000-ef5b52f29c3e225e.parquet",
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+ "sha256": "ef5b52f29c3e225e76dd7db347b514a06c0a9baa9de8c240169619bf3b7cdf46",
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297
+ "state_before": "R : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\n\u22a2 (lift f).range \u2264 S \u2194 NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra",
298
+ "tactic": "refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9"
299
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300
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301
+ "annotated_tactic": [
302
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303
+ "[]"
304
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305
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306
+ "state_before": "case refine_1\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : (lift f).range \u2264 S\n\u22a2 NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra",
307
+ "tactic": "rintro - \u27e8x, rfl\u27e9"
308
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309
+ {
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+ "annotated_tactic": [
311
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312
+ "[]"
313
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314
+ "state_after": "no goals",
315
+ "state_before": "case refine_1.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : (lift f).range \u2264 S\nx : A\n\u22a2 \u2191f x \u2208 S.toNonUnitalSubalgebra",
316
+ "tactic": "exact @h (f x) \u27e8x, by simp\u27e9"
317
+ },
318
+ {
319
+ "annotated_tactic": [
320
+ "\"simp\"",
321
+ "[]"
322
+ ],
323
+ "state_after": "no goals",
324
+ "state_before": "R : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : (lift f).range \u2264 S\nx : A\n\u22a2 (lift f).toRingHom \u2191x = f x",
325
+ "tactic": "simp"
326
+ },
327
+ {
328
+ "annotated_tactic": [
329
+ "\"rintro - \\u27e8x, rfl\\u27e9\"",
330
+ "[]"
331
+ ],
332
+ "state_after": "case refine_2.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra\nx : Unitization R A\n\u22a2 (lift f).toRingHom x \u2208 S",
333
+ "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra\n\u22a2 (lift f).range \u2264 S",
334
+ "tactic": "rintro - \u27e8x, rfl\u27e9"
335
+ },
336
+ {
337
+ "annotated_tactic": [
338
+ "\"induction x with\\n | _ r a => simpa using <a>add_mem<\\/a> (<a>algebraMap_mem<\\/a> S r) (h \\u27e8a, <a>rfl<\\/a>\\u27e9)\"",
339
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]},{\"full_name\":\"algebraMap_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,30]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
340
+ ],
341
+ "state_after": "no goals",
342
+ "state_before": "case refine_2.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra\nx : Unitization R A\n\u22a2 (lift f).toRingHom x \u2208 S",
343
+ "tactic": "induction x with\n| _ r a => simpa using add_mem (algebraMap_mem S r) (h \u27e8a, rfl\u27e9)"
344
+ },
345
+ {
346
+ "annotated_tactic": [
347
+ "\"simpa using <a>add_mem<\\/a> (<a>algebraMap_mem<\\/a> S r) (h \\u27e8a, <a>rfl<\\/a>\\u27e9)\"",
348
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]},{\"full_name\":\"algebraMap_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,30]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
349
+ ],
350
+ "state_after": "no goals",
351
+ "state_before": "case refine_2.intro.inl_add_inr\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : NonUnitalSemiring A\ninst\u271d\u2074 : Module R A\ninst\u271d\u00b3 : SMulCommClass R A A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\nf : A \u2192\u2099\u2090[R] C\nS : Subalgebra R C\nh : NonUnitalAlgHom.range f \u2264 S.toNonUnitalSubalgebra\nr : R\na : A\n\u22a2 (lift f).toRingHom (inl r + \u2191a) \u2208 S",
352
+ "tactic": "simpa using add_mem (algebraMap_mem S r) (h \u27e8a, rfl\u27e9)"
353
+ }
354
+ ],
355
+ "url": "https://github.com/leanprover-community/mathlib4"
356
+ },
357
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean"
358
+ },
359
+ {
360
+ "error": "no internal trace for tactic 'exact @h (f x) \u27e8x, by simp\u27e9'",
361
+ "full_name": "Unitization.starLift_range_le",
362
+ "id": "Unitization.starLift_range_le",
363
+ "reason": "import_failure",
364
+ "selection": {
365
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
366
+ "end": [
367
+ 364,
368
+ 69
369
+ ],
370
+ "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean",
371
+ "full_name": "Unitization.starLift_range_le",
372
+ "start": [
373
+ 356,
374
+ 1
375
+ ],
376
+ "traced_tactics": [
377
+ {
378
+ "annotated_tactic": [
379
+ "\"refine \\u27e8fun h \\u21a6 ?_, fun h \\u21a6 ?_\\u27e9\"",
380
+ "[]"
381
+ ],
382
+ "state_after": "case refine_1\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : (starLift f).range \u2264 S\n\u22a2 NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\n\ncase refine_2\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\n\u22a2 (starLift f).range \u2264 S",
383
+ "state_before": "R : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\n\u22a2 (starLift f).range \u2264 S \u2194 NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra",
384
+ "tactic": "refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9"
385
+ },
386
+ {
387
+ "annotated_tactic": [
388
+ "\"rintro - \\u27e8x, rfl\\u27e9\"",
389
+ "[]"
390
+ ],
391
+ "state_after": "case refine_1.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : (starLift f).range \u2264 S\nx : A\n\u22a2 \u2191(NonUnitalAlgHomClass.toNonUnitalAlgHom f) x \u2208 S.toNonUnitalStarSubalgebra",
392
+ "state_before": "case refine_1\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : (starLift f).range \u2264 S\n\u22a2 NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra",
393
+ "tactic": "rintro - \u27e8x, rfl\u27e9"
394
+ },
395
+ {
396
+ "annotated_tactic": [
397
+ "\"exact @h (f x) \\u27e8x, by simp\\u27e9\"",
398
+ "[]"
399
+ ],
400
+ "state_after": "no goals",
401
+ "state_before": "case refine_1.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : (starLift f).range \u2264 S\nx : A\n\u22a2 \u2191(NonUnitalAlgHomClass.toNonUnitalAlgHom f) x \u2208 S.toNonUnitalStarSubalgebra",
402
+ "tactic": "exact @h (f x) \u27e8x, by simp\u27e9"
403
+ },
404
+ {
405
+ "annotated_tactic": [
406
+ "\"simp\"",
407
+ "[]"
408
+ ],
409
+ "state_after": "no goals",
410
+ "state_before": "R : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : (starLift f).range \u2264 S\nx : A\n\u22a2 (starLift f).toRingHom \u2191x = f x",
411
+ "tactic": "simp"
412
+ },
413
+ {
414
+ "annotated_tactic": [
415
+ "\"rintro - \\u27e8x, rfl\\u27e9\"",
416
+ "[]"
417
+ ],
418
+ "state_after": "case refine_2.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\nx : Unitization R A\n\u22a2 (starLift f).toRingHom x \u2208 S",
419
+ "state_before": "case refine_2\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\n\u22a2 (starLift f).range \u2264 S",
420
+ "tactic": "rintro - \u27e8x, rfl\u27e9"
421
+ },
422
+ {
423
+ "annotated_tactic": [
424
+ "\"induction x with\\n | _ r a => simpa using <a>add_mem<\\/a> (<a>algebraMap_mem<\\/a> S r) (h \\u27e8a, <a>rfl<\\/a>\\u27e9)\"",
425
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]},{\"full_name\":\"algebraMap_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,30]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
426
+ ],
427
+ "state_after": "no goals",
428
+ "state_before": "case refine_2.intro\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\nx : Unitization R A\n\u22a2 (starLift f).toRingHom x \u2208 S",
429
+ "tactic": "induction x with\n| _ r a => simpa using add_mem (algebraMap_mem S r) (h \u27e8a, rfl\u27e9)"
430
+ },
431
+ {
432
+ "annotated_tactic": [
433
+ "\"simpa using <a>add_mem<\\/a> (<a>algebraMap_mem<\\/a> S r) (h \\u27e8a, <a>rfl<\\/a>\\u27e9)\"",
434
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]},{\"full_name\":\"algebraMap_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,30]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
435
+ ],
436
+ "state_after": "no goals",
437
+ "state_before": "case refine_2.intro.inl_add_inr\nR : Type u_1\nA : Type u_2\nC : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : NonUnitalSemiring A\ninst\u271d\u2079 : StarRing R\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : Module R A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : Semiring C\ninst\u271d\u00b2 : StarRing C\ninst\u271d\u00b9 : Algebra R C\ninst\u271d : StarModule R C\nf : A \u2192\u22c6\u2099\u2090[R] C\nS : StarSubalgebra R C\nh : NonUnitalStarAlgHom.range f \u2264 S.toNonUnitalStarSubalgebra\nr : R\na : A\n\u22a2 (starLift f).toRingHom (inl r + \u2191a) \u2208 S",
438
+ "tactic": "simpa using add_mem (algebraMap_mem S r) (h \u27e8a, rfl\u27e9)"
439
+ }
440
+ ],
441
+ "url": "https://github.com/leanprover-community/mathlib4"
442
+ },
443
+ "source_file": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean"
444
+ },
445
+ {
446
+ "error": "theorem has no selected tactic transitions",
447
+ "full_name": "AlgEquiv.coe_restrictScalars",
448
+ "id": "AlgEquiv.coe_restrictScalars",
449
+ "reason": "import_failure",
450
+ "selection": {
451
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
452
+ "end": [
453
+ 244,
454
+ 89
455
+ ],
456
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
457
+ "full_name": "AlgEquiv.coe_restrictScalars",
458
+ "start": [
459
+ 244,
460
+ 1
461
+ ],
462
+ "traced_tactics": [],
463
+ "url": "https://github.com/leanprover-community/mathlib4"
464
+ },
465
+ "source_file": "Mathlib/Algebra/Algebra/Tower.lean"
466
+ },
467
+ {
468
+ "error": "theorem has no selected tactic transitions",
469
+ "full_name": "AlgEquiv.coe_restrictScalars'",
470
+ "id": "AlgEquiv.coe_restrictScalars'",
471
+ "reason": "import_failure",
472
+ "selection": {
473
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
474
+ "end": [
475
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476
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556
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644
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663
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667
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686
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688
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689
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691
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710
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711
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712
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713
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714
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752
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754
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774
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798
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850
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851
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852
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853
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854
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855
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856
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857
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864
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865
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866
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867
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868
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870
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871
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872
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873
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874
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875
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876
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877
+ "annotated_tactic": [
878
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879
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880
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881
+ "state_after": "case inl_add_inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhx : (algebraMap R A) r = -\u2191a\n\u22a2 Unitization.inl r = -\u2191a",
882
+ "state_before": "case inl_add_inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhx : (algebraMap R A) r + \u2191a = 0\n\u22a2 Unitization.inl r + \u2191a = 0",
883
+ "tactic": "rw [add_eq_zero_iff_eq_neg] at hx \u22a2"
884
+ },
885
+ {
886
+ "annotated_tactic": [
887
+ "\"by_cases hr : r = 0\"",
888
+ "[]"
889
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890
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891
+ "state_before": "case inl_add_inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhx : (algebraMap R A) r = -\u2191a\n\u22a2 Unitization.inl r = -\u2191a",
892
+ "tactic": "by_cases hr : r = 0"
893
+ },
894
+ {
895
+ "annotated_tactic": [
896
+ "\"ext <;> simp [hr] at hx \\u22a2\"",
897
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898
+ ],
899
+ "state_after": "case pos.h2.a\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhr : r = 0\nhx : a = 0\n\u22a2 a = 0",
900
+ "state_before": "case pos\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhx : (algebraMap R A) r = -\u2191a\nhr : r = 0\n\u22a2 Unitization.inl r = -\u2191a",
901
+ "tactic": "ext <;> simp [hr] at hx \u22a2"
902
+ },
903
+ {
904
+ "annotated_tactic": [
905
+ "\"exact hx\"",
906
+ "[]"
907
+ ],
908
+ "state_after": "no goals",
909
+ "state_before": "case pos.h2.a\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhr : r = 0\nhx : a = 0\n\u22a2 a = 0",
910
+ "tactic": "exact hx"
911
+ },
912
+ {
913
+ "annotated_tactic": [
914
+ "\"exact (h r hr <| hx \\u25b8 (<a>neg_mem<\\/a> a.property)).<a>elim<\\/a>\"",
915
+ "[{\"full_name\":\"NegMemClass.neg_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subgroup\\/Basic.lean\",\"def_pos\":[108,3],\"def_end_pos\":[108,10]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
916
+ ],
917
+ "state_after": "no goals",
918
+ "state_before": "case neg\nF : Type u_1\nR : Type u_2\nS : Type u_3\nA : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : SetLike S A\nhSA : NonUnitalSubringClass S A\nhSRA : SMulMemClass S R A\ns : S\nh : \u2200 (r : R), r \u2260 0 \u2192 (algebraMap R A) r \u2209 s\ninst\u271d\u00b9 : FunLike F (Unitization R \u21a5s) A\ninst\u271d : AlgHomClass F R (Unitization R \u21a5s) A\nf : F\nhf : \u2200 (x : \u21a5s), f \u2191x = \u2191x\nr : R\na : \u21a5s\nhx : (algebraMap R A) r = -\u2191a\nhr : \u00acr = 0\n\u22a2 Unitization.inl r = -\u2191a",
919
+ "tactic": "exact (h r hr <| hx \u25b8 (neg_mem a.property)).elim"
920
+ }
921
+ ],
922
+ "url": "https://github.com/leanprover-community/mathlib4"
923
+ }
924
+ },
925
+ "theorem_name": "AlgHomClass.unitization_injective'"
926
+ },
927
+ {
928
+ "errors": [
929
+ "internal transition boundary mismatch"
930
+ ],
931
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
932
+ "proof_id": "mathlib:IsScalarTower.Algebra.ext",
933
+ "provenance": {
934
+ "selection": {
935
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
936
+ "end": [
937
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938
+ 97
939
+ ],
940
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
941
+ "full_name": "IsScalarTower.Algebra.ext",
942
+ "start": [
943
+ 135,
944
+ 1
945
+ ],
946
+ "traced_tactics": [
947
+ {
948
+ "annotated_tactic": [
949
+ "\"have I := h1\"",
950
+ "[]"
951
+ ],
952
+ "state_after": "R : Type u\nS\u271d : Type v\nA\u271d : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : CommSemiring S\u271d\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R S\u271d\ninst\u271d\u2077 : Algebra S\u271d A\u271d\ninst\u271d\u2076 : Algebra S\u271d B\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : IsScalarTower R S\u271d A\u271d\ninst\u271d\u00b2 : IsScalarTower R S\u271d B\nS : Type u\nA : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n\u22a2 A",
953
+ "state_before": "R : Type u\nS\u271d : Type v\nA\u271d : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : CommSemiring S\u271d\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R S\u271d\ninst\u271d\u2077 : Algebra S\u271d A\u271d\ninst\u271d\u2076 : Algebra S\u271d B\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : IsScalarTower R S\u271d A\u271d\ninst\u271d\u00b2 : IsScalarTower R S\u271d B\nS : Type u\nA : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\n\u22a2 A",
954
+ "tactic": "have I := h1"
955
+ },
956
+ {
957
+ "annotated_tactic": [
958
+ "\"exact r \\u2022 x\"",
959
+ "[]"
960
+ ],
961
+ "state_after": "no goals",
962
+ "state_before": "R : Type u\nS\u271d : Type v\nA\u271d : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : CommSemiring S\u271d\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R S\u271d\ninst\u271d\u2077 : Algebra S\u271d A\u271d\ninst\u271d\u2076 : Algebra S\u271d B\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : IsScalarTower R S\u271d A\u271d\ninst\u271d\u00b2 : IsScalarTower R S\u271d B\nS : Type u\nA : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Semiring A\nh1 h2 : Algebra S A\nr : S\nx : A\nI : Algebra S A\n\u22a2 A",
963
+ "tactic": "exact r \u2022 x"
964
+ },
965
+ {
966
+ "annotated_tactic": [
967
+ "\"simpa only [@<a>Algebra.smul_def<\\/a> _ _ _ _ h1, @<a>Algebra.smul_def<\\/a> _ _ _ _ h2, <a>mul_one<\\/a>] using h r 1\"",
968
+ "[{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]},{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
969
+ ],
970
+ "state_after": "no goals",
971
+ "state_before": "R : Type u\nS\u271d : Type v\nA\u271d : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : CommSemiring S\u271d\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R S\u271d\ninst\u271d\u2077 : Algebra S\u271d A\u271d\ninst\u271d\u2076 : Algebra S\u271d B\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : IsScalarTower R S\u271d A\u271d\ninst\u271d\u00b2 : IsScalarTower R S\u271d B\nS : Type u\nA : Type v\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Semiring A\nh1 h2 : Algebra S A\nh :\n \u2200 (r : S) (x : A),\n (let_fun I := h1;\n r \u2022 x) =\n r \u2022 x\nr : S\n\u22a2 (algebraMap S A) r = (algebraMap S A) r",
972
+ "tactic": "simpa only [@Algebra.smul_def _ _ _ _ h1, @Algebra.smul_def _ _ _ _ h2, mul_one] using h r 1"
973
+ }
974
+ ],
975
+ "url": "https://github.com/leanprover-community/mathlib4"
976
+ }
977
+ },
978
+ "theorem_name": "IsScalarTower.Algebra.ext"
979
+ },
980
+ {
981
+ "errors": [
982
+ "internal transition boundary mismatch",
983
+ "internal transition boundary mismatch"
984
+ ],
985
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
986
+ "proof_id": "mathlib:Submodule.smul_mem_span_smul_of_mem",
987
+ "provenance": {
988
+ "selection": {
989
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
990
+ "end": [
991
+ 301,
992
+ 74
993
+ ],
994
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
995
+ "full_name": "Submodule.smul_mem_span_smul_of_mem",
996
+ "start": [
997
+ 296,
998
+ 1
999
+ ],
1000
+ "traced_tactics": [
1001
+ {
1002
+ "annotated_tactic": [
1003
+ "\"rw [<a>zero_smul<\\/a>]\"",
1004
+ "[{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
1005
+ ],
1006
+ "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\n\u22a2 0 \u2208 span R (s \u2022 t)",
1007
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\n\u22a2 0 \u2022 x \u2208 span R (s \u2022 t)",
1008
+ "tactic": "rw [zero_smul]"
1009
+ },
1010
+ {
1011
+ "annotated_tactic": [
1012
+ "\"exact <a>zero_mem<\\/a> _\"",
1013
+ "[{\"full_name\":\"ZeroMemClass.zero_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[80,3],\"def_end_pos\":[80,11]}]"
1014
+ ],
1015
+ "state_after": "no goals",
1016
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\n\u22a2 0 \u2208 span R (s \u2022 t)",
1017
+ "tactic": "exact zero_mem _"
1018
+ },
1019
+ {
1020
+ "annotated_tactic": [
1021
+ "\"rw [<a>add_smul<\\/a>]\"",
1022
+ "[{\"full_name\":\"add_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,17]}]"
1023
+ ],
1024
+ "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nc\u2081 c\u2082 : S\nih\u2081 : c\u2081 \u2022 x \u2208 span R (s \u2022 t)\nih\u2082 : c\u2082 \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 c\u2081 \u2022 x + c\u2082 \u2022 x \u2208 span R (s \u2022 t)",
1025
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nc\u2081 c\u2082 : S\nih\u2081 : c\u2081 \u2022 x \u2208 span R (s \u2022 t)\nih\u2082 : c\u2082 \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 (c\u2081 + c\u2082) \u2022 x \u2208 span R (s \u2022 t)",
1026
+ "tactic": "rw [add_smul]"
1027
+ },
1028
+ {
1029
+ "annotated_tactic": [
1030
+ "\"exact <a>add_mem<\\/a> ih\\u2081 ih\\u2082\"",
1031
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]}]"
1032
+ ],
1033
+ "state_after": "no goals",
1034
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nc\u2081 c\u2082 : S\nih\u2081 : c\u2081 \u2022 x \u2208 span R (s \u2022 t)\nih\u2082 : c\u2082 \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 c\u2081 \u2022 x + c\u2082 \u2022 x \u2208 span R (s \u2022 t)",
1035
+ "tactic": "exact add_mem ih\u2081 ih\u2082"
1036
+ },
1037
+ {
1038
+ "annotated_tactic": [
1039
+ "\"rw [<a>IsScalarTower.smul_assoc<\\/a>]\"",
1040
+ "[{\"full_name\":\"IsScalarTower.smul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[251,3],\"def_end_pos\":[251,13]}]"
1041
+ ],
1042
+ "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nb : R\nc : S\nhc : c \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 b \u2022 c \u2022 x \u2208 span R (s \u2022 t)",
1043
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nb : R\nc : S\nhc : c \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 (b \u2022 c) \u2022 x \u2208 span R (s \u2022 t)",
1044
+ "tactic": "rw [IsScalarTower.smul_assoc]"
1045
+ },
1046
+ {
1047
+ "annotated_tactic": [
1048
+ "\"exact <a>smul_mem<\\/a> _ _ hc\"",
1049
+ "[{\"full_name\":\"Submodule.smul_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,17]}]"
1050
+ ],
1051
+ "state_after": "no goals",
1052
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k \u2208 span R s\nx : A\nhx : x \u2208 t\nb : R\nc : S\nhc : c \u2022 x \u2208 span R (s \u2022 t)\n\u22a2 b \u2022 c \u2022 x \u2208 span R (s \u2022 t)",
1053
+ "tactic": "exact smul_mem _ _ hc"
1054
+ }
1055
+ ],
1056
+ "url": "https://github.com/leanprover-community/mathlib4"
1057
+ }
1058
+ },
1059
+ "theorem_name": "Submodule.smul_mem_span_smul_of_mem"
1060
+ },
1061
+ {
1062
+ "errors": [
1063
+ "internal transition boundary mismatch",
1064
+ "internal transition boundary mismatch",
1065
+ "internal transition boundary mismatch"
1066
+ ],
1067
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
1068
+ "proof_id": "mathlib:Submodule.span_smul_of_span_eq_top",
1069
+ "provenance": {
1070
+ "selection": {
1071
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1072
+ "end": [
1073
+ 313,
1074
+ 76
1075
+ ],
1076
+ "file_path": "Mathlib/Algebra/Algebra/Tower.lean",
1077
+ "full_name": "Submodule.span_smul_of_span_eq_top",
1078
+ "start": [
1079
+ 304,
1080
+ 1
1081
+ ],
1082
+ "traced_tactics": [
1083
+ {
1084
+ "annotated_tactic": [
1085
+ "\"refine <a>span_induction<\\/a> (hs \\u25b8 <a>mem_top<\\/a> : s0 \\u2208 <a>span<\\/a> R s)\\n (fun x hx \\u21a6 <a>subset_span<\\/a> \\u27e8x, hx, y, hy, <a>rfl<\\/a>\\u27e9) ?_ ?_ ?_\"",
1086
+ "[{\"full_name\":\"Submodule.span_induction\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[182,9],\"def_end_pos\":[182,23]},{\"full_name\":\"Submodule.mem_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Lattice.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,16]},{\"full_name\":\"Submodule.span\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[51,5],\"def_end_pos\":[51,9]},{\"full_name\":\"Submodule.subset_span\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,20]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
1087
+ ],
1088
+ "state_after": "case refine_1\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 0 \u2022 y \u2208 span R (s \u2022 t)\n\ncase refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 \u2200 (x y_1 : S), x \u2022 y \u2208 span R (s \u2022 t) \u2192 y_1 \u2022 y \u2208 span R (s \u2022 t) \u2192 (x + y_1) \u2022 y \u2208 span R (s \u2022 t)\n\ncase refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 \u2200 (a : R) (x : S), x \u2022 y \u2208 span R (s \u2022 t) \u2192 (a \u2022 x) \u2022 y \u2208 span R (s \u2022 t)",
1089
+ "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 s0 \u2022 y \u2208 span R (s \u2022 t)",
1090
+ "tactic": "refine span_induction (hs \u25b8 mem_top : s0 \u2208 span R s)\n (fun x hx \u21a6 subset_span \u27e8x, hx, y, hy, rfl\u27e9) ?_ ?_ ?_"
1091
+ },
1092
+ {
1093
+ "annotated_tactic": [
1094
+ "\"rw [<a>zero_smul<\\/a>]\"",
1095
+ "[{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
1096
+ ],
1097
+ "state_after": "case refine_1\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 0 \u2208 span R (s \u2022 t)",
1098
+ "state_before": "case refine_1\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 0 \u2022 y \u2208 span R (s \u2022 t)",
1099
+ "tactic": "rw [zero_smul]"
1100
+ },
1101
+ {
1102
+ "annotated_tactic": [
1103
+ "\"apply <a>zero_mem<\\/a>\"",
1104
+ "[{\"full_name\":\"ZeroMemClass.zero_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Basic.lean\",\"def_pos\":[80,3],\"def_end_pos\":[80,11]}]"
1105
+ ],
1106
+ "state_after": "no goals",
1107
+ "state_before": "case refine_1\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 0 \u2208 span R (s \u2022 t)",
1108
+ "tactic": "apply zero_mem"
1109
+ },
1110
+ {
1111
+ "annotated_tactic": [
1112
+ "\"intro _ _\"",
1113
+ "[]"
1114
+ ],
1115
+ "state_after": "case refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\nx\u271d y\u271d : S\n\u22a2 x\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 y\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 (x\u271d + y\u271d) \u2022 y \u2208 span R (s \u2022 t)",
1116
+ "state_before": "case refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 \u2200 (x y_1 : S), x \u2022 y \u2208 span R (s \u2022 t) \u2192 y_1 \u2022 y \u2208 span R (s \u2022 t) \u2192 (x + y_1) \u2022 y \u2208 span R (s \u2022 t)",
1117
+ "tactic": "intro _ _"
1118
+ },
1119
+ {
1120
+ "annotated_tactic": [
1121
+ "\"rw [<a>add_smul<\\/a>]\"",
1122
+ "[{\"full_name\":\"add_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,17]}]"
1123
+ ],
1124
+ "state_after": "case refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\nx\u271d y\u271d : S\n\u22a2 x\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 y\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 x\u271d \u2022 y + y\u271d \u2022 y \u2208 span R (s \u2022 t)",
1125
+ "state_before": "case refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\nx\u271d y\u271d : S\n\u22a2 x\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 y\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 (x\u271d + y\u271d) \u2022 y \u2208 span R (s \u2022 t)",
1126
+ "tactic": "rw [add_smul]"
1127
+ },
1128
+ {
1129
+ "annotated_tactic": [
1130
+ "\"apply <a>add_mem<\\/a>\"",
1131
+ "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]}]"
1132
+ ],
1133
+ "state_after": "no goals",
1134
+ "state_before": "case refine_2\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\nx\u271d y\u271d : S\n\u22a2 x\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 y\u271d \u2022 y \u2208 span R (s \u2022 t) \u2192 x\u271d \u2022 y + y\u271d \u2022 y \u2208 span R (s \u2022 t)",
1135
+ "tactic": "apply add_mem"
1136
+ },
1137
+ {
1138
+ "annotated_tactic": [
1139
+ "\"intro r s0 hy\"",
1140
+ "[]"
1141
+ ],
1142
+ "state_after": "case refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0\u271d : S\ny : A\nhy\u271d : y \u2208 t\nr : R\ns0 : S\nhy : s0 \u2022 y \u2208 span R (s \u2022 t)\n\u22a2 (r \u2022 s0) \u2022 y \u2208 span R (s \u2022 t)",
1143
+ "state_before": "case refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0 : S\ny : A\nhy : y \u2208 t\n\u22a2 \u2200 (a : R) (x : S), x \u2022 y \u2208 span R (s \u2022 t) \u2192 (a \u2022 x) \u2022 y \u2208 span R (s \u2022 t)",
1144
+ "tactic": "intro r s0 hy"
1145
+ },
1146
+ {
1147
+ "annotated_tactic": [
1148
+ "\"rw [<a>IsScalarTower.smul_assoc<\\/a>]\"",
1149
+ "[{\"full_name\":\"IsScalarTower.smul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[251,3],\"def_end_pos\":[251,13]}]"
1150
+ ],
1151
+ "state_after": "case refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0\u271d : S\ny : A\nhy\u271d : y \u2208 t\nr : R\ns0 : S\nhy : s0 \u2022 y \u2208 span R (s \u2022 t)\n\u22a2 r \u2022 s0 \u2022 y \u2208 span R (s \u2022 t)",
1152
+ "state_before": "case refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0\u271d : S\ny : A\nhy\u271d : y \u2208 t\nr : R\ns0 : S\nhy : s0 \u2022 y \u2208 span R (s \u2022 t)\n\u22a2 (r \u2022 s0) \u2022 y \u2208 span R (s \u2022 t)",
1153
+ "tactic": "rw [IsScalarTower.smul_assoc]"
1154
+ },
1155
+ {
1156
+ "annotated_tactic": [
1157
+ "\"exact <a>smul_mem<\\/a> _ r hy\"",
1158
+ "[{\"full_name\":\"Submodule.smul_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,17]}]"
1159
+ ],
1160
+ "state_after": "no goals",
1161
+ "state_before": "case refine_3\nR : Type u\nS : Type v\nA : Type w\nB : Type u\u2081\nM : Type v\u2081\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module S A\ninst\u271d\u00b9 : Module R A\ninst\u271d : IsScalarTower R S A\ns : Set S\nhs : span R s = \u22a4\nt : Set A\np : A\nhp : p \u2208 restrictScalars R (span S t)\ns0\u271d : S\ny : A\nhy\u271d : y \u2208 t\nr : R\ns0 : S\nhy : s0 \u2022 y \u2208 span R (s \u2022 t)\n\u22a2 r \u2022 s0 \u2022 y \u2208 span R (s \u2022 t)",
1162
+ "tactic": "exact smul_mem _ r hy"
1163
+ }
1164
+ ],
1165
+ "url": "https://github.com/leanprover-community/mathlib4"
1166
+ }
1167
+ },
1168
+ "theorem_name": "Submodule.span_smul_of_span_eq_top"
1169
+ }
1170
+ ],
1171
+ "validation_rejected_proofs": 4,
1172
+ "zero_transition_proofs": 0
1173
+ },
1174
+ "shards": [
1175
+ {
1176
+ "path": "data/train/mathlib-000036-0000-4a6bfa4a9cba3b25.parquet",
1177
+ "sha256": "4a6bfa4a9cba3b25b2658a263ed413dd863ad35591488450950705b449024b70",
1178
+ "bytes": 242173,
1179
+ "rows": 46,
1180
+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
1181
+ },
1182
+ {
1183
+ "path": "data/dev/mathlib-000036-0000-dbba86fd626f835b.parquet",
1184
+ "sha256": "dbba86fd626f835b7a2a7baf08b823c2091e9b6a1bfbf4183fcbe87ce7eab472",
1185
+ "bytes": 55907,
1186
+ "rows": 2,
1187
+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
1188
+ },
1189
+ {
1190
+ "path": "data/internal_test/mathlib-000036-0000-864b695812da0c9d.parquet",
1191
+ "sha256": "864b695812da0c9de7e95b879dd8b714435e2cc1a6f8a47d84e075d78a4a6196",
1192
+ "bytes": 81161,
1193
+ "rows": 5,
1194
+ "revision": "20343734923b05bac9ee5e61e7fa8126c8532808"
1195
+ }
1196
+ ],
1197
+ "proof_ids": [
1198
+ "mathlib:AlgHom.map_algebraMap",
1199
+ "mathlib:AlgHomClass.unitization_injective",
1200
+ "mathlib:Algebra.adjoin_nonUnitalSubalgebra_eq_span",
1201
+ "mathlib:IsScalarTower.algebraMap_apply",
1202
+ "mathlib:IsScalarTower.algebraMap_eq",
1203
+ "mathlib:IsScalarTower.algebraMap_smul",
1204
+ "mathlib:IsScalarTower.of_algebraMap_eq",
1205
+ "mathlib:IsScalarTower.of_algebraMap_smul",
1206
+ "mathlib:NonUnitalStarSubalgebra.toStarSubalgebra_toNonUnitalStarSubalgebra",
1207
+ "mathlib:NonUnitalStarSubalgebra.unitization_injective",
1208
+ "mathlib:NonUnitalStarSubalgebra.unitization_range",
1209
+ "mathlib:NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra",
1210
+ "mathlib:NonUnitalSubalgebra.unitization_injective",
1211
+ "mathlib:NonUnitalSubalgebra.unitization_range",
1212
+ "mathlib:NonUnitalSubring.toSubring_toNonUnitalSubring",
1213
+ "mathlib:NonUnitalSubring.unitization_range",
1214
+ "mathlib:NonUnitalSubsemiring.toSubsemiring_toNonUnitalSubsemiring",
1215
+ "mathlib:NonUnitalSubsemiring.unitization_range",
1216
+ "mathlib:StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span",
1217
+ "mathlib:StarSubalgebra.toNonUnitalStarSubalgebra_toStarSubalgebra",
1218
+ "mathlib:Subalgebra.toNonUnitalSubalgebra_toSubalgebra",
1219
+ "mathlib:Submodule.map_mem_span_algebraMap_image",
1220
+ "mathlib:Submodule.restrictScalars_span",
1221
+ "mathlib:Submodule.smul_mem_span_smul",
1222
+ "mathlib:Submodule.smul_mem_span_smul'",
1223
+ "mathlib:Subring.toNonUnitalSubring_toSubring",
1224
+ "mathlib:Subsemiring.toNonUnitalSubsemiring_toSubsemiring",
1225
+ "mathlib:Unitization.lift_range",
1226
+ "mathlib:Unitization.starLift_range"
1227
+ ],
1228
+ "proof_keys": [
1229
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"AlgHomClass.unitization_injective\"]",
1230
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Algebra.adjoin_nonUnitalSubalgebra_eq_span\"]",
1231
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalStarSubalgebra.toStarSubalgebra_toNonUnitalStarSubalgebra\"]",
1232
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalStarSubalgebra.unitization_injective\"]",
1233
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalStarSubalgebra.unitization_range\"]",
1234
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubalgebra.toSubalgebra_toNonUnitalSubalgebra\"]",
1235
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubalgebra.unitization_injective\"]",
1236
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubalgebra.unitization_range\"]",
1237
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubring.toSubring_toNonUnitalSubring\"]",
1238
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubring.unitization_range\"]",
1239
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubsemiring.toSubsemiring_toNonUnitalSubsemiring\"]",
1240
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"NonUnitalSubsemiring.unitization_range\"]",
1241
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span\"]",
1242
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"StarSubalgebra.toNonUnitalStarSubalgebra_toStarSubalgebra\"]",
1243
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Subalgebra.toNonUnitalSubalgebra_toSubalgebra\"]",
1244
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Subring.toNonUnitalSubring_toSubring\"]",
1245
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Subsemiring.toNonUnitalSubsemiring_toSubsemiring\"]",
1246
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Unitization.lift_range\"]",
1247
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean\",null,null,\"Unitization.starLift_range\"]",
1248
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"AlgHom.map_algebraMap\"]",
1249
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"IsScalarTower.algebraMap_apply\"]",
1250
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"IsScalarTower.algebraMap_eq\"]",
1251
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"IsScalarTower.algebraMap_smul\"]",
1252
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"IsScalarTower.of_algebraMap_eq\"]",
1253
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"IsScalarTower.of_algebraMap_smul\"]",
1254
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"Submodule.map_mem_span_algebraMap_image\"]",
1255
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"Submodule.restrictScalars_span\"]",
1256
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"Submodule.smul_mem_span_smul\"]",
1257
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Algebra/Algebra/Tower.lean\",null,null,\"Submodule.smul_mem_span_smul'\"]"
1258
+ ]
1259
+ }