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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
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1
HyperCactus0 commited on
Commit
f73d2b8
·
verified ·
1 Parent(s): 809f84c

Record 20 verified HPC batch manifests

Browse files
Files changed (20) hide show
  1. manifests/batches/deepseek/000523.json +148 -0
  2. manifests/batches/deepseek/000524.json +148 -0
  3. manifests/batches/deepseek/000525.json +148 -0
  4. manifests/batches/deepseek/000526.json +148 -0
  5. manifests/batches/deepseek/000528.json +166 -0
  6. manifests/batches/deepseek/000529.json +148 -0
  7. manifests/batches/deepseek/000530.json +183 -0
  8. manifests/batches/mathlib/000452.json +0 -0
  9. manifests/batches/mathlib/000453.json +0 -0
  10. manifests/batches/mathlib/000454.json +1295 -0
  11. manifests/batches/mathlib/000455.json +0 -0
  12. manifests/batches/mathlib/000456.json +1060 -0
  13. manifests/batches/mathlib/000457.json +675 -0
  14. manifests/batches/mathlib/000458.json +0 -0
  15. manifests/batches/mathlib/000459.json +0 -0
  16. manifests/batches/numina/001213.json +52 -0
  17. manifests/batches/numina/001214.json +52 -0
  18. manifests/batches/numina/001216.json +51 -0
  19. manifests/batches/numina/001217.json +52 -0
  20. manifests/batches/numina/001219.json +52 -0
manifests/batches/deepseek/000523.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000523",
4
+ "batch_fingerprint": "fac184e0f338ec71316353adcea1993dbdb25eb2cb1e4da145acbbb7e152ceb3",
5
+ "evidence": {
6
+ "batch_fingerprint": "fac184e0f338ec71316353adcea1993dbdb25eb2cb1e4da145acbbb7e152ceb3",
7
+ "contamination_matches": [],
8
+ "failures": [],
9
+ "input_proofs": 50,
10
+ "input_units": 50,
11
+ "proofs": 50,
12
+ "published_proofs": 50,
13
+ "published_transitions": 118,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 118,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
20
+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000523-0000-f03654f6332b8867.parquet",
24
+ "sha256": "f03654f6332b886733e9c4064b857b2b214e3e382934143fe58df21ed32b9451",
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+ "bytes": 308479,
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+ "rows": 100,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
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+ "path": "data/dev/deepseek-000523-0000-de3e91e0dfd695aa.parquet",
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+ "sha256": "de3e91e0dfd695aae25ac26695f879f46c88440f2dade7bcf35ec2b82cf3347b",
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+ "bytes": 39585,
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+ "rows": 4,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
37
+ "path": "data/internal_test/deepseek-000523-0000-3294fe394eb7c9c9.parquet",
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+ "sha256": "3294fe394eb7c9c9f2db3334892071b0ccfbd22826a9f697c81a197302c9732b",
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+ "bytes": 64754,
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+ "rows": 14,
41
+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
42
+ }
43
+ ],
44
+ "proof_ids": [
45
+ "deepseek:thm_26198",
46
+ "deepseek:thm_26199",
47
+ "deepseek:thm_26200",
48
+ "deepseek:thm_26201",
49
+ "deepseek:thm_26202",
50
+ "deepseek:thm_26203",
51
+ "deepseek:thm_26204",
52
+ "deepseek:thm_26205",
53
+ "deepseek:thm_26206",
54
+ "deepseek:thm_26207",
55
+ "deepseek:thm_26208",
56
+ "deepseek:thm_26209",
57
+ "deepseek:thm_26210",
58
+ "deepseek:thm_26211",
59
+ "deepseek:thm_26212",
60
+ "deepseek:thm_26213",
61
+ "deepseek:thm_26214",
62
+ "deepseek:thm_26215",
63
+ "deepseek:thm_26216",
64
+ "deepseek:thm_26217",
65
+ "deepseek:thm_26218",
66
+ "deepseek:thm_26219",
67
+ "deepseek:thm_26220",
68
+ "deepseek:thm_26221",
69
+ "deepseek:thm_26222",
70
+ "deepseek:thm_26223",
71
+ "deepseek:thm_26224",
72
+ "deepseek:thm_26225",
73
+ "deepseek:thm_26226",
74
+ "deepseek:thm_26227",
75
+ "deepseek:thm_26228",
76
+ "deepseek:thm_26229",
77
+ "deepseek:thm_26230",
78
+ "deepseek:thm_26231",
79
+ "deepseek:thm_26232",
80
+ "deepseek:thm_26233",
81
+ "deepseek:thm_26234",
82
+ "deepseek:thm_26235",
83
+ "deepseek:thm_26236",
84
+ "deepseek:thm_26237",
85
+ "deepseek:thm_26238",
86
+ "deepseek:thm_26239",
87
+ "deepseek:thm_26240",
88
+ "deepseek:thm_26241",
89
+ "deepseek:thm_26242",
90
+ "deepseek:thm_26243",
91
+ "deepseek:thm_26244",
92
+ "deepseek:thm_26245",
93
+ "deepseek:thm_26246",
94
+ "deepseek:thm_26247"
95
+ ],
96
+ "proof_keys": [
97
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026150.lean\",\"f082acc6c03a93677d2205388ef449a3964b297ee42719b28411fcca03a15842\",0,\"thm_26198\"]",
98
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026151.lean\",\"c747ac224b0c3a9e997b4fb3929e658ba71de438cda836d38b24086079a2abc9\",0,\"thm_26199\"]",
99
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026152.lean\",\"fcf5fd7955a416279b57706c296e402d597f9e31d63e37ef04ca0ceeab661050\",0,\"thm_26200\"]",
100
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026153.lean\",\"2286fdd84a45c89cbe16a1e361c831984536d5ac9806f4da86c8627259e67983\",0,\"thm_26201\"]",
101
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026154.lean\",\"e364b054319331e4b3ed7025069d162f33c5ba7e9cb228f432e218d7a913f664\",0,\"thm_26202\"]",
102
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026155.lean\",\"b3a7e01fcee8abcc24efbabe0c12fb9a9eb1ed7c5cefb892cb85efae04c3529d\",0,\"thm_26203\"]",
103
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026156.lean\",\"319a69fbdb86047779aa1dcf9859332e0f9da3ee0d79523ac5402dff682c1882\",0,\"thm_26204\"]",
104
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026157.lean\",\"e92ba3569412e60d9e532ee493f93d25f4b958d66bac3b7e7ac0dfcbf9c9ce96\",0,\"thm_26205\"]",
105
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026158.lean\",\"d87c5a9e262d092bcfe65e5b7a8843ed95629600fa0c8e9e0775baf5f0869d6d\",0,\"thm_26206\"]",
106
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026159.lean\",\"d60f44a1e78b75f642cd17000ccee84d323436d3e2ed43a3c6aaef6ae2fd65c5\",0,\"thm_26207\"]",
107
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026160.lean\",\"306f36d8d4a054d6935c367c5cd9fd9014c9dc089eae6033b0e0e7a4d9b23f9c\",0,\"thm_26208\"]",
108
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026161.lean\",\"4b90e6a8abdf5c6532ce67582b1d988e9318c7a7719c68d262c961f025e5ee40\",0,\"thm_26209\"]",
109
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026162.lean\",\"2a15213b120bbf9a7aa207c81d07d5299b6b5929b8b7835cb48f6386cdffd3ad\",0,\"thm_26210\"]",
110
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026163.lean\",\"d9a073efe7308ae23e876ed2b365650cac281803aaf73ad99a7327e1b7558df0\",0,\"thm_26211\"]",
111
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026164.lean\",\"713ed4888a8ab9818ac222bab579304acc607e8f409a21d1627b6a0f85825368\",0,\"thm_26212\"]",
112
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026165.lean\",\"0d5e3ea4f8f50f68465d82674f601c85b8f78c7f8eb15ac30f9a13a7195d9c4a\",0,\"thm_26213\"]",
113
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026166.lean\",\"37cf3557e65356737a0dbcb542bb12e89bcb80d0c3607a1efdd3134f47e78581\",0,\"thm_26214\"]",
114
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026167.lean\",\"aeaaac3c57beb25c27ebf425b12a5090c83ae441cdcd37a4a7ca8059c73b0d4b\",0,\"thm_26215\"]",
115
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026168.lean\",\"d1b0c786530a36343169108151c6c8991848281570d7d50d87c77f02f65c8f4b\",0,\"thm_26216\"]",
116
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026169.lean\",\"9df98f80217f5a2cacdd00441749555d9e77f7d687325d270aa0112bd6574db6\",0,\"thm_26217\"]",
117
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026170.lean\",\"2571bad5a8769db267b080148d8ece5265e1ae7815b9403cfabd1e809a9695b9\",0,\"thm_26218\"]",
118
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026171.lean\",\"8d9b97d42e6d56291a4f6f1e38054168621a28f620d44c876b812d0463ee293d\",0,\"thm_26219\"]",
119
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026172.lean\",\"579d1db3a06391a571d9be196e3bae963f05b9394c8d557b388e844a59991d6d\",0,\"thm_26220\"]",
120
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026173.lean\",\"c6d96cfdfcbf2aa20268b81ae69b81ea4d558b176115b8c3d9cbf52bc88046d3\",0,\"thm_26221\"]",
121
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026174.lean\",\"dc98ca95be57615043b9929351dbd4d721363ab99d76b9d1499bbdb63e2abab2\",0,\"thm_26222\"]",
122
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026175.lean\",\"ff4547dd8b83122d8cd223e9883aabe6dd61867deac30be66d4ede26a58db958\",0,\"thm_26223\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026176.lean\",\"5999f0213aaaa18943499b27b050b93d44b2923dac104fc9334294ae72bd0359\",0,\"thm_26224\"]",
124
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026177.lean\",\"8a777fb91e4ddd8630be734a20e7395be1dde868d899ee2fc2bc460890c93591\",0,\"thm_26225\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026178.lean\",\"efc6dfadef31f07a59fbd2110a6a40565a73daebc6fb7c069745e2548847deec\",0,\"thm_26226\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026179.lean\",\"90e24e08cb8e1e10092e8f01c538cbcae3bf9f931d4591b761ae9dd6b7633c63\",0,\"thm_26227\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026180.lean\",\"a4073c6cbdad8f1306656290368115c333c5a435a0343d5e746e0b87884636cc\",0,\"thm_26228\"]",
128
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026181.lean\",\"00d4b50402a3bd114f0e35a74421279a1878c0ed3e5b08fd9440bef8ec9358e4\",0,\"thm_26229\"]",
129
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026182.lean\",\"723de175a9e1fbb0656b560b7bcc7b072f8c001c1b059433573b00ee61d1d9e5\",0,\"thm_26230\"]",
130
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026183.lean\",\"3bdd578f8b8c65902f0f231aa91dcc6d7e14bf346c4dd501290b7aef9e0a09d7\",0,\"thm_26231\"]",
131
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026184.lean\",\"365d3f8ec69c7fe818b35982ccec4e56210488899b8127c6025ae986262df57b\",0,\"thm_26232\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026185.lean\",\"e4a65656b60fc23befdc7342855a7c8a34629e38936e1b2857ce0ad9ffe60e9d\",0,\"thm_26233\"]",
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026186.lean\",\"ce14e85c21579d12c28c9dd3087130f6869a7c3276ec03b5f86ebf3b64b6727c\",0,\"thm_26234\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026187.lean\",\"cf5b84cf161ed57e17e102cfbf692b0b496d5a515b6a6d10734d9362b1695e17\",0,\"thm_26235\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026188.lean\",\"feb72ac91665a7790eb9f0e36bf3955bea8cf67b3e120698ce43c8cd8a848a97\",0,\"thm_26236\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026189.lean\",\"31509341250379ddf0e26fed8b18251ccc4624739835397ba5aca3577712bc0c\",0,\"thm_26237\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026190.lean\",\"a36f93133cd1a006e039f3e09c05089bfd8ec0f343cc4783f696842efa17996e\",0,\"thm_26238\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026191.lean\",\"0c98dccdbc5d5ef9970984757e7a85feb175764f889bc51ec9c1ed4215db00ba\",0,\"thm_26239\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026192.lean\",\"60070306ee9c58f6cde3cc082033897db0afb81b63f0fdacc9a09789ab344b32\",0,\"thm_26240\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026193.lean\",\"5c7a1a241c842f60946bffc7844f27e670d2ccd69597fbce2a0bab6c5161a3ff\",0,\"thm_26241\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026194.lean\",\"377b2d0d09b7f370a6783e2fa1bf83830f67537ffcf02998afcb28208379fb75\",0,\"thm_26242\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026195.lean\",\"4f53009d5cd5f9f45e0dc75eafad9cc51a9846160105886b5e1bc5e57bfb13c6\",0,\"thm_26243\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026196.lean\",\"ca30d0cead346bf43c969ec60ce6551adb79f1f5b2e1434e302d51699fa2c7d9\",0,\"thm_26244\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026197.lean\",\"5088b970738d6c4ff851cfbf35af91ae2a3589c3c1eee583f31fee41fde616b6\",0,\"thm_26245\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026198.lean\",\"d08e964fb62a8bc22006a872ab81aa83377015488d594e2fe5ec07336b509929\",0,\"thm_26246\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_026199.lean\",\"2ed915e14a72be0af66d4a6b36f3142f210bff8d53c310cb759e107e3d1de394\",0,\"thm_26247\"]"
147
+ ]
148
+ }
manifests/batches/deepseek/000524.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000524",
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+ "batch_fingerprint": "efd6a0ea0aa6ab2f1b7b27b0f7eb4fcaee27277260b9f6290831ca763e58f732",
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+ "evidence": {
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+ "batch_fingerprint": "efd6a0ea0aa6ab2f1b7b27b0f7eb4fcaee27277260b9f6290831ca763e58f732",
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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,
12
+ "published_proofs": 50,
13
+ "published_transitions": 124,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 124,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
20
+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000524-0000-f12972c0ab10d579.parquet",
24
+ "sha256": "f12972c0ab10d57967daa2249cece37a93ad4265c017f5379c794ec18a4c657c",
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+ "bytes": 344220,
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+ "rows": 100,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
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+ "path": "data/dev/deepseek-000524-0000-a326e0eabfb3cd87.parquet",
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+ "sha256": "a326e0eabfb3cd87f133ad1cdb4736a6dee91f9b434af65e49bc8106fa212631",
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+ "bytes": 49097,
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+ "rows": 9,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
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+ "path": "data/internal_test/deepseek-000524-0000-29ddfd0690525798.parquet",
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+ "sha256": "29ddfd069052579835c87a5e26faa3fb2a0b13986bbb8f38e235be2d9c35360e",
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+ "bytes": 56543,
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+ "rows": 15,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ ]
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+ }
manifests/batches/deepseek/000525.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000525",
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+ "batch_fingerprint": "3a58382c58243c6b4d493bbb126d9d205e1b8e5ea796490ac47bd2257f26834c",
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+ "evidence": {
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+ "contamination_matches": [],
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+ "published_transitions": 114,
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+ "replay_failures": [],
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+ "trace_mode": "individual",
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+ "transitions": 114,
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+ "validation_failures": [],
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+ "zero_transition_proofs": 0
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+ "path": "data/train/deepseek-000525-0000-8163aad72cbd409b.parquet",
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+ "path": "data/internal_test/deepseek-000525-0000-cbee061543d4feb3.parquet",
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+ ]
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+ }
manifests/batches/deepseek/000526.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "replay_failures": [],
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+ "validation_failures": [],
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+ ]
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+ }
manifests/batches/deepseek/000528.json ADDED
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+ "id": "thm_26457",
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+ ]
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+ }
manifests/batches/deepseek/000529.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000529",
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147
+ ]
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+ }
manifests/batches/deepseek/000530.json ADDED
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manifests/batches/mathlib/000452.json ADDED
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manifests/batches/mathlib/000453.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/000454.json ADDED
@@ -0,0 +1,1295 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000454",
4
+ "batch_fingerprint": "28ddfde842fa431f1373662268c0ef197ee472156fee52c68fcd697d2b618c2d",
5
+ "evidence": {
6
+ "batch_fingerprint": "28ddfde842fa431f1373662268c0ef197ee472156fee52c68fcd697d2b618c2d",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "convex_halfspace_im_gt",
12
+ "id": "convex_halfspace_im_gt",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 44,
18
+ 72
19
+ ],
20
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
21
+ "full_name": "convex_halfspace_im_gt",
22
+ "start": [
23
+ 43,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "convex_halfspace_im_le",
34
+ "id": "convex_halfspace_im_le",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 40,
40
+ 72
41
+ ],
42
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
43
+ "full_name": "convex_halfspace_im_le",
44
+ "start": [
45
+ 39,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "convex_halfspace_im_lt",
56
+ "id": "convex_halfspace_im_lt",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 36,
62
+ 72
63
+ ],
64
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
65
+ "full_name": "convex_halfspace_im_lt",
66
+ "start": [
67
+ 35,
68
+ 1
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "convex_halfspace_re_ge",
78
+ "id": "convex_halfspace_re_ge",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
82
+ "end": [
83
+ 32,
84
+ 72
85
+ ],
86
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
87
+ "full_name": "convex_halfspace_re_ge",
88
+ "start": [
89
+ 31,
90
+ 1
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
96
+ },
97
+ {
98
+ "error": "theorem has no selected tactic transitions",
99
+ "full_name": "convex_halfspace_re_gt",
100
+ "id": "convex_halfspace_re_gt",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 28,
106
+ 72
107
+ ],
108
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
109
+ "full_name": "convex_halfspace_re_gt",
110
+ "start": [
111
+ 27,
112
+ 1
113
+ ],
114
+ "traced_tactics": [],
115
+ "url": "https://github.com/leanprover-community/mathlib4"
116
+ },
117
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
118
+ },
119
+ {
120
+ "error": "theorem has no selected tactic transitions",
121
+ "full_name": "convex_halfspace_re_le",
122
+ "id": "convex_halfspace_re_le",
123
+ "reason": "import_failure",
124
+ "selection": {
125
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
126
+ "end": [
127
+ 24,
128
+ 72
129
+ ],
130
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
131
+ "full_name": "convex_halfspace_re_le",
132
+ "start": [
133
+ 23,
134
+ 1
135
+ ],
136
+ "traced_tactics": [],
137
+ "url": "https://github.com/leanprover-community/mathlib4"
138
+ },
139
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
140
+ },
141
+ {
142
+ "error": "theorem has no selected tactic transitions",
143
+ "full_name": "convex_halfspace_re_lt",
144
+ "id": "convex_halfspace_re_lt",
145
+ "reason": "import_failure",
146
+ "selection": {
147
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
148
+ "end": [
149
+ 20,
150
+ 72
151
+ ],
152
+ "file_path": "Mathlib/Analysis/Convex/Complex.lean",
153
+ "full_name": "convex_halfspace_re_lt",
154
+ "start": [
155
+ 19,
156
+ 1
157
+ ],
158
+ "traced_tactics": [],
159
+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/Analysis/Convex/Complex.lean"
162
+ },
163
+ {
164
+ "error": "no internal trace for tactic \"rwa [smul_smul, inv_mul_cancel hc.ne', one_smul]\"",
165
+ "full_name": "Convex.mem_toCone'",
166
+ "id": "Convex.mem_toCone'",
167
+ "reason": "import_failure",
168
+ "selection": {
169
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
170
+ "end": [
171
+ 650,
172
+ 90
173
+ ],
174
+ "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean",
175
+ "full_name": "Convex.mem_toCone'",
176
+ "start": [
177
+ 645,
178
+ 1
179
+ ],
180
+ "traced_tactics": [
181
+ {
182
+ "annotated_tactic": [
183
+ "\"refine hs.mem_toCone.trans \\u27e8?_, ?_\\u27e9\"",
184
+ "[]"
185
+ ],
186
+ "state_after": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 (\u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x) \u2192 \u2203 c, 0 < c \u2227 c \u2022 x \u2208 s\n\ncase refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 (\u2203 c, 0 < c \u2227 c \u2022 x \u2208 s) \u2192 \u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x",
187
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 x \u2208 toCone s hs \u2194 \u2203 c, 0 < c \u2227 c \u2022 x \u2208 s",
188
+ "tactic": "refine hs.mem_toCone.trans \u27e8?_, ?_\u27e9"
189
+ },
190
+ {
191
+ "annotated_tactic": [
192
+ "\"rintro \\u27e8c, hc, y, hy, rfl\\u27e9\"",
193
+ "[]"
194
+ ],
195
+ "state_after": "case refine_1.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nc : \ud835\udd5c\nhc : 0 < c\ny : E\nhy : y \u2208 s\n\u22a2 \u2203 c_1, 0 < c_1 \u2227 c_1 \u2022 c \u2022 y \u2208 s",
196
+ "state_before": "case refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 (\u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x) \u2192 \u2203 c, 0 < c \u2227 c \u2022 x \u2208 s",
197
+ "tactic": "rintro \u27e8c, hc, y, hy, rfl\u27e9"
198
+ },
199
+ {
200
+ "annotated_tactic": [
201
+ "\"exact \\u27e8c\\u207b\\u00b9, <a>inv_pos<\\/a>.2 hc, by rwa [<a>smul_smul<\\/a>, <a>inv_mul_cancel<\\/a> hc.ne', <a>one_smul<\\/a>]\\u27e9\"",
202
+ "[{\"full_name\":\"inv_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[47,15],\"def_end_pos\":[47,22]},{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
203
+ ],
204
+ "state_after": "no goals",
205
+ "state_before": "case refine_1.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nc : \ud835\udd5c\nhc : 0 < c\ny : E\nhy : y \u2208 s\n\u22a2 \u2203 c_1, 0 < c_1 \u2227 c_1 \u2022 c \u2022 y \u2208 s",
206
+ "tactic": "exact \u27e8c\u207b\u00b9, inv_pos.2 hc, by rwa [smul_smul, inv_mul_cancel hc.ne', one_smul]\u27e9"
207
+ },
208
+ {
209
+ "annotated_tactic": [
210
+ "\"rwa [<a>smul_smul<\\/a>, <a>inv_mul_cancel<\\/a> hc.ne', <a>one_smul<\\/a>]\"",
211
+ "[{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
212
+ ],
213
+ "state_after": "no goals",
214
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nc : \ud835\udd5c\nhc : 0 < c\ny : E\nhy : y \u2208 s\n\u22a2 c\u207b\u00b9 \u2022 c \u2022 y \u2208 s",
215
+ "tactic": "rwa [smul_smul, inv_mul_cancel hc.ne', one_smul]"
216
+ },
217
+ {
218
+ "annotated_tactic": [
219
+ "\"rintro \\u27e8c, hc, hcx\\u27e9\"",
220
+ "[]"
221
+ ],
222
+ "state_after": "case refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nc : \ud835\udd5c\nhc : 0 < c\nhcx : c \u2022 x \u2208 s\n\u22a2 \u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x",
223
+ "state_before": "case refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 (\u2203 c, 0 < c \u2227 c \u2022 x \u2208 s) \u2192 \u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x",
224
+ "tactic": "rintro \u27e8c, hc, hcx\u27e9"
225
+ },
226
+ {
227
+ "annotated_tactic": [
228
+ "\"exact \\u27e8c\\u207b\\u00b9, <a>inv_pos<\\/a>.2 hc, _, hcx, by rw [<a>smul_smul<\\/a>, <a>inv_mul_cancel<\\/a> hc.ne', <a>one_smul<\\/a>]\\u27e9\"",
229
+ "[{\"full_name\":\"inv_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[47,15],\"def_end_pos\":[47,22]},{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
230
+ ],
231
+ "state_after": "no goals",
232
+ "state_before": "case refine_2.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nc : \ud835\udd5c\nhc : 0 < c\nhcx : c \u2022 x \u2208 s\n\u22a2 \u2203 c, 0 < c \u2227 \u2203 y \u2208 s, c \u2022 y = x",
233
+ "tactic": "exact \u27e8c\u207b\u00b9, inv_pos.2 hc, _, hcx, by rw [smul_smul, inv_mul_cancel hc.ne', one_smul]\u27e9"
234
+ },
235
+ {
236
+ "annotated_tactic": [
237
+ "\"rw [<a>smul_smul<\\/a>, <a>inv_mul_cancel<\\/a> hc.ne', <a>one_smul<\\/a>]\"",
238
+ "[{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
239
+ ],
240
+ "state_after": "no goals",
241
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nc : \ud835\udd5c\nhc : 0 < c\nhcx : c \u2022 x \u2208 s\n\u22a2 c\u207b\u00b9 \u2022 c \u2022 x = x",
242
+ "tactic": "rw [smul_smul, inv_mul_cancel hc.ne', one_smul]"
243
+ }
244
+ ],
245
+ "url": "https://github.com/leanprover-community/mathlib4"
246
+ },
247
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+ "mathlib:Convex.subset_toCone",
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+ "mathlib:Convex.toCone_isLeast",
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+ "mathlib:ConvexCone.Blunt.salient",
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+ "mathlib:ConvexCone.Flat.pointed",
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+ "mathlib:ConvexCone.blunt_iff_not_pointed",
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+ "mathlib:ConvexCone.coe_iInf",
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+ "mathlib:ConvexCone.mem_iInf",
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+ "mathlib:ConvexCone.pointed_zero",
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+ "mathlib:ConvexCone.salient_iff_not_flat",
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+ "mathlib:ConvexCone.to_orderedSMul",
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+ "mathlib:convexHull_toCone_isLeast"
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+ ],
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+ "proof_keys": [
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1283
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1284
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1285
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1286
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.Flat.pointed\"]",
1287
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.blunt_iff_not_pointed\"]",
1288
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.coe_iInf\"]",
1289
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.mem_iInf\"]",
1290
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.pointed_zero\"]",
1291
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.salient_iff_not_flat\"]",
1292
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"ConvexCone.to_orderedSMul\"]",
1293
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Basic.lean\",null,null,\"convexHull_toCone_isLeast\"]"
1294
+ ]
1295
+ }
manifests/batches/mathlib/000455.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/000456.json ADDED
@@ -0,0 +1,1060 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000456",
4
+ "batch_fingerprint": "8b7a3a5867a4e95a0aa686006eecc7e96a0bc6d8759a2c9153b659444a6a9a79",
5
+ "evidence": {
6
+ "batch_fingerprint": "8b7a3a5867a4e95a0aa686006eecc7e96a0bc6d8759a2c9153b659444a6a9a79",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "no internal trace for tactic 'rw [sub_ne_zero]'",
11
+ "full_name": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem",
12
+ "id": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 200,
18
+ 49
19
+ ],
20
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
21
+ "full_name": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem",
22
+ "start": [
23
+ 170,
24
+ 1
25
+ ],
26
+ "traced_tactics": [
27
+ {
28
+ "annotated_tactic": [
29
+ "\"obtain \\u27e8z, hzK, infi\\u27e9 := <a>exists_norm_eq_iInf_of_complete_convex<\\/a> ne hc.isComplete K.convex b\"",
30
+ "[{\"full_name\":\"exists_norm_eq_iInf_of_complete_convex\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Projection.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,47]}]"
31
+ ],
32
+ "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 x \u2208 K, 0 \u2264 \u27eax, y\u27eb_\u211d) \u2227 \u27eay, b\u27eb_\u211d < 0",
33
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\n\u22a2 \u2203 y, (\u2200 x \u2208 K, 0 \u2264 \u27eax, y\u27eb_\u211d) \u2227 \u27eay, b\u27eb_\u211d < 0",
34
+ "tactic": "obtain \u27e8z, hzK, infi\u27e9 := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b"
35
+ },
36
+ {
37
+ "annotated_tactic": [
38
+ "\"have hinner := (<a>norm_eq_iInf_iff_real_inner_le_zero<\\/a> K.convex hzK).1 infi\"",
39
+ "[{\"full_name\":\"norm_eq_iInf_iff_real_inner_le_zero\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Projection.lean\",\"def_pos\":[182,9],\"def_end_pos\":[182,44]}]"
40
+ ],
41
+ "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u2203 y, (\u2200 x \u2208 K, 0 \u2264 \u27eax, y\u27eb_\u211d) \u2227 \u27eay, b\u27eb_\u211d < 0",
42
+ "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 x \u2208 K, 0 \u2264 \u27eax, y\u27eb_\u211d) \u2227 \u27eay, b\u27eb_\u211d < 0",
43
+ "tactic": "have hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi"
44
+ },
45
+ {
46
+ "annotated_tactic": [
47
+ "\"use z - b\"",
48
+ "[]"
49
+ ],
50
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 (\u2200 x \u2208 K, 0 \u2264 \u27eax, z - b\u27eb_\u211d) \u2227 \u27eaz - b, b\u27eb_\u211d < 0",
51
+ "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u2203 y, (\u2200 x \u2208 K, 0 \u2264 \u27eax, y\u27eb_\u211d) \u2227 \u27eay, b\u27eb_\u211d < 0",
52
+ "tactic": "use z - b"
53
+ },
54
+ {
55
+ "annotated_tactic": [
56
+ "\"constructor\"",
57
+ "[]"
58
+ ],
59
+ "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u2200 x \u2208 K, 0 \u2264 \u27eax, z - b\u27eb_\u211d\n\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
60
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 (\u2200 x \u2208 K, 0 \u2264 \u27eax, z - b\u27eb_\u211d) \u2227 \u27eaz - b, b\u27eb_\u211d < 0",
61
+ "tactic": "constructor"
62
+ },
63
+ {
64
+ "annotated_tactic": [
65
+ "\"rintro x hxK\"",
66
+ "[]"
67
+ ],
68
+ "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 \u27eax, z - b\u27eb_\u211d",
69
+ "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u2200 x \u2208 K, 0 \u2264 \u27eax, z - b\u27eb_\u211d",
70
+ "tactic": "rintro x hxK"
71
+ },
72
+ {
73
+ "annotated_tactic": [
74
+ "\"specialize hinner _ (K.add_mem hxK hzK)\"",
75
+ "[]"
76
+ ],
77
+ "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : \u27eab - z, x + z - z\u27eb_\u211d \u2264 0\n\u22a2 0 \u2264 \u27eax, z - b\u27eb_\u211d",
78
+ "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 \u27eax, z - b\u27eb_\u211d",
79
+ "tactic": "specialize hinner _ (K.add_mem hxK hzK)"
80
+ },
81
+ {
82
+ "annotated_tactic": [
83
+ "\"rwa [<a>add_sub_cancel_right<\\/a>, <a>real_inner_comm<\\/a>, \\u2190 <a>neg_nonneg<\\/a>, <a>neg_eq_neg_one_mul<\\/a>,\\n \\u2190 <a>real_inner_smul_right<\\/a>, <a>neg_smul<\\/a>, <a>one_smul<\\/a>, <a>neg_sub<\\/a>] at hinner\"",
84
+ "[{\"full_name\":\"add_sub_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1008,3],\"def_end_pos\":[1008,14]},{\"full_name\":\"real_inner_comm\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,24]},{\"full_name\":\"neg_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[671,24],\"def_end_pos\":[671,34]},{\"full_name\":\"neg_eq_neg_one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[347,9],\"def_end_pos\":[347,27]},{\"full_name\":\"real_inner_smul_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,30]},{\"full_name\":\"neg_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]},{\"full_name\":\"neg_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[553,3],\"def_end_pos\":[553,14]}]"
85
+ ],
86
+ "state_after": "no goals",
87
+ "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : \u27eab - z, x + z - z\u27eb_\u211d \u2264 0\n\u22a2 0 \u2264 \u27eax, z - b\u27eb_\u211d",
88
+ "tactic": "rwa [add_sub_cancel_right, real_inner_comm, \u2190 neg_nonneg, neg_eq_neg_one_mul,\n \u2190 real_inner_smul_right, neg_smul, one_smul, neg_sub] at hinner"
89
+ },
90
+ {
91
+ "annotated_tactic": [
92
+ "\"have hinner\\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc)\"",
93
+ "[]"
94
+ ],
95
+ "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : \u27eab - z, 0 - z\u27eb_\u211d \u2264 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
96
+ "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
97
+ "tactic": "have hinner\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc)"
98
+ },
99
+ {
100
+ "annotated_tactic": [
101
+ "\"rw [<a>zero_sub<\\/a>, <a>inner_neg_right<\\/a>, <a>Right.neg_nonpos_iff<\\/a>] at hinner\\u2080\"",
102
+ "[{\"full_name\":\"zero_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[467,3],\"def_end_pos\":[467,14]},{\"full_name\":\"inner_neg_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[626,9],\"def_end_pos\":[626,24]},{\"full_name\":\"Right.neg_nonpos_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[214,3],\"def_end_pos\":[214,14]}]"
103
+ ],
104
+ "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
105
+ "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : \u27eab - z, 0 - z\u27eb_\u211d \u2264 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
106
+ "tactic": "rw [zero_sub, inner_neg_right, Right.neg_nonpos_iff] at hinner\u2080"
107
+ },
108
+ {
109
+ "annotated_tactic": [
110
+ "\"have hbz : b - z \\u2260 0 := by\\n rw [<a>sub_ne_zero<\\/a>]\\n contrapose! hzK\\n rwa [\\u2190 hzK]\"",
111
+ "[{\"full_name\":\"sub_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1082,3],\"def_end_pos\":[1082,14]}]"
112
+ ],
113
+ "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhbz : b - z \u2260 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
114
+ "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
115
+ "tactic": "have hbz : b - z \u2260 0 := by\n rw [sub_ne_zero]\n contrapose! hzK\n rwa [\u2190 hzK]"
116
+ },
117
+ {
118
+ "annotated_tactic": [
119
+ "\"rw [\\u2190 <a>neg_zero<\\/a>, <a>lt_neg<\\/a>, \\u2190 <a>neg_one_mul<\\/a>, \\u2190 <a>real_inner_smul_left<\\/a>, <a>smul_sub<\\/a>, <a>neg_smul<\\/a>, <a>one_smul<\\/a>,\\n <a>neg_smul<\\/a>, <a>neg_sub_neg<\\/a>, <a>one_smul<\\/a>]\"",
120
+ "[{\"full_name\":\"neg_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1117,3],\"def_end_pos\":[1117,14]},{\"full_name\":\"lt_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[393,15],\"def_end_pos\":[393,21]},{\"full_name\":\"neg_one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[357,9],\"def_end_pos\":[357,20]},{\"full_name\":\"real_inner_smul_left\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[468,9],\"def_end_pos\":[468,29]},{\"full_name\":\"smul_sub\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,17]},{\"full_name\":\"neg_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]},{\"full_name\":\"neg_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,17]},{\"full_name\":\"neg_sub_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[750,3],\"def_end_pos\":[750,14]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
121
+ ],
122
+ "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhbz : b - z \u2260 0\n\u22a2 0 < \u27eab - z, b\u27eb_\u211d",
123
+ "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhbz : b - z \u2260 0\n\u22a2 \u27eaz - b, b\u27eb_\u211d < 0",
124
+ "tactic": "rw [\u2190 neg_zero, lt_neg, \u2190 neg_one_mul, \u2190 real_inner_smul_left, smul_sub, neg_smul, one_smul,\n neg_smul, neg_sub_neg, one_smul]"
125
+ },
126
+ {
127
+ "annotated_tactic": [
128
+ "\"calc\\n 0 < \\u27eab - z, b - z\\u27eb_\\u211d := <a>lt_of_not_le<\\/a> ((<a>Iff.not<\\/a> <a>real_inner_self_nonpos<\\/a>).2 hbz)\\n _ = \\u27eab - z, b - z\\u27eb_\\u211d + 0 := (<a>add_zero<\\/a> _).<a>symm<\\/a>\\n _ \\u2264 \\u27eab - z, b - z\\u27eb_\\u211d + \\u27eab - z, z\\u27eb_\\u211d := <a>add_le_add<\\/a> rfl.ge hinner\\u2080\\n _ = \\u27eab - z, b - z + z\\u27eb_\\u211d := (<a>inner_add_right<\\/a> _ _ _).<a>symm<\\/a>\\n _ = \\u27eab - z, b\\u27eb_\\u211d := by rw [<a>sub_add_cancel<\\/a>]\"",
129
+ "[{\"full_name\":\"lt_of_not_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[455,9],\"def_end_pos\":[455,21]},{\"full_name\":\"Iff.not\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[279,7],\"def_end_pos\":[279,14]},{\"full_name\":\"real_inner_self_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[612,9],\"def_end_pos\":[612,31]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"add_le_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[205,32],\"def_end_pos\":[205,42]},{\"full_name\":\"inner_add_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[453,9],\"def_end_pos\":[453,24]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"sub_add_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[997,3],\"def_end_pos\":[997,14]}]"
130
+ ],
131
+ "state_after": "no goals",
132
+ "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhbz : b - z \u2260 0\n\u22a2 0 < \u27eab - z, b\u27eb_\u211d",
133
+ "tactic": "calc\n 0 < \u27eab - z, b - z\u27eb_\u211d := lt_of_not_le ((Iff.not real_inner_self_nonpos).2 hbz)\n _ = \u27eab - z, b - z\u27eb_\u211d + 0 := (add_zero _).symm\n _ \u2264 \u27eab - z, b - z\u27eb_\u211d + \u27eab - z, z\u27eb_\u211d := add_le_add rfl.ge hinner\u2080\n _ = \u27eab - z, b - z + z\u27eb_\u211d := (inner_add_right _ _ _).symm\n _ = \u27eab - z, b\u27eb_\u211d := by rw [sub_add_cancel]"
134
+ },
135
+ {
136
+ "annotated_tactic": [
137
+ "\"rw [<a>sub_ne_zero<\\/a>]\"",
138
+ "[{\"full_name\":\"sub_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1082,3],\"def_end_pos\":[1082,14]}]"
139
+ ],
140
+ "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\n\u22a2 b \u2260 z",
141
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\n\u22a2 b - z \u2260 0",
142
+ "tactic": "rw [sub_ne_zero]"
143
+ },
144
+ {
145
+ "annotated_tactic": [
146
+ "\"contrapose! hzK\"",
147
+ "[]"
148
+ ],
149
+ "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhzK : b = z\n\u22a2 z \u2209 \u2191K",
150
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\n\u22a2 b \u2260 z",
151
+ "tactic": "contrapose! hzK"
152
+ },
153
+ {
154
+ "annotated_tactic": [
155
+ "\"rwa [\\u2190 hzK]\"",
156
+ "[]"
157
+ ],
158
+ "state_after": "no goals",
159
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhzK : b = z\n\u22a2 z \u2209 \u2191K",
160
+ "tactic": "rwa [\u2190 hzK]"
161
+ },
162
+ {
163
+ "annotated_tactic": [
164
+ "\"rw [<a>sub_add_cancel<\\/a>]\"",
165
+ "[{\"full_name\":\"sub_add_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[997,3],\"def_end_pos\":[997,14]}]"
166
+ ],
167
+ "state_after": "no goals",
168
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nb : H\ndisj : b \u2209 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 w \u2208 \u2191K, \u27eab - z, w - z\u27eb_\u211d \u2264 0\nhinner\u2080 : 0 \u2264 \u27eab - z, z\u27eb_\u211d\nhbz : b - z \u2260 0\n\u22a2 \u27eab - z, b - z + z\u27eb_\u211d = \u27eab - z, b\u27eb_\u211d",
169
+ "tactic": "rw [sub_add_cancel]"
170
+ }
171
+ ],
172
+ "url": "https://github.com/leanprover-community/mathlib4"
173
+ },
174
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
175
+ },
176
+ {
177
+ "error": "no internal trace for tactic 'rintro _ \u27e8_, h, rfl\u27e9'",
178
+ "full_name": "ConvexCone.pointed_of_nonempty_of_isClosed",
179
+ "id": "ConvexCone.pointed_of_nonempty_of_isClosed",
180
+ "reason": "import_failure",
181
+ "selection": {
182
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
183
+ "end": [
184
+ 161,
185
+ 93
186
+ ],
187
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
188
+ "full_name": "ConvexCone.pointed_of_nonempty_of_isClosed",
189
+ "start": [
190
+ 144,
191
+ 1
192
+ ],
193
+ "traced_tactics": [
194
+ {
195
+ "annotated_tactic": [
196
+ "\"obtain \\u27e8x, hx\\u27e9 := ne\"",
197
+ "[]"
198
+ ],
199
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\n\u22a2 K.Pointed",
200
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\n\u22a2 K.Pointed",
201
+ "tactic": "obtain \u27e8x, hx\u27e9 := ne"
202
+ },
203
+ {
204
+ "annotated_tactic": [
205
+ "\"have fI : f '' <a>Set.Ioi<\\/a> 0 \\u2286 (K : <a>Set<\\/a> H) := by\\n rintro _ \\u27e8_, h, rfl\\u27e9\\n exact K.smul_mem (<a>Set.mem_Ioi<\\/a>.1 h) hx\"",
206
+ "[{\"full_name\":\"Set.Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,8]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Set.mem_Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,16]}]"
207
+ ],
208
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\n\u22a2 K.Pointed",
209
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\n\u22a2 K.Pointed",
210
+ "tactic": "have fI : f '' Set.Ioi 0 \u2286 (K : Set H) := by\n rintro _ \u27e8_, h, rfl\u27e9\n exact K.smul_mem (Set.mem_Ioi.1 h) hx"
211
+ },
212
+ {
213
+ "annotated_tactic": [
214
+ "\"have clf : <a>closure<\\/a> (f '' <a>Set.Ioi<\\/a> 0) \\u2286 (K : <a>Set<\\/a> H) := hc.closure_subset_iff.2 fI\"",
215
+ "[{\"full_name\":\"closure\",\"def_path\":\"Mathlib\\/Topology\\/Defs\\/Basic.lean\",\"def_pos\":[116,5],\"def_end_pos\":[116,12]},{\"full_name\":\"Set.Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,8]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
216
+ ],
217
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\n\u22a2 K.Pointed",
218
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\n\u22a2 K.Pointed",
219
+ "tactic": "have clf : closure (f '' Set.Ioi 0) \u2286 (K : Set H) := hc.closure_subset_iff.2 fI"
220
+ },
221
+ {
222
+ "annotated_tactic": [
223
+ "\"have fc : <a>ContinuousWithinAt<\\/a> f (<a>Set.Ioi<\\/a> (0 : \\u211d)) 0 :=\\n (continuous_id.smul <a>continuous_const<\\/a>).<a>continuousWithinAt<\\/a>\"",
224
+ "[{\"full_name\":\"ContinuousWithinAt\",\"def_path\":\"Mathlib\\/Topology\\/Defs\\/Filter.lean\",\"def_pos\":[163,5],\"def_end_pos\":[163,23]},{\"full_name\":\"Set.Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,8]},{\"full_name\":\"continuous_const\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1675,9],\"def_end_pos\":[1675,25]},{\"full_name\":\"Continuous.continuousWithinAt\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[973,9],\"def_end_pos\":[973,38]}]"
225
+ ],
226
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\n\u22a2 K.Pointed",
227
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\n\u22a2 K.Pointed",
228
+ "tactic": "have fc : ContinuousWithinAt f (Set.Ioi (0 : \u211d)) 0 :=\n (continuous_id.smul continuous_const).continuousWithinAt"
229
+ },
230
+ {
231
+ "annotated_tactic": [
232
+ "\"have mem\\u2080 := fc.mem_closure_image (by rw [<a>closure_Ioi<\\/a> (0 : \\u211d), <a>mem_Ici<\\/a>])\"",
233
+ "[{\"full_name\":\"closure_Ioi\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/DenselyOrdered.lean\",\"def_pos\":[34,9],\"def_end_pos\":[34,20]},{\"full_name\":\"Set.mem_Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[146,9],\"def_end_pos\":[146,16]}]"
234
+ ],
235
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\n\u22a2 K.Pointed",
236
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\n\u22a2 K.Pointed",
237
+ "tactic": "have mem\u2080 := fc.mem_closure_image (by rw [closure_Ioi (0 : \u211d), mem_Ici])"
238
+ },
239
+ {
240
+ "annotated_tactic": [
241
+ "\"have f\\u2080 : f 0 = 0 := <a>zero_smul<\\/a> \\u211d x\"",
242
+ "[{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
243
+ ],
244
+ "state_after": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\nf\u2080 : f 0 = 0\n\u22a2 K.Pointed",
245
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\n\u22a2 K.Pointed",
246
+ "tactic": "have f\u2080 : f 0 = 0 := zero_smul \u211d x"
247
+ },
248
+ {
249
+ "annotated_tactic": [
250
+ "\"simpa only [f\\u2080, <a>ConvexCone.Pointed<\\/a>, \\u2190 <a>SetLike.mem_coe<\\/a>] using <a>mem_of_subset_of_mem<\\/a> clf mem\\u2080\"",
251
+ "[{\"full_name\":\"ConvexCone.Pointed\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[327,5],\"def_end_pos\":[327,12]},{\"full_name\":\"SetLike.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[180,9],\"def_end_pos\":[180,16]},{\"full_name\":\"Set.mem_of_subset_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[360,9],\"def_end_pos\":[360,29]}]"
252
+ ],
253
+ "state_after": "no goals",
254
+ "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\nf\u2080 : f 0 = 0\n\u22a2 K.Pointed",
255
+ "tactic": "simpa only [f\u2080, ConvexCone.Pointed, \u2190 SetLike.mem_coe] using mem_of_subset_of_mem clf mem\u2080"
256
+ },
257
+ {
258
+ "annotated_tactic": [
259
+ "\"rintro _ \\u27e8_, h, rfl\\u27e9\"",
260
+ "[]"
261
+ ],
262
+ "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nw\u271d : \u211d\nh : w\u271d \u2208 Ioi 0\n\u22a2 f w\u271d \u2208 \u2191K",
263
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\n\u22a2 f '' Ioi 0 \u2286 \u2191K",
264
+ "tactic": "rintro _ \u27e8_, h, rfl\u27e9"
265
+ },
266
+ {
267
+ "annotated_tactic": [
268
+ "\"exact K.smul_mem (<a>Set.mem_Ioi<\\/a>.1 h) hx\"",
269
+ "[{\"full_name\":\"Set.mem_Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,16]}]"
270
+ ],
271
+ "state_after": "no goals",
272
+ "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nw\u271d : \u211d\nh : w\u271d \u2208 Ioi 0\n\u22a2 f w\u271d \u2208 \u2191K",
273
+ "tactic": "exact K.smul_mem (Set.mem_Ioi.1 h) hx"
274
+ },
275
+ {
276
+ "annotated_tactic": [
277
+ "\"rw [<a>closure_Ioi<\\/a> (0 : \\u211d), <a>mem_Ici<\\/a>]\"",
278
+ "[{\"full_name\":\"closure_Ioi\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/DenselyOrdered.lean\",\"def_pos\":[34,9],\"def_end_pos\":[34,20]},{\"full_name\":\"Set.mem_Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[146,9],\"def_end_pos\":[146,16]}]"
279
+ ],
280
+ "state_after": "no goals",
281
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\n\u22a2 0 \u2208 closure (Ioi 0)",
282
+ "tactic": "rw [closure_Ioi (0 : \u211d), mem_Ici]"
283
+ }
284
+ ],
285
+ "url": "https://github.com/leanprover-community/mathlib4"
286
+ },
287
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
288
+ },
289
+ {
290
+ "error": "theorem has no selected tactic transitions",
291
+ "full_name": "innerDualCone_empty",
292
+ "id": "innerDualCone_empty",
293
+ "reason": "import_failure",
294
+ "selection": {
295
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
296
+ "end": [
297
+ 67,
298
+ 41
299
+ ],
300
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
301
+ "full_name": "innerDualCone_empty",
302
+ "start": [
303
+ 66,
304
+ 1
305
+ ],
306
+ "traced_tactics": [],
307
+ "url": "https://github.com/leanprover-community/mathlib4"
308
+ },
309
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
310
+ },
311
+ {
312
+ "error": "theorem has no selected tactic transitions",
313
+ "full_name": "innerDualCone_le_innerDualCone",
314
+ "id": "innerDualCone_le_innerDualCone",
315
+ "reason": "import_failure",
316
+ "selection": {
317
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
318
+ "end": [
319
+ 86,
320
+ 31
321
+ ],
322
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
323
+ "full_name": "innerDualCone_le_innerDualCone",
324
+ "start": [
325
+ 85,
326
+ 1
327
+ ],
328
+ "traced_tactics": [],
329
+ "url": "https://github.com/leanprover-community/mathlib4"
330
+ },
331
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
332
+ },
333
+ {
334
+ "error": "theorem has no selected tactic transitions",
335
+ "full_name": "innerDualCone_singleton",
336
+ "id": "innerDualCone_singleton",
337
+ "reason": "import_failure",
338
+ "selection": {
339
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
340
+ "end": [
341
+ 96,
342
+ 36
343
+ ],
344
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
345
+ "full_name": "innerDualCone_singleton",
346
+ "start": [
347
+ 94,
348
+ 1
349
+ ],
350
+ "traced_tactics": [],
351
+ "url": "https://github.com/leanprover-community/mathlib4"
352
+ },
353
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
354
+ },
355
+ {
356
+ "error": "theorem has no selected tactic transitions",
357
+ "full_name": "innerDualCone_union",
358
+ "id": "innerDualCone_union",
359
+ "reason": "import_failure",
360
+ "selection": {
361
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
362
+ "end": [
363
+ 102,
364
+ 43
365
+ ],
366
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
367
+ "full_name": "innerDualCone_union",
368
+ "start": [
369
+ 99,
370
+ 1
371
+ ],
372
+ "traced_tactics": [],
373
+ "url": "https://github.com/leanprover-community/mathlib4"
374
+ },
375
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
376
+ },
377
+ {
378
+ "error": "no internal trace for tactic 'apply SetLike.coe_injective'",
379
+ "full_name": "innerDualCone_univ",
380
+ "id": "innerDualCone_univ",
381
+ "reason": "import_failure",
382
+ "selection": {
383
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
384
+ "end": [
385
+ 82,
386
+ 83
387
+ ],
388
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
389
+ "full_name": "innerDualCone_univ",
390
+ "start": [
391
+ 78,
392
+ 1
393
+ ],
394
+ "traced_tactics": [
395
+ {
396
+ "annotated_tactic": [
397
+ "\"suffices \\u2200 x : H, x \\u2208 (<a>univ<\\/a> : <a>Set<\\/a> H).<a>innerDualCone<\\/a> \\u2192 x = 0 by\\n apply <a>SetLike.coe_injective<\\/a>\\n exact eq_singleton_iff_unique_mem.mpr \\u27e8fun x _ => (<a>inner_zero_right<\\/a> _).<a>ge<\\/a>, this\\u27e9\"",
398
+ "[{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Set.innerDualCone\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[50,5],\"def_end_pos\":[50,22]},{\"full_name\":\"SetLike.coe_injective\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,22]},{\"full_name\":\"inner_zero_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,25]},{\"full_name\":\"Eq.ge\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[218,19],\"def_end_pos\":[218,21]}]"
399
+ ],
400
+ "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 x \u2208 univ.innerDualCone, x = 0",
401
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 univ.innerDualCone = 0",
402
+ "tactic": "suffices \u2200 x : H, x \u2208 (univ : Set H).innerDualCone \u2192 x = 0 by\n apply SetLike.coe_injective\n exact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9"
403
+ },
404
+ {
405
+ "annotated_tactic": [
406
+ "\"exact fun x hx => by simpa [\\u2190 <a>real_inner_self_nonpos<\\/a>] using hx (-x) (<a>mem_univ<\\/a> _)\"",
407
+ "[{\"full_name\":\"real_inner_self_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[612,9],\"def_end_pos\":[612,31]},{\"full_name\":\"Set.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[82,29],\"def_end_pos\":[82,37]}]"
408
+ ],
409
+ "state_after": "no goals",
410
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 x \u2208 univ.innerDualCone, x = 0",
411
+ "tactic": "exact fun x hx => by simpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _)"
412
+ },
413
+ {
414
+ "annotated_tactic": [
415
+ "\"apply <a>SetLike.coe_injective<\\/a>\"",
416
+ "[{\"full_name\":\"SetLike.coe_injective\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,22]}]"
417
+ ],
418
+ "state_after": "case a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nthis : \u2200 x \u2208 univ.innerDualCone, x = 0\n\u22a2 \u2191univ.innerDualCone = \u21910",
419
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nthis : \u2200 x \u2208 univ.innerDualCone, x = 0\n\u22a2 univ.innerDualCone = 0",
420
+ "tactic": "apply SetLike.coe_injective"
421
+ },
422
+ {
423
+ "annotated_tactic": [
424
+ "\"exact eq_singleton_iff_unique_mem.mpr \\u27e8fun x _ => (<a>inner_zero_right<\\/a> _).<a>ge<\\/a>, this\\u27e9\"",
425
+ "[{\"full_name\":\"inner_zero_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,25]},{\"full_name\":\"Eq.ge\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[218,19],\"def_end_pos\":[218,21]}]"
426
+ ],
427
+ "state_after": "no goals",
428
+ "state_before": "case a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nthis : \u2200 x \u2208 univ.innerDualCone, x = 0\n\u22a2 \u2191univ.innerDualCone = \u21910",
429
+ "tactic": "exact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9"
430
+ },
431
+ {
432
+ "annotated_tactic": [
433
+ "\"simpa [\\u2190 <a>real_inner_self_nonpos<\\/a>] using hx (-x) (<a>mem_univ<\\/a> _)\"",
434
+ "[{\"full_name\":\"real_inner_self_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[612,9],\"def_end_pos\":[612,31]},{\"full_name\":\"Set.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[82,29],\"def_end_pos\":[82,37]}]"
435
+ ],
436
+ "state_after": "no goals",
437
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : H\nhx : x \u2208 univ.innerDualCone\n\u22a2 x = 0",
438
+ "tactic": "simpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _)"
439
+ }
440
+ ],
441
+ "url": "https://github.com/leanprover-community/mathlib4"
442
+ },
443
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
444
+ },
445
+ {
446
+ "error": "theorem has no selected tactic transitions",
447
+ "full_name": "innerDualCone_zero",
448
+ "id": "innerDualCone_zero",
449
+ "reason": "import_failure",
450
+ "selection": {
451
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
452
+ "end": [
453
+ 73,
454
+ 76
455
+ ],
456
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
457
+ "full_name": "innerDualCone_zero",
458
+ "start": [
459
+ 72,
460
+ 1
461
+ ],
462
+ "traced_tactics": [],
463
+ "url": "https://github.com/leanprover-community/mathlib4"
464
+ },
465
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
466
+ },
467
+ {
468
+ "error": "no internal trace for tactic 'rw [innerDualCone_singleton, ConvexCone.coe_comap, ConvexCone.coe_positive, inner\u209b\u2097_apply_coe]'",
469
+ "full_name": "isClosed_innerDualCone",
470
+ "id": "isClosed_innerDualCone",
471
+ "reason": "import_failure",
472
+ "selection": {
473
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
474
+ "end": [
475
+ 140,
476
+ 70
477
+ ],
478
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
479
+ "full_name": "isClosed_innerDualCone",
480
+ "start": [
481
+ 130,
482
+ 1
483
+ ],
484
+ "traced_tactics": [
485
+ {
486
+ "annotated_tactic": [
487
+ "\"rw [<a>innerDualCone_eq_iInter_innerDualCone_singleton<\\/a>]\"",
488
+ "[{\"full_name\":\"innerDualCone_eq_iInter_innerDualCone_singleton\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,56]}]"
489
+ ],
490
+ "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 IsClosed (\u22c2 i, \u2191{\u2191i}.innerDualCone)",
491
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 IsClosed \u2191s.innerDualCone",
492
+ "tactic": "rw [innerDualCone_eq_iInter_innerDualCone_singleton]"
493
+ },
494
+ {
495
+ "annotated_tactic": [
496
+ "\"apply <a>isClosed_iInter<\\/a>\"",
497
+ "[{\"full_name\":\"isClosed_iInter\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,24]}]"
498
+ ],
499
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 (i : \u2191s), IsClosed \u2191{\u2191i}.innerDualCone",
500
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 IsClosed (\u22c2 i, \u2191{\u2191i}.innerDualCone)",
501
+ "tactic": "apply isClosed_iInter"
502
+ },
503
+ {
504
+ "annotated_tactic": [
505
+ "\"intro x\"",
506
+ "[]"
507
+ ],
508
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\n\u22a2 IsClosed \u2191{\u2191x}.innerDualCone",
509
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 (i : \u2191s), IsClosed \u2191{\u2191i}.innerDualCone",
510
+ "tactic": "intro x"
511
+ },
512
+ {
513
+ "annotated_tactic": [
514
+ "\"have h : ({\\u2191x} : <a>Set<\\/a> H).<a>innerDualCone<\\/a> = (<a>inner<\\/a> x : H \\u2192 \\u211d) \\u207b\\u00b9' <a>Set.Ici<\\/a> 0 := by\\n rw [<a>innerDualCone_singleton<\\/a>, <a>ConvexCone.coe_comap<\\/a>, <a>ConvexCone.coe_positive<\\/a>, <a>inner\\u209b\\u2097_apply_coe<\\/a>]\"",
515
+ "[{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Set.innerDualCone\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[50,5],\"def_end_pos\":[50,22]},{\"full_name\":\"Inner.inner\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[81,3],\"def_end_pos\":[81,8]},{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]},{\"full_name\":\"innerDualCone_singleton\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[94,9],\"def_end_pos\":[94,32]},{\"full_name\":\"ConvexCone.coe_comap\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[258,9],\"def_end_pos\":[258,18]},{\"full_name\":\"ConvexCone.coe_positive\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[563,9],\"def_end_pos\":[563,21]},{\"full_name\":\"inner\\u209b\\u2097_apply_coe\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[1753,9],\"def_end_pos\":[1753,26]}]"
516
+ ],
517
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191{\u2191x}.innerDualCone = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed \u2191{\u2191x}.innerDualCone",
518
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\n\u22a2 IsClosed \u2191{\u2191x}.innerDualCone",
519
+ "tactic": "have h : ({\u2191x} : Set H).innerDualCone = (inner x : H \u2192 \u211d) \u207b\u00b9' Set.Ici 0 := by\n rw [innerDualCone_singleton, ConvexCone.coe_comap, ConvexCone.coe_positive, inner\u209b\u2097_apply_coe]"
520
+ },
521
+ {
522
+ "annotated_tactic": [
523
+ "\"rw [h]\"",
524
+ "[]"
525
+ ],
526
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191{\u2191x}.innerDualCone = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed (inner \u2191x \u207b\u00b9' Ici 0)",
527
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191{\u2191x}.innerDualCone = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed \u2191{\u2191x}.innerDualCone",
528
+ "tactic": "rw [h]"
529
+ },
530
+ {
531
+ "annotated_tactic": [
532
+ "\"exact isClosed_Ici.preimage (continuous_const.inner <a>continuous_id'<\\/a>)\"",
533
+ "[{\"full_name\":\"continuous_id'\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1617,9],\"def_end_pos\":[1617,23]}]"
534
+ ],
535
+ "state_after": "no goals",
536
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191{\u2191x}.innerDualCone = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed (inner \u2191x \u207b\u00b9' Ici 0)",
537
+ "tactic": "exact isClosed_Ici.preimage (continuous_const.inner continuous_id')"
538
+ },
539
+ {
540
+ "annotated_tactic": [
541
+ "\"rw [<a>innerDualCone_singleton<\\/a>, <a>ConvexCone.coe_comap<\\/a>, <a>ConvexCone.coe_positive<\\/a>, <a>inner\\u209b\\u2097_apply_coe<\\/a>]\"",
542
+ "[{\"full_name\":\"innerDualCone_singleton\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[94,9],\"def_end_pos\":[94,32]},{\"full_name\":\"ConvexCone.coe_comap\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[258,9],\"def_end_pos\":[258,18]},{\"full_name\":\"ConvexCone.coe_positive\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[563,9],\"def_end_pos\":[563,21]},{\"full_name\":\"inner\\u209b\\u2097_apply_coe\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[1753,9],\"def_end_pos\":[1753,26]}]"
543
+ ],
544
+ "state_after": "no goals",
545
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\n\u22a2 \u2191{\u2191x}.innerDualCone = inner \u2191x \u207b\u00b9' Ici 0",
546
+ "tactic": "rw [innerDualCone_singleton, ConvexCone.coe_comap, ConvexCone.coe_positive, inner\u209b\u2097_apply_coe]"
547
+ }
548
+ ],
549
+ "url": "https://github.com/leanprover-community/mathlib4"
550
+ },
551
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
552
+ },
553
+ {
554
+ "error": "theorem has no selected tactic transitions",
555
+ "full_name": "mem_innerDualCone",
556
+ "id": "mem_innerDualCone",
557
+ "reason": "import_failure",
558
+ "selection": {
559
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
560
+ "end": [
561
+ 62,
562
+ 10
563
+ ],
564
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
565
+ "full_name": "mem_innerDualCone",
566
+ "start": [
567
+ 61,
568
+ 1
569
+ ],
570
+ "traced_tactics": [],
571
+ "url": "https://github.com/leanprover-community/mathlib4"
572
+ },
573
+ "source_file": "Mathlib/Analysis/Convex/Cone/InnerDual.lean"
574
+ },
575
+ {
576
+ "error": "theorem has no selected tactic transitions",
577
+ "full_name": "ConvexCone.coe_toPointedCone",
578
+ "id": "ConvexCone.coe_toPointedCone",
579
+ "reason": "import_failure",
580
+ "selection": {
581
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
582
+ "end": [
583
+ 84,
584
+ 6
585
+ ],
586
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
587
+ "full_name": "ConvexCone.coe_toPointedCone",
588
+ "start": [
589
+ 81,
590
+ 1
591
+ ],
592
+ "traced_tactics": [],
593
+ "url": "https://github.com/leanprover-community/mathlib4"
594
+ },
595
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
596
+ },
597
+ {
598
+ "error": "theorem has no selected tactic transitions",
599
+ "full_name": "ConvexCone.mem_toPointedCone",
600
+ "id": "ConvexCone.mem_toPointedCone",
601
+ "reason": "import_failure",
602
+ "selection": {
603
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
604
+ "end": [
605
+ 79,
606
+ 10
607
+ ],
608
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
609
+ "full_name": "ConvexCone.mem_toPointedCone",
610
+ "start": [
611
+ 76,
612
+ 1
613
+ ],
614
+ "traced_tactics": [],
615
+ "url": "https://github.com/leanprover-community/mathlib4"
616
+ },
617
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
618
+ },
619
+ {
620
+ "error": "theorem has no selected tactic transitions",
621
+ "full_name": "PointedCone.coe_comap",
622
+ "id": "PointedCone.coe_comap",
623
+ "reason": "import_failure",
624
+ "selection": {
625
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
626
+ "end": [
627
+ 141,
628
+ 6
629
+ ],
630
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
631
+ "full_name": "PointedCone.coe_comap",
632
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633
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634
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635
+ ],
636
+ "traced_tactics": [],
637
+ "url": "https://github.com/leanprover-community/mathlib4"
638
+ },
639
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
640
+ },
641
+ {
642
+ "error": "theorem has no selected tactic transitions",
643
+ "full_name": "PointedCone.coe_map",
644
+ "id": "PointedCone.coe_map",
645
+ "reason": "import_failure",
646
+ "selection": {
647
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
648
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649
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650
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651
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652
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
653
+ "full_name": "PointedCone.coe_map",
654
+ "start": [
655
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656
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657
+ ],
658
+ "traced_tactics": [],
659
+ "url": "https://github.com/leanprover-community/mathlib4"
660
+ },
661
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
662
+ },
663
+ {
664
+ "error": "theorem has no selected tactic transitions",
665
+ "full_name": "PointedCone.comap_comap",
666
+ "id": "PointedCone.comap_comap",
667
+ "reason": "import_failure",
668
+ "selection": {
669
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
670
+ "end": [
671
+ 149,
672
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673
+ ],
674
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
675
+ "full_name": "PointedCone.comap_comap",
676
+ "start": [
677
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678
+ 1
679
+ ],
680
+ "traced_tactics": [],
681
+ "url": "https://github.com/leanprover-community/mathlib4"
682
+ },
683
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
684
+ },
685
+ {
686
+ "error": "theorem has no selected tactic transitions",
687
+ "full_name": "PointedCone.comap_id",
688
+ "id": "PointedCone.comap_id",
689
+ "reason": "import_failure",
690
+ "selection": {
691
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
692
+ "end": [
693
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694
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695
+ ],
696
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
697
+ "full_name": "PointedCone.comap_id",
698
+ "start": [
699
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700
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701
+ ],
702
+ "traced_tactics": [],
703
+ "url": "https://github.com/leanprover-community/mathlib4"
704
+ },
705
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
706
+ },
707
+ {
708
+ "error": "theorem has no selected tactic transitions",
709
+ "full_name": "PointedCone.ext",
710
+ "id": "PointedCone.ext",
711
+ "reason": "import_failure",
712
+ "selection": {
713
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
714
+ "end": [
715
+ 56,
716
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717
+ ],
718
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
719
+ "full_name": "PointedCone.ext",
720
+ "start": [
721
+ 55,
722
+ 1
723
+ ],
724
+ "traced_tactics": [],
725
+ "url": "https://github.com/leanprover-community/mathlib4"
726
+ },
727
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
728
+ },
729
+ {
730
+ "error": "theorem has no selected tactic transitions",
731
+ "full_name": "PointedCone.map_id",
732
+ "id": "PointedCone.map_id",
733
+ "reason": "import_failure",
734
+ "selection": {
735
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
736
+ "end": [
737
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738
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739
+ ],
740
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
741
+ "full_name": "PointedCone.map_id",
742
+ "start": [
743
+ 132,
744
+ 1
745
+ ],
746
+ "traced_tactics": [],
747
+ "url": "https://github.com/leanprover-community/mathlib4"
748
+ },
749
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
750
+ },
751
+ {
752
+ "error": "theorem has no selected tactic transitions",
753
+ "full_name": "PointedCone.map_map",
754
+ "id": "PointedCone.map_map",
755
+ "reason": "import_failure",
756
+ "selection": {
757
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
758
+ "end": [
759
+ 129,
760
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761
+ ],
762
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
763
+ "full_name": "PointedCone.map_map",
764
+ "start": [
765
+ 127,
766
+ 1
767
+ ],
768
+ "traced_tactics": [],
769
+ "url": "https://github.com/leanprover-community/mathlib4"
770
+ },
771
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
772
+ },
773
+ {
774
+ "error": "theorem has no selected tactic transitions",
775
+ "full_name": "PointedCone.mem_comap",
776
+ "id": "PointedCone.mem_comap",
777
+ "reason": "import_failure",
778
+ "selection": {
779
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
780
+ "end": [
781
+ 153,
782
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783
+ ],
784
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
785
+ "full_name": "PointedCone.mem_comap",
786
+ "start": [
787
+ 152,
788
+ 1
789
+ ],
790
+ "traced_tactics": [],
791
+ "url": "https://github.com/leanprover-community/mathlib4"
792
+ },
793
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
794
+ },
795
+ {
796
+ "error": "theorem has no selected tactic transitions",
797
+ "full_name": "PointedCone.mem_map",
798
+ "id": "PointedCone.mem_map",
799
+ "reason": "import_failure",
800
+ "selection": {
801
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
802
+ "end": [
803
+ 125,
804
+ 10
805
+ ],
806
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
807
+ "full_name": "PointedCone.mem_map",
808
+ "start": [
809
+ 124,
810
+ 1
811
+ ],
812
+ "traced_tactics": [],
813
+ "url": "https://github.com/leanprover-community/mathlib4"
814
+ },
815
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
816
+ },
817
+ {
818
+ "error": "theorem has no selected tactic transitions",
819
+ "full_name": "PointedCone.mem_positive",
820
+ "id": "PointedCone.mem_positive",
821
+ "reason": "import_failure",
822
+ "selection": {
823
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
824
+ "end": [
825
+ 170,
826
+ 10
827
+ ],
828
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
829
+ "full_name": "PointedCone.mem_positive",
830
+ "start": [
831
+ 169,
832
+ 1
833
+ ],
834
+ "traced_tactics": [],
835
+ "url": "https://github.com/leanprover-community/mathlib4"
836
+ },
837
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
838
+ },
839
+ {
840
+ "error": "theorem has no selected tactic transitions",
841
+ "full_name": "PointedCone.toConvexCone_dual",
842
+ "id": "PointedCone.toConvexCone_dual",
843
+ "reason": "import_failure",
844
+ "selection": {
845
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
846
+ "end": [
847
+ 187,
848
+ 6
849
+ ],
850
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
851
+ "full_name": "PointedCone.toConvexCone_dual",
852
+ "start": [
853
+ 186,
854
+ 1
855
+ ],
856
+ "traced_tactics": [],
857
+ "url": "https://github.com/leanprover-community/mathlib4"
858
+ },
859
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
860
+ },
861
+ {
862
+ "error": "theorem has no selected tactic transitions",
863
+ "full_name": "PointedCone.toConvexCone_map",
864
+ "id": "PointedCone.toConvexCone_map",
865
+ "reason": "import_failure",
866
+ "selection": {
867
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
868
+ "end": [
869
+ 117,
870
+ 6
871
+ ],
872
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
873
+ "full_name": "PointedCone.toConvexCone_map",
874
+ "start": [
875
+ 115,
876
+ 1
877
+ ],
878
+ "traced_tactics": [],
879
+ "url": "https://github.com/leanprover-community/mathlib4"
880
+ },
881
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
882
+ },
883
+ {
884
+ "error": "theorem has no selected tactic transitions",
885
+ "full_name": "PointedCone.toConvexCone_positive",
886
+ "id": "PointedCone.toConvexCone_positive",
887
+ "reason": "import_failure",
888
+ "selection": {
889
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
890
+ "end": [
891
+ 174,
892
+ 6
893
+ ],
894
+ "file_path": "Mathlib/Analysis/Convex/Cone/Pointed.lean",
895
+ "full_name": "PointedCone.toConvexCone_positive",
896
+ "start": [
897
+ 173,
898
+ 1
899
+ ],
900
+ "traced_tactics": [],
901
+ "url": "https://github.com/leanprover-community/mathlib4"
902
+ },
903
+ "source_file": "Mathlib/Analysis/Convex/Cone/Pointed.lean"
904
+ }
905
+ ],
906
+ "input_proofs": 34,
907
+ "input_units": 2,
908
+ "proofs": 9,
909
+ "published_proofs": 8,
910
+ "published_transitions": 12,
911
+ "replay_failures": [],
912
+ "transitions": 20,
913
+ "validation_failures": [
914
+ {
915
+ "errors": [
916
+ "internal transition boundary mismatch",
917
+ "internal transition boundary mismatch"
918
+ ],
919
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
920
+ "proof_id": "mathlib:ConvexCone.innerDualCone_of_innerDualCone_eq_self",
921
+ "provenance": {
922
+ "selection": {
923
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
924
+ "end": [
925
+ 215,
926
+ 26
927
+ ],
928
+ "file_path": "Mathlib/Analysis/Convex/Cone/InnerDual.lean",
929
+ "full_name": "ConvexCone.innerDualCone_of_innerDualCone_eq_self",
930
+ "start": [
931
+ 205,
932
+ 1
933
+ ],
934
+ "traced_tactics": [
935
+ {
936
+ "annotated_tactic": [
937
+ "\"ext x\"",
938
+ "[]"
939
+ ],
940
+ "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone \u2194 x \u2208 K",
941
+ "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\n\u22a2 (\u2191(\u2191K).innerDualCone).innerDualCone = K",
942
+ "tactic": "ext x"
943
+ },
944
+ {
945
+ "annotated_tactic": [
946
+ "\"constructor\"",
947
+ "[]"
948
+ ],
949
+ "state_after": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone \u2192 x \u2208 K\n\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 K \u2192 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone",
950
+ "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone \u2194 x \u2208 K",
951
+ "tactic": "constructor"
952
+ },
953
+ {
954
+ "annotated_tactic": [
955
+ "\"rw [<a>mem_innerDualCone<\\/a>, \\u2190 <a>SetLike.mem_coe<\\/a>]\"",
956
+ "[{\"full_name\":\"mem_innerDualCone\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,26]},{\"full_name\":\"SetLike.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[180,9],\"def_end_pos\":[180,16]}]"
957
+ ],
958
+ "state_after": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 (\u2200 x_1 \u2208 \u2191(\u2191K).innerDualCone, 0 \u2264 \u27eax_1, x\u27eb_\u211d) \u2192 x \u2208 \u2191K",
959
+ "state_before": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone \u2192 x \u2208 K",
960
+ "tactic": "rw [mem_innerDualCone, \u2190 SetLike.mem_coe]"
961
+ },
962
+ {
963
+ "annotated_tactic": [
964
+ "\"contrapose!\"",
965
+ "[]"
966
+ ],
967
+ "state_after": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2209 \u2191K \u2192 \u2203 x_1 \u2208 \u2191(\u2191K).innerDualCone, \u27eax_1, x\u27eb_\u211d < 0",
968
+ "state_before": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 (\u2200 x_1 \u2208 \u2191(\u2191K).innerDualCone, 0 \u2264 \u27eax_1, x\u27eb_\u211d) \u2192 x \u2208 \u2191K",
969
+ "tactic": "contrapose!"
970
+ },
971
+ {
972
+ "annotated_tactic": [
973
+ "\"exact K.hyperplane_separation_of_nonempty_of_isClosed_of_nmem ne hc\"",
974
+ "[]"
975
+ ],
976
+ "state_after": "no goals",
977
+ "state_before": "case h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2209 \u2191K \u2192 \u2203 x_1 \u2208 \u2191(\u2191K).innerDualCone, \u27eax_1, x\u27eb_\u211d < 0",
978
+ "tactic": "exact K.hyperplane_separation_of_nonempty_of_isClosed_of_nmem ne hc"
979
+ },
980
+ {
981
+ "annotated_tactic": [
982
+ "\"rintro hxK y h\"",
983
+ "[]"
984
+ ],
985
+ "state_after": "case h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : y \u2208 \u2191(\u2191K).innerDualCone\n\u22a2 0 \u2264 \u27eay, x\u27eb_\u211d",
986
+ "state_before": "case h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 K \u2192 x \u2208 (\u2191(\u2191K).innerDualCone).innerDualCone",
987
+ "tactic": "rintro hxK y h"
988
+ },
989
+ {
990
+ "annotated_tactic": [
991
+ "\"specialize h x hxK\"",
992
+ "[]"
993
+ ],
994
+ "state_after": "case h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : 0 \u2264 \u27eax, y\u27eb_\u211d\n\u22a2 0 \u2264 \u27eay, x\u27eb_\u211d",
995
+ "state_before": "case h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : y \u2208 \u2191(\u2191K).innerDualCone\n\u22a2 0 \u2264 \u27eay, x\u27eb_\u211d",
996
+ "tactic": "specialize h x hxK"
997
+ },
998
+ {
999
+ "annotated_tactic": [
1000
+ "\"rwa [<a>real_inner_comm<\\/a>]\"",
1001
+ "[{\"full_name\":\"real_inner_comm\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,24]}]"
1002
+ ],
1003
+ "state_after": "no goals",
1004
+ "state_before": "case h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : (\u2191K).Nonempty\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : 0 \u2264 \u27eax, y\u27eb_\u211d\n\u22a2 0 \u2264 \u27eay, x\u27eb_\u211d",
1005
+ "tactic": "rwa [real_inner_comm]"
1006
+ }
1007
+ ],
1008
+ "url": "https://github.com/leanprover-community/mathlib4"
1009
+ }
1010
+ },
1011
+ "theorem_name": "ConvexCone.innerDualCone_of_innerDualCone_eq_self"
1012
+ }
1013
+ ],
1014
+ "validation_rejected_proofs": 1,
1015
+ "zero_transition_proofs": 0
1016
+ },
1017
+ "shards": [
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+ {
1019
+ "path": "data/train/mathlib-000456-0000-16c883ed2d2b4e1b.parquet",
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+ "sha256": "16c883ed2d2b4e1b51ca65464d273718935415b5e585f79f7204dabe68c0a120",
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+ "bytes": 59146,
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+ "rows": 10,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ "sha256": "8a8cbb2b6f97b123c6c7da04d900a476d248a43f0c438e224e8de33a7103d283",
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
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+ "path": "data/internal_test/mathlib-000456-0000-c80cf2e2ec15134c.parquet",
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+ "sha256": "c80cf2e2ec15134c326cbd65a0cfcc4b856e00d2871d62cfce1be2df31129c76",
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+ "bytes": 34668,
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+ "rows": 1,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ }
1039
+ ],
1040
+ "proof_ids": [
1041
+ "mathlib:PointedCone.mem_dual",
1042
+ "mathlib:PointedCone.toConvexCone_injective",
1043
+ "mathlib:PointedCone.toConvexCone_pointed",
1044
+ "mathlib:innerDualCone_eq_iInter_innerDualCone_singleton",
1045
+ "mathlib:innerDualCone_iUnion",
1046
+ "mathlib:innerDualCone_insert",
1047
+ "mathlib:innerDualCone_sUnion",
1048
+ "mathlib:pointed_innerDualCone"
1049
+ ],
1050
+ "proof_keys": [
1051
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/InnerDual.lean\",null,null,\"innerDualCone_eq_iInter_innerDualCone_singleton\"]",
1052
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/InnerDual.lean\",null,null,\"innerDualCone_iUnion\"]",
1053
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/InnerDual.lean\",null,null,\"innerDualCone_insert\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/InnerDual.lean\",null,null,\"innerDualCone_sUnion\"]",
1055
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/InnerDual.lean\",null,null,\"pointed_innerDualCone\"]",
1056
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Pointed.lean\",null,null,\"PointedCone.mem_dual\"]",
1057
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Pointed.lean\",null,null,\"PointedCone.toConvexCone_injective\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Pointed.lean\",null,null,\"PointedCone.toConvexCone_pointed\"]"
1059
+ ]
1060
+ }
manifests/batches/mathlib/000457.json ADDED
@@ -0,0 +1,675 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "mathlib/000457",
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+ "batch_fingerprint": "2947db3fe847eb4a5db4bae3da775474e15c4f29ec8589a3356d56edbff18fd3",
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+ "full_name": "ProperCone.coe_comap",
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+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
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+ },
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+ {
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+ "error": "theorem has no selected tactic transitions",
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
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+ },
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ },
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+ {
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+ "id": "ProperCone.coe_positive",
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+ "selection": {
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+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
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+ },
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+ {
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+ "error": "theorem has no selected tactic transitions",
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+ "full_name": "ProperCone.coe_zero",
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+ "id": "ProperCone.coe_zero",
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+ },
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+ },
119
+ {
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121
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+ "id": "ProperCone.comap_id",
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
139
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
140
+ },
141
+ {
142
+ "error": "theorem has no selected tactic transitions",
143
+ "full_name": "ProperCone.dual_dual",
144
+ "id": "ProperCone.dual_dual",
145
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146
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+ "start": [
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+ ],
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+ "traced_tactics": [],
159
+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
162
+ },
163
+ {
164
+ "error": "theorem has no selected tactic transitions",
165
+ "full_name": "ProperCone.ext",
166
+ "id": "ProperCone.ext",
167
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168
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182
+ },
183
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
184
+ },
185
+ {
186
+ "error": "theorem has no selected tactic transitions",
187
+ "full_name": "ProperCone.isClosed",
188
+ "id": "ProperCone.isClosed",
189
+ "reason": "import_failure",
190
+ "selection": {
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+ "full_name": "ProperCone.isClosed",
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203
+ "url": "https://github.com/leanprover-community/mathlib4"
204
+ },
205
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
206
+ },
207
+ {
208
+ "error": "theorem has no selected tactic transitions",
209
+ "full_name": "ProperCone.mem_coe",
210
+ "id": "ProperCone.mem_coe",
211
+ "reason": "import_failure",
212
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+ "traced_tactics": [],
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226
+ },
227
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
228
+ },
229
+ {
230
+ "error": "theorem has no selected tactic transitions",
231
+ "full_name": "ProperCone.mem_comap",
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+ "id": "ProperCone.mem_comap",
233
+ "reason": "import_failure",
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+ "url": "https://github.com/leanprover-community/mathlib4"
248
+ },
249
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
250
+ },
251
+ {
252
+ "error": "theorem has no selected tactic transitions",
253
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+ "id": "ProperCone.mem_map",
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+ },
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+ },
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+ {
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+ },
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+ {
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+ "id": "ProperCone.mem_zero",
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314
+ },
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316
+ },
317
+ {
318
+ "error": "theorem has no selected tactic transitions",
319
+ "full_name": "ProperCone.pointed",
320
+ "id": "ProperCone.pointed",
321
+ "reason": "import_failure",
322
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336
+ },
337
+ "source_file": "Mathlib/Analysis/Convex/Cone/Proper.lean"
338
+ },
339
+ {
340
+ "error": "theorem has no selected tactic transitions",
341
+ "full_name": "Convex.contractibleSpace",
342
+ "id": "Convex.contractibleSpace",
343
+ "reason": "import_failure",
344
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350
+ "file_path": "Mathlib/Analysis/Convex/Contractible.lean",
351
+ "full_name": "Convex.contractibleSpace",
352
+ "start": [
353
+ 42,
354
+ 11
355
+ ],
356
+ "traced_tactics": [],
357
+ "url": "https://github.com/leanprover-community/mathlib4"
358
+ },
359
+ "source_file": "Mathlib/Analysis/Convex/Contractible.lean"
360
+ },
361
+ {
362
+ "error": "no internal trace for tactic 'ext1'",
363
+ "full_name": "StarConvex.contractibleSpace",
364
+ "id": "StarConvex.contractibleSpace",
365
+ "reason": "import_failure",
366
+ "selection": {
367
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
368
+ "end": [
369
+ 38,
370
+ 9
371
+ ],
372
+ "file_path": "Mathlib/Analysis/Convex/Contractible.lean",
373
+ "full_name": "StarConvex.contractibleSpace",
374
+ "start": [
375
+ 23,
376
+ 11
377
+ ],
378
+ "traced_tactics": [
379
+ {
380
+ "annotated_tactic": [
381
+ "\"refine\\n (<a>contractible_iff_id_nullhomotopic<\\/a> s).2\\n \\u27e8\\u27e8x, h.mem hne\\u27e9,\\n \\u27e8\\u27e8\\u27e8fun p => \\u27e8p.1.1 \\u2022 x + (1 - p.1.1) \\u2022 (p.2 : E), ?_\\u27e9, ?_\\u27e9, fun x => ?_, fun x => ?_\\u27e9\\u27e9\\u27e9\"",
382
+ "[{\"full_name\":\"contractible_iff_id_nullhomotopic\",\"def_path\":\"Mathlib\\/Topology\\/Homotopy\\/Contractible.lean\",\"def_pos\":[68,9],\"def_end_pos\":[68,42]}]"
383
+ ],
384
+ "state_after": "case refine_1\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : s.Nonempty\np : \u2191unitInterval \u00d7 \u2191s\n\u22a2 \u2191p.1 \u2022 x + (1 - \u2191p.1) \u2022 \u2191p.2 \u2208 s\n\ncase refine_2\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : s.Nonempty\n\u22a2 Continuous fun p => \u27e8\u2191p.1 \u2022 x + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9\n\ncase refine_3\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 { toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := ?refine_2 }.toFun (0, x) =\n (ContinuousMap.id \u2191s) x\n\ncase refine_4\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 { toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := ?refine_2 }.toFun (1, x) =\n (ContinuousMap.const \u2191s \u27e8x\u271d, \u22ef\u27e9) x",
385
+ "state_before": "E : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : s.Nonempty\n\u22a2 ContractibleSpace \u2191s",
386
+ "tactic": "refine\n (contractible_iff_id_nullhomotopic s).2\n \u27e8\u27e8x, h.mem hne\u27e9,\n \u27e8\u27e8\u27e8fun p => \u27e8p.1.1 \u2022 x + (1 - p.1.1) \u2022 (p.2 : E), ?_\u27e9, ?_\u27e9, fun x => ?_, fun x => ?_\u27e9\u27e9\u27e9"
387
+ },
388
+ {
389
+ "annotated_tactic": [
390
+ "\"exact h p.2.2 p.1.2.1 (<a>sub_nonneg<\\/a>.2 p.1.2.2) (<a>add_sub_cancel<\\/a> _ _)\"",
391
+ "[{\"full_name\":\"sub_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[720,30],\"def_end_pos\":[720,40]},{\"full_name\":\"add_sub_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]}]"
392
+ ],
393
+ "state_after": "no goals",
394
+ "state_before": "case refine_1\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : s.Nonempty\np : \u2191unitInterval \u00d7 \u2191s\n\u22a2 \u2191p.1 \u2022 x + (1 - \u2191p.1) \u2022 \u2191p.2 \u2208 s",
395
+ "tactic": "exact h p.2.2 p.1.2.1 (sub_nonneg.2 p.1.2.2) (add_sub_cancel _ _)"
396
+ },
397
+ {
398
+ "annotated_tactic": [
399
+ "\"exact\\n ((continuous_subtype_val.fst'.smul <a>continuous_const<\\/a>).<a>add<\\/a>\\n ((continuous_const.sub continuous_subtype_val.fst').<a>smul<\\/a>\\n continuous_subtype_val.snd')).<a>subtype_mk<\\/a>\\n _\"",
400
+ "[{\"full_name\":\"continuous_const\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1675,9],\"def_end_pos\":[1675,25]},{\"full_name\":\"Continuous.add\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Monoid.lean\",\"def_pos\":[96,3],\"def_end_pos\":[96,14]},{\"full_name\":\"Continuous.smul\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/MulAction.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,24]},{\"full_name\":\"Continuous.subtype_mk\",\"def_path\":\"Mathlib\\/Topology\\/Constructions.lean\",\"def_pos\":[1122,9],\"def_end_pos\":[1122,30]}]"
401
+ ],
402
+ "state_after": "no goals",
403
+ "state_before": "case refine_2\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : s.Nonempty\n\u22a2 Continuous fun p => \u27e8\u2191p.1 \u2022 x + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9",
404
+ "tactic": "exact\n ((continuous_subtype_val.fst'.smul continuous_const).add\n ((continuous_const.sub continuous_subtype_val.fst').smul\n continuous_subtype_val.snd')).subtype_mk\n _"
405
+ },
406
+ {
407
+ "annotated_tactic": [
408
+ "\"ext1\"",
409
+ "[]"
410
+ ],
411
+ "state_after": "case refine_3.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 \u2191({ toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (0, x)) =\n \u2191((ContinuousMap.id \u2191s) x)",
412
+ "state_before": "case refine_3\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 { toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (0, x) = (ContinuousMap.id \u2191s) x",
413
+ "tactic": "ext1"
414
+ },
415
+ {
416
+ "annotated_tactic": [
417
+ "\"simp\"",
418
+ "[]"
419
+ ],
420
+ "state_after": "no goals",
421
+ "state_before": "case refine_3.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 \u2191({ toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (0, x)) =\n \u2191((ContinuousMap.id \u2191s) x)",
422
+ "tactic": "simp"
423
+ },
424
+ {
425
+ "annotated_tactic": [
426
+ "\"ext1\"",
427
+ "[]"
428
+ ],
429
+ "state_after": "case refine_4.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 \u2191({ toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (1, x)) =\n \u2191((ContinuousMap.const \u2191s \u27e8x\u271d, \u22ef\u27e9) x)",
430
+ "state_before": "case refine_4\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 { toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (1, x) =\n (ContinuousMap.const \u2191s \u27e8x\u271d, \u22ef\u27e9) x",
431
+ "tactic": "ext1"
432
+ },
433
+ {
434
+ "annotated_tactic": [
435
+ "\"simp\"",
436
+ "[]"
437
+ ],
438
+ "state_after": "no goals",
439
+ "state_before": "case refine_4.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : s.Nonempty\nx : \u2191s\n\u22a2 \u2191({ toFun := fun p => \u27e8\u2191p.1 \u2022 x\u271d + (1 - \u2191p.1) \u2022 \u2191p.2, \u22ef\u27e9, continuous_toFun := \u22ef }.toFun (1, x)) =\n \u2191((ContinuousMap.const \u2191s \u27e8x\u271d, \u22ef\u27e9) x)",
440
+ "tactic": "simp"
441
+ }
442
+ ],
443
+ "url": "https://github.com/leanprover-community/mathlib4"
444
+ },
445
+ "source_file": "Mathlib/Analysis/Convex/Contractible.lean"
446
+ }
447
+ ],
448
+ "input_proofs": 25,
449
+ "input_units": 2,
450
+ "proofs": 8,
451
+ "published_proofs": 7,
452
+ "published_transitions": 10,
453
+ "replay_failures": [],
454
+ "transitions": 28,
455
+ "validation_failures": [
456
+ {
457
+ "errors": [
458
+ "internal transition boundary mismatch"
459
+ ],
460
+ "file_path": "Mathlib/Analysis/Convex/Cone/Proper.lean",
461
+ "proof_id": "mathlib:ProperCone.hyperplane_separation",
462
+ "provenance": {
463
+ "selection": {
464
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
465
+ "end": [
466
+ 283,
467
+ 27
468
+ ],
469
+ "file_path": "Mathlib/Analysis/Convex/Cone/Proper.lean",
470
+ "full_name": "ProperCone.hyperplane_separation",
471
+ "start": [
472
+ 243,
473
+ 1
474
+ ],
475
+ "traced_tactics": [
476
+ {
477
+ "annotated_tactic": [
478
+ "\"simp_rw [<a>mem_map<\\/a>, <a>PointedCone.mem_closure<\\/a>, <a>PointedCone.coe_map<\\/a>, <a>coe_coe<\\/a>,\\n <a>mem_closure_iff_seq_limit<\\/a>, <a>mem_image<\\/a>, <a>SetLike.mem_coe<\\/a>, <a>mem_coe<\\/a>, <a>mem_dual<\\/a>,\\n <a>adjoint_inner_right<\\/a>, <a>forall_exists_index<\\/a>, <a>and_imp<\\/a>]\"",
479
+ "[{\"full_name\":\"ProperCone.mem_map\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,16]},{\"full_name\":\"PointedCone.mem_closure\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Closure.lean\",\"def_pos\":[70,19],\"def_end_pos\":[70,30]},{\"full_name\":\"PointedCone.coe_map\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Pointed.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,16]},{\"full_name\":\"ContinuousLinearMap.coe_coe\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[527,9],\"def_end_pos\":[527,16]},{\"full_name\":\"mem_closure_iff_seq_limit\",\"def_path\":\"Mathlib\\/Topology\\/Sequences.lean\",\"def_pos\":[113,9],\"def_end_pos\":[113,34]},{\"full_name\":\"Set.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"SetLike.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[180,9],\"def_end_pos\":[180,16]},{\"full_name\":\"ProperCone.mem_coe\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,16]},{\"full_name\":\"ProperCone.mem_dual\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[193,9],\"def_end_pos\":[193,17]},{\"full_name\":\"ContinuousLinearMap.adjoint_inner_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Adjoint.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,28]},{\"full_name\":\"forall_exists_index\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[189,17],\"def_end_pos\":[189,36]},{\"full_name\":\"and_imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[115,17],\"def_end_pos\":[115,24]}]"
480
+ ],
481
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\n\u22a2 \u2200 (x : \u2115 \u2192 F),\n (\u2200 (n : \u2115), \u2203 x_1 \u2208 K, f x_1 = x n) \u2192\n Tendsto x atTop (nhds b) \u2192 \u2200 (y : F), (\u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d) \u2192 0 \u2264 \u27eay, b\u27eb_\u211d",
482
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\n\u22a2 b \u2208 map f K \u2192 \u2200 (y : F), (adjoint f) y \u2208 K.dual \u2192 0 \u2264 \u27eay, b\u27eb_\u211d",
483
+ "tactic": "simp_rw [mem_map, PointedCone.mem_closure, PointedCone.coe_map, coe_coe,\n mem_closure_iff_seq_limit, mem_image, SetLike.mem_coe, mem_coe, mem_dual,\n adjoint_inner_right, forall_exists_index, and_imp]"
484
+ },
485
+ {
486
+ "annotated_tactic": [
487
+ "\"rintro seq hmem htends y hinner\"",
488
+ "[]"
489
+ ],
490
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\n\u22a2 0 \u2264 \u27eay, b\u27eb_\u211d",
491
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\n\u22a2 \u2200 (x : \u2115 \u2192 F),\n (\u2200 (n : \u2115), \u2203 x_1 \u2208 K, f x_1 = x n) \u2192\n Tendsto x atTop (nhds b) \u2192 \u2200 (y : F), (\u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d) \u2192 0 \u2264 \u27eay, b\u27eb_\u211d",
492
+ "tactic": "rintro seq hmem htends y hinner"
493
+ },
494
+ {
495
+ "annotated_tactic": [
496
+ "\"suffices h : \\u2200 n, 0 \\u2264 \\u27eay, seq n\\u27eb_\\u211d from\\n <a>ge_of_tendsto'<\\/a>\\n (<a>Continuous.seqContinuous<\\/a> (<a>Continuous.inner<\\/a> (@<a>continuous_const<\\/a> _ _ _ _ y) <a>continuous_id<\\/a>)\\n htends)\\n h\"",
497
+ "[{\"full_name\":\"ge_of_tendsto'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[399,9],\"def_end_pos\":[399,23]},{\"full_name\":\"Continuous.seqContinuous\",\"def_path\":\"Mathlib\\/Topology\\/Sequences.lean\",\"def_pos\":[194,19],\"def_end_pos\":[194,43]},{\"full_name\":\"Continuous.inner\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[2294,9],\"def_end_pos\":[2294,25]},{\"full_name\":\"continuous_const\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1675,9],\"def_end_pos\":[1675,25]},{\"full_name\":\"continuous_id\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1611,9],\"def_end_pos\":[1611,22]}]"
498
+ ],
499
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\n\u22a2 \u2200 (n : \u2115), 0 \u2264 \u27eay, seq n\u27eb_\u211d",
500
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\n\u22a2 0 \u2264 \u27eay, b\u27eb_\u211d",
501
+ "tactic": "suffices h : \u2200 n, 0 \u2264 \u27eay, seq n\u27eb_\u211d from\n ge_of_tendsto'\n (Continuous.seqContinuous (Continuous.inner (@continuous_const _ _ _ _ y) continuous_id)\n htends)\n h"
502
+ },
503
+ {
504
+ "annotated_tactic": [
505
+ "\"intro n\"",
506
+ "[]"
507
+ ],
508
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\nn : \u2115\n\u22a2 0 \u2264 \u27eay, seq n\u27eb_\u211d",
509
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\n\u22a2 \u2200 (n : \u2115), 0 \u2264 \u27eay, seq n\u27eb_\u211d",
510
+ "tactic": "intro n"
511
+ },
512
+ {
513
+ "annotated_tactic": [
514
+ "\"obtain \\u27e8_, h, hseq\\u27e9 := hmem n\"",
515
+ "[]"
516
+ ],
517
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\nn : \u2115\nw\u271d : E\nh : w\u271d \u2208 K\nhseq : f w\u271d = seq n\n\u22a2 0 \u2264 \u27eay, seq n\u27eb_\u211d",
518
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\nn : \u2115\n\u22a2 0 \u2264 \u27eay, seq n\u27eb_\u211d",
519
+ "tactic": "obtain \u27e8_, h, hseq\u27e9 := hmem n"
520
+ },
521
+ {
522
+ "annotated_tactic": [
523
+ "\"simpa only [\\u2190 hseq, <a>real_inner_comm<\\/a>] using hinner h\"",
524
+ "[{\"full_name\":\"real_inner_comm\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Basic.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,24]}]"
525
+ ],
526
+ "state_after": "no goals",
527
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nseq : \u2115 \u2192 F\nhmem : \u2200 (n : \u2115), \u2203 x \u2208 K, f x = seq n\nhtends : Tendsto seq atTop (nhds b)\ny : F\nhinner : \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d\nn : \u2115\nw\u271d : E\nh : w\u271d \u2208 K\nhseq : f w\u271d = seq n\n\u22a2 0 \u2264 \u27eay, seq n\u27eb_\u211d",
528
+ "tactic": "simpa only [\u2190 hseq, real_inner_comm] using hinner h"
529
+ },
530
+ {
531
+ "annotated_tactic": [
532
+ "\"intro h\"",
533
+ "[]"
534
+ ],
535
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : \u2200 (y : F), (adjoint f) y \u2208 K.dual \u2192 0 \u2264 \u27eay, b\u27eb_\u211d\n\u22a2 b \u2208 map f K",
536
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\n\u22a2 (\u2200 (y : F), (adjoint f) y \u2208 K.dual \u2192 0 \u2264 \u27eay, b\u27eb_\u211d) \u2192 b \u2208 map f K",
537
+ "tactic": "intro h"
538
+ },
539
+ {
540
+ "annotated_tactic": [
541
+ "\"contrapose! h\"",
542
+ "[]"
543
+ ],
544
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
545
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : \u2200 (y : F), (adjoint f) y \u2208 K.dual \u2192 0 \u2264 \u27eay, b\u27eb_\u211d\n\u22a2 b \u2208 map f K",
546
+ "tactic": "contrapose! h"
547
+ },
548
+ {
549
+ "annotated_tactic": [
550
+ "\"let C := @<a>PointedCone.toConvexCone<\\/a> \\u211d F _ _ _ (K.map f)\"",
551
+ "[{\"full_name\":\"PointedCone.toConvexCone\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Pointed.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]}]"
552
+ ],
553
+ "state_after": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
554
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
555
+ "tactic": "let C := @PointedCone.toConvexCone \u211d F _ _ _ (K.map f)"
556
+ },
557
+ {
558
+ "annotated_tactic": [
559
+ "\"obtain \\u27e8y, hxy, hyb\\u27e9 :=\\n @<a>ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem<\\/a>\\n _ _ _ _ C (K.map f).<a>nonempty<\\/a> (K.map f).<a>isClosed<\\/a> b h\"",
560
+ "[{\"full_name\":\"ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/InnerDual.lean\",\"def_pos\":[170,9],\"def_end_pos\":[170,73]},{\"full_name\":\"ProperCone.nonempty\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[87,19],\"def_end_pos\":[87,27]},{\"full_name\":\"ProperCone.isClosed\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[91,19],\"def_end_pos\":[91,27]}]"
561
+ ],
562
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
563
+ "state_before": "E : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
564
+ "tactic": "obtain \u27e8y, hxy, hyb\u27e9 :=\n @ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem\n _ _ _ _ C (K.map f).nonempty (K.map f).isClosed b h"
565
+ },
566
+ {
567
+ "annotated_tactic": [
568
+ "\"refine \\u27e8y, ?_, hyb\\u27e9\"",
569
+ "[]"
570
+ ],
571
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 (adjoint f) y \u2208 K.dual",
572
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 \u2203 y, (adjoint f) y \u2208 K.dual \u2227 \u27eay, b\u27eb_\u211d < 0",
573
+ "tactic": "refine \u27e8y, ?_, hyb\u27e9"
574
+ },
575
+ {
576
+ "annotated_tactic": [
577
+ "\"simp_rw [<a>ProperCone.mem_dual<\\/a>, <a>adjoint_inner_right<\\/a>]\"",
578
+ "[{\"full_name\":\"ProperCone.mem_dual\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[193,9],\"def_end_pos\":[193,17]},{\"full_name\":\"ContinuousLinearMap.adjoint_inner_right\",\"def_path\":\"Mathlib\\/Analysis\\/InnerProductSpace\\/Adjoint.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,28]}]"
579
+ ],
580
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d",
581
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 (adjoint f) y \u2208 K.dual",
582
+ "tactic": "simp_rw [ProperCone.mem_dual, adjoint_inner_right]"
583
+ },
584
+ {
585
+ "annotated_tactic": [
586
+ "\"intro x hxK\"",
587
+ "[]"
588
+ ],
589
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 0 \u2264 \u27eaf x, y\u27eb_\u211d",
590
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\n\u22a2 \u2200 \u2983x : E\u2984, x \u2208 K \u2192 0 \u2264 \u27eaf x, y\u27eb_\u211d",
591
+ "tactic": "intro x hxK"
592
+ },
593
+ {
594
+ "annotated_tactic": [
595
+ "\"apply hxy (f x)\"",
596
+ "[]"
597
+ ],
598
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 C",
599
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 0 \u2264 \u27eaf x, y\u27eb_\u211d",
600
+ "tactic": "apply hxy (f x)"
601
+ },
602
+ {
603
+ "annotated_tactic": [
604
+ "\"simp_rw [C, <a>coe_map<\\/a>]\"",
605
+ "[{\"full_name\":\"ProperCone.coe_map\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Proper.lean\",\"def_pos\":[165,9],\"def_end_pos\":[165,16]}]"
606
+ ],
607
+ "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 \u2191(PointedCone.map \u2191f \u2191K).closure",
608
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 C",
609
+ "tactic": "simp_rw [C, coe_map]"
610
+ },
611
+ {
612
+ "annotated_tactic": [
613
+ "\"apply <a>subset_closure<\\/a>\"",
614
+ "[{\"full_name\":\"subset_closure\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[403,9],\"def_end_pos\":[403,23]}]"
615
+ ],
616
+ "state_after": "case intro.intro.a\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 \u2191\u2191(PointedCone.map \u2191f \u2191K)",
617
+ "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 \u2191(PointedCone.map \u2191f \u2191K).closure",
618
+ "tactic": "apply subset_closure"
619
+ },
620
+ {
621
+ "annotated_tactic": [
622
+ "\"simp_rw [<a>PointedCone.toConvexCone_map<\\/a>, <a>ConvexCone.coe_map<\\/a>, <a>coe_coe<\\/a>, <a>mem_image<\\/a>,\\n <a>SetLike.mem_coe<\\/a>]\"",
623
+ "[{\"full_name\":\"PointedCone.toConvexCone_map\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Pointed.lean\",\"def_pos\":[115,9],\"def_end_pos\":[115,25]},{\"full_name\":\"ConvexCone.coe_map\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Cone\\/Basic.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,16]},{\"full_name\":\"ContinuousLinearMap.coe_coe\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[527,9],\"def_end_pos\":[527,16]},{\"full_name\":\"Set.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"SetLike.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[180,9],\"def_end_pos\":[180,16]}]"
624
+ ],
625
+ "state_after": "case intro.intro.a\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 \u2203 x_1 \u2208 \u2191\u2191K, f x_1 = f x",
626
+ "state_before": "case intro.intro.a\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 f x \u2208 \u2191\u2191(PointedCone.map \u2191f \u2191K)",
627
+ "tactic": "simp_rw [PointedCone.toConvexCone_map, ConvexCone.coe_map, coe_coe, mem_image,\n SetLike.mem_coe]"
628
+ },
629
+ {
630
+ "annotated_tactic": [
631
+ "\"exact \\u27e8x, hxK, <a>rfl<\\/a>\\u27e9\"",
632
+ "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
633
+ ],
634
+ "state_after": "no goals",
635
+ "state_before": "case intro.intro.a\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nK : ProperCone \u211d E\nf : E \u2192L[\u211d] F\nb : F\nh : b \u2209 map f K\nC : ConvexCone \u211d F := \u2191\u2191(map f K)\ny : F\nhxy : \u2200 x \u2208 C, 0 \u2264 \u27eax, y\u27eb_\u211d\nhyb : \u27eay, b\u27eb_\u211d < 0\nx : E\nhxK : x \u2208 K\n\u22a2 \u2203 x_1 \u2208 \u2191\u2191K, f x_1 = f x",
636
+ "tactic": "exact \u27e8x, hxK, rfl\u27e9"
637
+ }
638
+ ],
639
+ "url": "https://github.com/leanprover-community/mathlib4"
640
+ }
641
+ },
642
+ "theorem_name": "ProperCone.hyperplane_separation"
643
+ }
644
+ ],
645
+ "validation_rejected_proofs": 1,
646
+ "zero_transition_proofs": 0
647
+ },
648
+ "shards": [
649
+ {
650
+ "path": "data/train/mathlib-000457-0000-5e66fb2909d6bd00.parquet",
651
+ "sha256": "5e66fb2909d6bd00d15aaf365b29435de0484b6e8d41b5c05b71ef82ac2dbe41",
652
+ "bytes": 85284,
653
+ "rows": 10,
654
+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
655
+ }
656
+ ],
657
+ "proof_ids": [
658
+ "mathlib:ProperCone.comap_comap",
659
+ "mathlib:ProperCone.hyperplane_separation_of_nmem",
660
+ "mathlib:ProperCone.map_id",
661
+ "mathlib:ProperCone.mem_dual",
662
+ "mathlib:ProperCone.nonempty",
663
+ "mathlib:ProperCone.pointed_zero",
664
+ "mathlib:ProperCone.toPointedCone_injective"
665
+ ],
666
+ "proof_keys": [
667
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.comap_comap\"]",
668
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.hyperplane_separation_of_nmem\"]",
669
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.map_id\"]",
670
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.mem_dual\"]",
671
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.nonempty\"]",
672
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.pointed_zero\"]",
673
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Convex/Cone/Proper.lean\",null,null,\"ProperCone.toPointedCone_injective\"]"
674
+ ]
675
+ }
manifests/batches/mathlib/000458.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/mathlib/000459.json ADDED
The diff for this file is too large to render. See raw diff
 
manifests/batches/numina/001213.json ADDED
@@ -0,0 +1,52 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "numina/001213",
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+ "batch_fingerprint": "5783adf67605f47fd05278f6f82f221a5d4e0886ed308dce0866ee3c7d5028be",
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+ "evidence": {
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+ "batch_fingerprint": "5783adf67605f47fd05278f6f82f221a5d4e0886ed308dce0866ee3c7d5028be",
7
+ "bundle_error": "helper declarations require isolated extraction",
8
+ "contamination_matches": [],
9
+ "failures": [],
10
+ "input_proofs": 5,
11
+ "input_units": 5,
12
+ "proofs": 5,
13
+ "published_proofs": 5,
14
+ "published_transitions": 26,
15
+ "replay_failures": [],
16
+ "trace_mode": "individual",
17
+ "transitions": 26,
18
+ "validation_failures": [],
19
+ "validation_rejected_proofs": 0,
20
+ "zero_transition_proofs": 0
21
+ },
22
+ "shards": [
23
+ {
24
+ "path": "data/train/numina-001213-0000-56d62a9c65f7e17a.parquet",
25
+ "sha256": "56d62a9c65f7e17a13713287aeffef4fc199cffcfdd7830392959db94a50ad0c",
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+ "bytes": 375459,
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+ "rows": 20,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
29
+ },
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+ {
31
+ "path": "data/internal_test/numina-001213-0000-8f80d9a4c89de4ad.parquet",
32
+ "sha256": "8f80d9a4c89de4adc25da8d3c375da5878b71a7fc8f831617fabf6faba30b6b8",
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+ "bytes": 45866,
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35
+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ }
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+ ],
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+ "proof_ids": [
39
+ "numina:28b6fa94-a3e0-5bea-827f-1e57cf031415",
40
+ "numina:4be7b59f-2fbe-5d86-9149-9678b74f239d",
41
+ "numina:97d1bc37-be52-5922-89eb-457a1a08d06c",
42
+ "numina:aee8b880-db55-5f87-9783-c4feb64140b8",
43
+ "numina:c7760dc8-24c8-504d-9924-80f2568f6238"
44
+ ],
45
+ "proof_keys": [
46
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006065.lean\",\"b6d0b30f6ce9f774851c11cded384e8150198f2d3ba666ece5cb1fceed9772c2\",0,\"inequalities_319894\"]",
47
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006066.lean\",\"3d582690f969fa4aa75bd4595ba2591c8e7413fd824ebf4de44fcce9ae0e5f68\",2,\"algebra_170356\"]",
48
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006067.lean\",\"07f4403e20404fffba1d626386a0c8b8ecbb6bd2b28dfa2a23b929964faaa826\",0,\"number_theory_599878\"]",
49
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006068.lean\",\"5db9a70389ae8078438ec536d48401c9aedbd555ad2106190d10ebc97bae2db0\",0,\"algebra_170339\"]",
50
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006069.lean\",\"51f61de26d323990b9587e51fc29f710b3019962f66e1ca27e3265cad4e56b58\",0,\"number_theory_325554\"]"
51
+ ]
52
+ }
manifests/batches/numina/001214.json ADDED
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1
+ {
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+ "format": "leangpt-parquet-v2",
3
+ "batch": "numina/001214",
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+ "batch_fingerprint": "2ec646b6ed5af6003ad426ee0e32b097dcf0a52acdfcdc6f313cecf4c0b5706d",
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+ "evidence": {
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+ "batch_fingerprint": "2ec646b6ed5af6003ad426ee0e32b097dcf0a52acdfcdc6f313cecf4c0b5706d",
7
+ "bundle_error": "helper declarations require isolated extraction",
8
+ "contamination_matches": [],
9
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10
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11
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12
+ "proofs": 5,
13
+ "published_proofs": 5,
14
+ "published_transitions": 24,
15
+ "replay_failures": [],
16
+ "trace_mode": "individual",
17
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18
+ "validation_failures": [],
19
+ "validation_rejected_proofs": 0,
20
+ "zero_transition_proofs": 0
21
+ },
22
+ "shards": [
23
+ {
24
+ "path": "data/train/numina-001214-0000-c51b4bb382277d95.parquet",
25
+ "sha256": "c51b4bb382277d95eb8403329732277c67e7086ae5ca2ce7f42ed216b4a8500b",
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+ "bytes": 488622,
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+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ },
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+ {
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+ "path": "data/dev/numina-001214-0000-76785508fe3cd5be.parquet",
32
+ "sha256": "76785508fe3cd5be252478add98e095e6365f5c8ec00649a49d89c399ca5dabb",
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+ "bytes": 179841,
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35
+ "revision": "809f84c93ea19a28b9c7bbe8a10557e5f2d7fb8d"
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+ }
37
+ ],
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+ "proof_ids": [
39
+ "numina:26252760-eff1-5bb1-91a1-d3ca42ec6a3c",
40
+ "numina:80cfd283-05d0-5cbd-9255-b643c6ad1f52",
41
+ "numina:a9192722-e656-5fe0-bb99-2ef4807292a0",
42
+ "numina:d0ec9154-0f8a-53b2-8f99-cf8e2cbd3293",
43
+ "numina:e0b048af-e52d-5542-856b-36ada5f6b664"
44
+ ],
45
+ "proof_keys": [
46
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006070.lean\",\"3f10b8acf5efd23c03f353cc89767ca7539baa6f82120cc93b7e556d2a49c81a\",0,\"number_theory_175779\"]",
47
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006071.lean\",\"3dfbe305f70ce9242237897bb77e3b011029b3f09fe134ae1098f6fb1a1628f2\",0,\"algebra_170868\"]",
48
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006072.lean\",\"d7b0dc86cb01c7115430f8084355610af0d8d796d21d4c5f5f9e834d3992c996\",1,\"number_theory_199932\"]",
49
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006073.lean\",\"52ff4d54ef6919c5bafbea03af044751d2f38a909be80db13547db871901293e\",0,\"number_theory_600025\"]",
50
+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006074.lean\",\"fa44badfa8735107ef3a5d728230f0cd63395652d4520ab07cc94209f7ddda2d\",3,\"number_theory_142167\"]"
51
+ ]
52
+ }
manifests/batches/numina/001216.json ADDED
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1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "numina/001216",
4
+ "batch_fingerprint": "feab94c48b53189cf7c4156d8b6b238e44a7bb179ca7e25bfa0b7111fcd07f0f",
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