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Datasets:
HyperCactus0
/
LeanTransitionCorpus

Tasks:
Text Generation
Tags:
lean
lean4
automated-theorem-proving
theorem-proving
formal-mathematics
mathematics
License:
Dataset card Data Studio Files
xet
 Community
1
HyperCactus0 commited on
Commit
4850bc9
·
verified ·
1 Parent(s): 2bac0d6

Record 17 verified HPC batch manifests

Browse files
Files changed (17) hide show
  1. manifests/batches/deepseek/000476.json +159 -0
  2. manifests/batches/deepseek/000477.json +148 -0
  3. manifests/batches/deepseek/000478.json +166 -0
  4. manifests/batches/deepseek/000480.json +148 -0
  5. manifests/batches/deepseek/000481.json +148 -0
  6. manifests/batches/mathlib/000404.json +1411 -0
  7. manifests/batches/mathlib/000405.json +2270 -0
  8. manifests/batches/mathlib/000406.json +1240 -0
  9. manifests/batches/mathlib/000407.json +421 -0
  10. manifests/batches/mathlib/000408.json +1144 -0
  11. manifests/batches/mathlib/000409.json +873 -0
  12. manifests/batches/mathlib/000410.json +335 -0
  13. manifests/batches/mathlib/000411.json +0 -0
  14. manifests/batches/numina/001164.json +58 -0
  15. manifests/batches/numina/001165.json +45 -0
  16. manifests/batches/numina/001166.json +44 -0
  17. manifests/batches/numina/001170.json +45 -0
manifests/batches/deepseek/000476.json ADDED
@@ -0,0 +1,159 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "deepseek/000476",
4
+ "batch_fingerprint": "d3ad9b6081a43f694fa4b17d82bc1c09d2345f1a14a1b42e4bf1a3717c7b23bf",
5
+ "evidence": {
6
+ "batch_fingerprint": "d3ad9b6081a43f694fa4b17d82bc1c09d2345f1a14a1b42e4bf1a3717c7b23bf",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "final replayable tactic does not close all goals",
11
+ "id": "thm_23887",
12
+ "index": 23840,
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "file_path": "Full/Proof_023840.lean",
16
+ "id": "thm_23887",
17
+ "index": 23840,
18
+ "metadata": {
19
+ "declaration_kind": "theorem",
20
+ "source_dataset": "deepseek",
21
+ "theorem_name": "thm_23887",
22
+ "theorem_ordinal": 0
23
+ },
24
+ "source_sha256": "ee87b355104d535de78b7df282cd0a2349077617eebaed31f1002e026b80f54a"
25
+ },
26
+ "source_file": "Full/Proof_023840.lean"
27
+ }
28
+ ],
29
+ "input_proofs": 50,
30
+ "input_units": 50,
31
+ "proofs": 49,
32
+ "published_proofs": 49,
33
+ "published_transitions": 119,
34
+ "replay_failures": [],
35
+ "trace_mode": "individual",
36
+ "transitions": 119,
37
+ "validation_failures": [],
38
+ "validation_rejected_proofs": 0,
39
+ "zero_transition_proofs": 0
40
+ },
41
+ "shards": [
42
+ {
43
+ "path": "data/train/deepseek-000476-0000-833e0b71f0a6074c.parquet",
44
+ "sha256": "833e0b71f0a6074c3cfe7226060c5639bd4361d33b59de1a915d23b007c22f82",
45
+ "bytes": 373770,
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+ "rows": 116,
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+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
48
+ },
49
+ {
50
+ "path": "data/internal_test/deepseek-000476-0000-f8154f1d1aff2adf.parquet",
51
+ "sha256": "f8154f1d1aff2adf99b4b03c9c8bfdc9498dd23a522e484fa02d56a10878d01d",
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+ "bytes": 41018,
53
+ "rows": 3,
54
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
55
+ }
56
+ ],
57
+ "proof_ids": [
58
+ "deepseek:thm_23847",
59
+ "deepseek:thm_23848",
60
+ "deepseek:thm_23849",
61
+ "deepseek:thm_23850",
62
+ "deepseek:thm_23851",
63
+ "deepseek:thm_23852",
64
+ "deepseek:thm_23853",
65
+ "deepseek:thm_23854",
66
+ "deepseek:thm_23855",
67
+ "deepseek:thm_23856",
68
+ "deepseek:thm_23857",
69
+ "deepseek:thm_23858",
70
+ "deepseek:thm_23859",
71
+ "deepseek:thm_23860",
72
+ "deepseek:thm_23861",
73
+ "deepseek:thm_23862",
74
+ "deepseek:thm_23863",
75
+ "deepseek:thm_23864",
76
+ "deepseek:thm_23865",
77
+ "deepseek:thm_23866",
78
+ "deepseek:thm_23867",
79
+ "deepseek:thm_23868",
80
+ "deepseek:thm_23869",
81
+ "deepseek:thm_23870",
82
+ "deepseek:thm_23871",
83
+ "deepseek:thm_23872",
84
+ "deepseek:thm_23873",
85
+ "deepseek:thm_23874",
86
+ "deepseek:thm_23875",
87
+ "deepseek:thm_23876",
88
+ "deepseek:thm_23877",
89
+ "deepseek:thm_23878",
90
+ "deepseek:thm_23879",
91
+ "deepseek:thm_23880",
92
+ "deepseek:thm_23881",
93
+ "deepseek:thm_23882",
94
+ "deepseek:thm_23883",
95
+ "deepseek:thm_23884",
96
+ "deepseek:thm_23885",
97
+ "deepseek:thm_23886",
98
+ "deepseek:thm_23888",
99
+ "deepseek:thm_23889",
100
+ "deepseek:thm_23890",
101
+ "deepseek:thm_23891",
102
+ "deepseek:thm_23892",
103
+ "deepseek:thm_23893",
104
+ "deepseek:thm_23894",
105
+ "deepseek:thm_23895",
106
+ "deepseek:thm_23896"
107
+ ],
108
+ "proof_keys": [
109
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023800.lean\",\"de9dea78b7691715756e4ad2697171a797f8e6a74736f48051d254862924c5c6\",0,\"thm_23847\"]",
110
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023801.lean\",\"0ccd656753e6f93c54ca12996d612415030dddd610a0a5247016c86bef4c27d4\",0,\"thm_23848\"]",
111
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023802.lean\",\"2f1f0c4343a05c33d83104db0694d912e9a77f4bd74b48bfac0a646edd605c88\",0,\"thm_23849\"]",
112
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023803.lean\",\"6a64d04fe882736a64e1571ca3f0dbfaaa04276721069cb1b8d5a467a981fa59\",0,\"thm_23850\"]",
113
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023804.lean\",\"99c4e97fe26e22cb0cb75eafa07fb50ba4dc6d7acc74106142c1592a81ecd894\",0,\"thm_23851\"]",
114
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023805.lean\",\"1b3a11c1536e89e2c01c02248e5c23a2cdcd289db1210375186ec712b673c037\",0,\"thm_23852\"]",
115
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023806.lean\",\"fc4880f8673aa279c1e79cd2b271b8e606c61e9d1da5b5d5aa5b32414d5d930c\",0,\"thm_23853\"]",
116
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023807.lean\",\"5518eb3cbf3cd71d79077acaf9344a044f9f0c86ac9b813c9ae6d1945ca33a01\",0,\"thm_23854\"]",
117
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023808.lean\",\"c5d90d4d4e02c9bcd9dec69f963a5d5732469cb75c92091745c52c6f03abaf49\",0,\"thm_23855\"]",
118
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023809.lean\",\"726b5764b12f559298761b44e2439b9ee8217667f0121e6327ea7d945b8e5503\",0,\"thm_23856\"]",
119
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023810.lean\",\"84326175595f84217ad97ee28f895b7f0c320ef1782b5ad097149ae66880caf7\",0,\"thm_23857\"]",
120
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023811.lean\",\"5f2c1854aae3fb8d5c7d982691488abb8a0932544d36879eb246d9889338fee4\",0,\"thm_23858\"]",
121
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023812.lean\",\"3af6ad11b5b9198a25d6a3e14cfb5904e4a3d9cf8b690b2a3d964d110e59d5f4\",0,\"thm_23859\"]",
122
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023813.lean\",\"541ca592f5f71ac4d7cb1bba925e3103ec83ca698b3ed351636a0e3cfc548272\",0,\"thm_23860\"]",
123
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023814.lean\",\"c7c7e71120474fae7358767f9b7d20e6cfb21e445205a3734ff0a1b1b64c73b5\",0,\"thm_23861\"]",
124
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023815.lean\",\"bf4633906470afc0b3d064d124c5a86daa10522d0ed42d19682e291eb37c27c5\",0,\"thm_23862\"]",
125
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023816.lean\",\"5ac9fd8c8ea8d01acd94ff56af4e9c236fdf9296c1ba41d35da0b3934287e96d\",0,\"thm_23863\"]",
126
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023817.lean\",\"953cb9ef01588a7ad8ebc36801a472fed55fd1c6bea37c82b3c6bac2a10919b2\",0,\"thm_23864\"]",
127
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023818.lean\",\"9deed52f263ea1737a65c462bfa8118d131ea0c98269616d0867a9fec7ee0f11\",0,\"thm_23865\"]",
128
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023819.lean\",\"9e75108b7c57dc6a98c7849eae6314e79702ce8a17e993fafe2fa000b68ad6cf\",0,\"thm_23866\"]",
129
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023820.lean\",\"ad9fb6ef016367a9e3b333a67baf7f8b49a2901fab9b494101c13be5772b7e90\",0,\"thm_23867\"]",
130
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023821.lean\",\"a482d351e3d3b5253f3caa7d9dffb0a87beefd91e461a002b397ff436b345371\",0,\"thm_23868\"]",
131
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023822.lean\",\"4733b96001cdb9ce9dd4023dc5c22ac661fd18b1e43008eb2772748efda88ad1\",0,\"thm_23869\"]",
132
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023823.lean\",\"e2ea73c2ca9979d72c3606121c37e4a47db138a29f918b4b99b868543a113aef\",0,\"thm_23870\"]",
133
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023824.lean\",\"2c431d5d4bc90859f1495d057546e15d6e317a271796f48ea796f0c4334b1c16\",0,\"thm_23871\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023825.lean\",\"25e27502b538e42f26920836bea7419738611965a0d0315853cfce6710c8ff94\",0,\"thm_23872\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023826.lean\",\"3d5585245a1a973b86f4ec076d98d2021b1e5cf2875fcbaecd59b95ab59b5659\",0,\"thm_23873\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023827.lean\",\"cfbec8b93f77d3b43cd327beaff206612fc0609a8f57eaf8976208fb1ea5b559\",0,\"thm_23874\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023828.lean\",\"14e8334cc887ab26f57409cba4c4eedd9944aa49a72ac6d4721a54b5150ffa61\",0,\"thm_23875\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023829.lean\",\"6b6004b40304810c9121babfe13ff946a0d57bde7ce54527ff2e014848c7cbc5\",0,\"thm_23876\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023830.lean\",\"737ef218b81b1d1d6d0f8fe3d82feb177f643e0131dd8a55d72bff1f60b10da9\",0,\"thm_23877\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023831.lean\",\"4cef0679e5fff5239ca95b82de7a93878e3bbb7298866199bd955b0055767973\",0,\"thm_23878\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023832.lean\",\"df623b47e4100cb09410f3b1dd32de56131ba299a7e46dc75a58d1f67bb6b9a3\",0,\"thm_23879\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023833.lean\",\"66e689ae140e13d9dc088dc4c1664ee41a5e81cda3a17024c4dc8c5c1065aa8c\",0,\"thm_23880\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023834.lean\",\"afd15cc4bb8415af72eeb4a09f7abce09ea25e43353c3c4b0a0cb011e16045e0\",0,\"thm_23881\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023835.lean\",\"76c476c9c3404775d387cf105072b16856ea53342edb14471dabbb20ff90a06b\",0,\"thm_23882\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023836.lean\",\"e97424d27e65328cc8c598dd7e81b763e4cf5455564e3d142b3c3c51d2be5115\",0,\"thm_23883\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023837.lean\",\"e1fee9ab0ff3c17c88a87cf7024ea78b3d96c3f2191c35ac563ec69f6e0f2c19\",0,\"thm_23884\"]",
147
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023838.lean\",\"9f8424474b66ff0b976f7c4217b1ce64fb31171dc1084914f6dbf0d7ddcc3973\",0,\"thm_23885\"]",
148
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023839.lean\",\"90b98bb91a49b197b6f8fb4b2399952af910d6081b7520d504493fc9b87d4ca9\",0,\"thm_23886\"]",
149
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023841.lean\",\"e0f7e4a0aec2fa8793b58e23dbcb3a56e0e39a33d0407f851c26b227d6ff4896\",0,\"thm_23888\"]",
150
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023842.lean\",\"22b7fc44f93619642783f83c166707a23536311838814caa8f8c54cedb73143e\",0,\"thm_23889\"]",
151
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023843.lean\",\"8e1bb2e761655e493cdc25724976f8913d272eccf35923e48f967b2172bd9fea\",0,\"thm_23890\"]",
152
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023844.lean\",\"ff7acf70c30113d62c43a6a157ee1418bd027c280343e5b17d6d8707b96b0df8\",0,\"thm_23891\"]",
153
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023845.lean\",\"d71a5c7deebb88cfaf49b53f9516a2457910bad93f72e1c929f9c0890198a3a8\",0,\"thm_23892\"]",
154
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023846.lean\",\"1ad2288fceda97b233581e62f26b2642f28fd18ed47e1ac8bf13b84c8f6791c9\",0,\"thm_23893\"]",
155
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023847.lean\",\"d6e88bd64becfe4a642d02361cc989ad5f68172cb79101bad3284a71fe213764\",0,\"thm_23894\"]",
156
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023848.lean\",\"a48a057e8a6e1cc1decad2f6c766553ba2f7c00b6a2fad939930e44274a770f1\",0,\"thm_23895\"]",
157
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023849.lean\",\"dba780377fa42eff6ff5224a4529111326d37d949a02d5a73f950a331d12aa02\",0,\"thm_23896\"]"
158
+ ]
159
+ }
manifests/batches/deepseek/000477.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000477",
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+ "batch_fingerprint": "848dada77b8bb13dfeeb35a7dd444852c1390f43e4569d4d0e0d7581782d85eb",
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+ "evidence": {
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+ "batch_fingerprint": "848dada77b8bb13dfeeb35a7dd444852c1390f43e4569d4d0e0d7581782d85eb",
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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": 116,
14
+ "replay_failures": [],
15
+ "trace_mode": "individual",
16
+ "transitions": 116,
17
+ "validation_failures": [],
18
+ "validation_rejected_proofs": 0,
19
+ "zero_transition_proofs": 0
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+ },
21
+ "shards": [
22
+ {
23
+ "path": "data/train/deepseek-000477-0000-35bb0499b6999d49.parquet",
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+ "sha256": "35bb0499b6999d49e32265508f6034cc27983ca607fd517a3fc66cd53fff9aa0",
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+ "bytes": 348071,
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+ "rows": 105,
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+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
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+ },
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+ {
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+ "path": "data/dev/deepseek-000477-0000-28e8e1b2c82b0b03.parquet",
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+ "sha256": "28e8e1b2c82b0b032e3b27d96cdf5131e29ea1e05a5bbf939f89719cef4a0047",
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+ "bytes": 33253,
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+ "rows": 4,
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+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
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+ },
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+ {
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+ "path": "data/internal_test/deepseek-000477-0000-eb597870c8a797e9.parquet",
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+ "sha256": "eb597870c8a797e9f45ccc5341cf14ea825c53f549fc5b925642a2ece3c93014",
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+ "bytes": 44033,
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+ "rows": 7,
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+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
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+ }
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+ ],
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+ "proof_ids": [
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+ "deepseek:thm_23897",
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+ "deepseek:thm_23898",
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+ "deepseek:thm_23899",
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+ "deepseek:thm_23900",
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+ "deepseek:thm_23901",
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+ "deepseek:thm_23902",
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+ "deepseek:thm_23903",
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+ "deepseek:thm_23904",
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+ "deepseek:thm_23905",
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+ "deepseek:thm_23906",
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+ "deepseek:thm_23907",
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+ "deepseek:thm_23908",
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+ "deepseek:thm_23909",
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+ "deepseek:thm_23910",
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+ "deepseek:thm_23911",
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+ "deepseek:thm_23913",
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+ "deepseek:thm_23914",
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+ "deepseek:thm_23915",
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+ "deepseek:thm_23916",
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+ "deepseek:thm_23926",
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+ "deepseek:thm_23929",
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+ "deepseek:thm_23931",
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+ "deepseek:thm_23934",
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+ "deepseek:thm_23936",
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+ "deepseek:thm_23937",
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+ "deepseek:thm_23938",
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+ "deepseek:thm_23941",
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+ "deepseek:thm_23942",
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+ "deepseek:thm_23943",
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+ "deepseek:thm_23944",
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+ "deepseek:thm_23945",
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+ "deepseek:thm_23946"
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+ ],
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+ "proof_keys": [
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023893.lean\",\"2057a82ee91b9325993c897a032e26da2151fa4e1ca3f198e7b681cab9aea9a7\",0,\"thm_23940\"]",
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+ ]
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+ }
manifests/batches/deepseek/000478.json ADDED
@@ -0,0 +1,166 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "format": "leangpt-parquet-v2",
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+ "batch": "deepseek/000478",
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+ "batch_fingerprint": "555e5df882182d8a5173515fca87e13f493550a8d68596a41ca671a1bb298b1b",
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+ "evidence": {
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+ "contamination_matches": [],
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+ "failures": [
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+ {
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+ "error": "final replayable tactic does not close all goals",
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+ "id": "thm_23964",
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+ "index": 23917,
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+ "reason": "import_failure",
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+ "selection": {
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+ "file_path": "Full/Proof_023917.lean",
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+ "id": "thm_23964",
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+ "index": 23917,
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+ "metadata": {
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+ "declaration_kind": "theorem",
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+ "source_dataset": "deepseek",
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+ "theorem_name": "thm_23964",
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+ "theorem_ordinal": 0
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+ "source_file": "Full/Proof_023917.lean"
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+ "input_units": 50,
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+ "published_transitions": 106,
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+ "replay_failures": [],
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+ "trace_mode": "individual",
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+ "transitions": 106,
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+ "validation_failures": [],
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+ ],
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+ "proof_keys": [
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_023948.lean\",\"d733797f1763dd1ea577257d386d8b40b39afaaa594bb2b785b4a649644acbc7\",0,\"thm_23995\"]",
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165
+ ]
166
+ }
manifests/batches/deepseek/000480.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ {
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+ "evidence": {
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+ "contamination_matches": [],
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+ "replay_failures": [],
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+ "trace_mode": "individual",
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+ "transitions": 116,
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+ "validation_failures": [],
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+ "zero_transition_proofs": 0
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+ "path": "data/train/deepseek-000480-0000-ba57122ac425d2d9.parquet",
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+ "proof_keys": [
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+ ]
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+ }
manifests/batches/deepseek/000481.json ADDED
@@ -0,0 +1,148 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024085.lean\",\"360a2bfbf157ddbd4f0f0bed14a82b18dc78ec2a985544faab4874e3fd189aa8\",0,\"thm_24132\"]",
133
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024086.lean\",\"2e56de0c3f0e873223deeef63670d6ac1b439e4bff11cdee733bd6c105dae2e9\",0,\"thm_24133\"]",
134
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024087.lean\",\"44ff2a66c8f7a0830d2104cf0e9f52a3bd381c47750db216ae7755e6e849e97e\",0,\"thm_24134\"]",
135
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024088.lean\",\"45abb0cb620302628539e6243e3742f882cb02f2345cefebcbf836ea9e69dcf3\",0,\"thm_24135\"]",
136
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024089.lean\",\"4a3d983444e79acaeab3097c67189fddfb768ae2389763914b93afb53762a40e\",0,\"thm_24136\"]",
137
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024090.lean\",\"311fc40ba045b7759dc75663446a14bde8a3ca8240483371539bc23af092cac9\",0,\"thm_24137\"]",
138
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024091.lean\",\"7733656964badf07e743cc889bcc402c44442fa1f1cff211020a660573f99a60\",0,\"thm_24138\"]",
139
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024092.lean\",\"736a61aeb7de9794cada26b80a8ed91a3e839f3d5d323e0b8817acd02bb27780\",0,\"thm_24139\"]",
140
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024093.lean\",\"ee105b7908049c5bf1ffda81546669a737dc99634fa3ebbebe818cff9fdddd91\",0,\"thm_24140\"]",
141
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024094.lean\",\"65db3cbd7b3d2dce5afcc4de924a462a28008638411b60f832c4de0a3b08d226\",0,\"thm_24141\"]",
142
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024095.lean\",\"47cc9f0d177b5e5cdeb49cecc48faae917b6007536a69bd6ce319c4766ab400b\",0,\"thm_24142\"]",
143
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024096.lean\",\"1e11b1d56d53a4823583653e3d35b6bab8542fce97a69a4c36e38ed5778f55d4\",0,\"thm_24143\"]",
144
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024097.lean\",\"aee9229fd0b5449e54fe0faba18ad0eb3c635331d12ccefec8b41ef10691c73d\",0,\"thm_24144\"]",
145
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024098.lean\",\"50e4a93c518fb5adedf1e120f7cd14a31e634c1438b84f9ff5a726ecf0d575f1\",0,\"thm_24145\"]",
146
+ "[\"deepseek\",\"1ee889f608fb12ba3596757ee91a60acd663ea81\",\"Full/Proof_024099.lean\",\"e0b79fd2a20ed200fd87dd341f512569d63cf61ee418391f95686f93e16c5fc5\",0,\"thm_24146\"]"
147
+ ]
148
+ }
manifests/batches/mathlib/000404.json ADDED
@@ -0,0 +1,1411 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ {
2
+ "format": "leangpt-parquet-v2",
3
+ "batch": "mathlib/000404",
4
+ "batch_fingerprint": "661335640cf14bfe3f66ac1fa346a5111f983062f58a44b3910e3546a176ad89",
5
+ "evidence": {
6
+ "batch_fingerprint": "661335640cf14bfe3f66ac1fa346a5111f983062f58a44b3910e3546a176ad89",
7
+ "contamination_matches": [],
8
+ "failures": [
9
+ {
10
+ "error": "theorem has no selected tactic transitions",
11
+ "full_name": "Convex.image_deriv",
12
+ "id": "Convex.image_deriv",
13
+ "reason": "import_failure",
14
+ "selection": {
15
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
16
+ "end": [
17
+ 125,
18
+ 42
19
+ ],
20
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
21
+ "full_name": "Convex.image_deriv",
22
+ "start": [
23
+ 123,
24
+ 1
25
+ ],
26
+ "traced_tactics": [],
27
+ "url": "https://github.com/leanprover-community/mathlib4"
28
+ },
29
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
30
+ },
31
+ {
32
+ "error": "theorem has no selected tactic transitions",
33
+ "full_name": "Convex.image_derivWithin",
34
+ "id": "Convex.image_derivWithin",
35
+ "reason": "import_failure",
36
+ "selection": {
37
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
38
+ "end": [
39
+ 118,
40
+ 48
41
+ ],
42
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
43
+ "full_name": "Convex.image_derivWithin",
44
+ "start": [
45
+ 116,
46
+ 1
47
+ ],
48
+ "traced_tactics": [],
49
+ "url": "https://github.com/leanprover-community/mathlib4"
50
+ },
51
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
52
+ },
53
+ {
54
+ "error": "theorem has no selected tactic transitions",
55
+ "full_name": "Convex.image_hasDerivWithinAt",
56
+ "id": "Convex.image_hasDerivWithinAt",
57
+ "reason": "import_failure",
58
+ "selection": {
59
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
60
+ "end": [
61
+ 111,
62
+ 53
63
+ ],
64
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
65
+ "full_name": "Convex.image_hasDerivWithinAt",
66
+ "start": [
67
+ 109,
68
+ 1
69
+ ],
70
+ "traced_tactics": [],
71
+ "url": "https://github.com/leanprover-community/mathlib4"
72
+ },
73
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
74
+ },
75
+ {
76
+ "error": "theorem has no selected tactic transitions",
77
+ "full_name": "Set.OrdConnected.image_derivWithin",
78
+ "id": "Set.OrdConnected.image_derivWithin",
79
+ "reason": "import_failure",
80
+ "selection": {
81
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
82
+ "end": [
83
+ 97,
84
+ 67
85
+ ],
86
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
87
+ "full_name": "Set.OrdConnected.image_derivWithin",
88
+ "start": [
89
+ 95,
90
+ 1
91
+ ],
92
+ "traced_tactics": [],
93
+ "url": "https://github.com/leanprover-community/mathlib4"
94
+ },
95
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
96
+ },
97
+ {
98
+ "error": "no internal trace for tactic 'have : Icc b a \u2286 s := hs.out hb ha'",
99
+ "full_name": "Set.OrdConnected.image_hasDerivWithinAt",
100
+ "id": "Set.OrdConnected.image_hasDerivWithinAt",
101
+ "reason": "import_failure",
102
+ "selection": {
103
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
104
+ "end": [
105
+ 90,
106
+ 52
107
+ ],
108
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
109
+ "full_name": "Set.OrdConnected.image_hasDerivWithinAt",
110
+ "start": [
111
+ 76,
112
+ 1
113
+ ],
114
+ "traced_tactics": [
115
+ {
116
+ "annotated_tactic": [
117
+ "\"apply <a>ordConnected_of_Ioo<\\/a>\"",
118
+ "[{\"full_name\":\"Set.ordConnected_of_Ioo\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnected.lean\",\"def_pos\":[57,9],\"def_end_pos\":[57,28]}]"
119
+ ],
120
+ "state_after": "case hs\na b : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\n\u22a2 \u2200 x \u2208 f' '' s, \u2200 y \u2208 f' '' s, x < y \u2192 Ioo x y \u2286 f' '' s",
121
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\n\u22a2 (f' '' s).OrdConnected",
122
+ "tactic": "apply ordConnected_of_Ioo"
123
+ },
124
+ {
125
+ "annotated_tactic": [
126
+ "\"rintro _ \\u27e8a, ha, rfl\\u27e9 _ \\u27e8b, hb, rfl\\u27e9 - m \\u27e8hma, hmb\\u27e9\"",
127
+ "[]"
128
+ ],
129
+ "state_after": "case hs.intro.intro.intro.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\n\u22a2 m \u2208 f' '' s",
130
+ "state_before": "case hs\na b : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\n\u22a2 \u2200 x \u2208 f' '' s, \u2200 y \u2208 f' '' s, x < y \u2192 Ioo x y \u2286 f' '' s",
131
+ "tactic": "rintro _ \u27e8a, ha, rfl\u27e9 _ \u27e8b, hb, rfl\u27e9 - m \u27e8hma, hmb\u27e9"
132
+ },
133
+ {
134
+ "annotated_tactic": [
135
+ "\"rcases <a>le_total<\\/a> a b with hab | hab\"",
136
+ "[{\"full_name\":\"le_total\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,17]}]"
137
+ ],
138
+ "state_after": "case hs.intro.intro.intro.intro.intro.inl\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\n\u22a2 m \u2208 f' '' s\n\ncase hs.intro.intro.intro.intro.intro.inr\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\n\u22a2 m \u2208 f' '' s",
139
+ "state_before": "case hs.intro.intro.intro.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\n\u22a2 m \u2208 f' '' s",
140
+ "tactic": "rcases le_total a b with hab | hab"
141
+ },
142
+ {
143
+ "annotated_tactic": [
144
+ "\"have : <a>Icc<\\/a> a b \\u2286 s := hs.out ha hb\"",
145
+ "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]}]"
146
+ ],
147
+ "state_after": "case hs.intro.intro.intro.intro.intro.inl\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\nthis : Icc a b \u2286 s\n\u22a2 m \u2208 f' '' s",
148
+ "state_before": "case hs.intro.intro.intro.intro.intro.inl\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\n\u22a2 m \u2208 f' '' s",
149
+ "tactic": "have : Icc a b \u2286 s := hs.out ha hb"
150
+ },
151
+ {
152
+ "annotated_tactic": [
153
+ "\"rcases <a>exists_hasDerivWithinAt_eq_of_gt_of_lt<\\/a> hab (fun x hx => (hf x <| this hx).<a>mono<\\/a> this) hma\\n hmb with\\n \\u27e8c, cmem, hc\\u27e9\"",
154
+ "[{\"full_name\":\"exists_hasDerivWithinAt_eq_of_gt_of_lt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Darboux.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,47]},{\"full_name\":\"HasDerivWithinAt.mono\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,30]}]"
155
+ ],
156
+ "state_after": "case hs.intro.intro.intro.intro.intro.inl.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\nthis : Icc a b \u2286 s\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : f' c = m\n\u22a2 m \u2208 f' '' s",
157
+ "state_before": "case hs.intro.intro.intro.intro.intro.inl\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\nthis : Icc a b \u2286 s\n\u22a2 m \u2208 f' '' s",
158
+ "tactic": "rcases exists_hasDerivWithinAt_eq_of_gt_of_lt hab (fun x hx => (hf x <| this hx).mono this) hma\n hmb with\n \u27e8c, cmem, hc\u27e9"
159
+ },
160
+ {
161
+ "annotated_tactic": [
162
+ "\"exact \\u27e8c, this <| <a>Ioo_subset_Icc_self<\\/a> cmem, hc\\u27e9\"",
163
+ "[{\"full_name\":\"Set.Ioo_subset_Icc_self\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[532,9],\"def_end_pos\":[532,28]}]"
164
+ ],
165
+ "state_after": "no goals",
166
+ "state_before": "case hs.intro.intro.intro.intro.intro.inl.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : a \u2264 b\nthis : Icc a b \u2286 s\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : f' c = m\n\u22a2 m \u2208 f' '' s",
167
+ "tactic": "exact \u27e8c, this <| Ioo_subset_Icc_self cmem, hc\u27e9"
168
+ },
169
+ {
170
+ "annotated_tactic": [
171
+ "\"have : <a>Icc<\\/a> b a \\u2286 s := hs.out hb ha\"",
172
+ "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]}]"
173
+ ],
174
+ "state_after": "case hs.intro.intro.intro.intro.intro.inr\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\nthis : Icc b a \u2286 s\n\u22a2 m \u2208 f' '' s",
175
+ "state_before": "case hs.intro.intro.intro.intro.intro.inr\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\n\u22a2 m \u2208 f' '' s",
176
+ "tactic": "have : Icc b a \u2286 s := hs.out hb ha"
177
+ },
178
+ {
179
+ "annotated_tactic": [
180
+ "\"rcases <a>exists_hasDerivWithinAt_eq_of_lt_of_gt<\\/a> hab (fun x hx => (hf x <| this hx).<a>mono<\\/a> this) hmb\\n hma with\\n \\u27e8c, cmem, hc\\u27e9\"",
181
+ "[{\"full_name\":\"exists_hasDerivWithinAt_eq_of_lt_of_gt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Darboux.lean\",\"def_pos\":[65,9],\"def_end_pos\":[65,47]},{\"full_name\":\"HasDerivWithinAt.mono\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,30]}]"
182
+ ],
183
+ "state_after": "case hs.intro.intro.intro.intro.intro.inr.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\nthis : Icc b a \u2286 s\nc : \u211d\ncmem : c \u2208 Ioo b a\nhc : f' c = m\n\u22a2 m \u2208 f' '' s",
184
+ "state_before": "case hs.intro.intro.intro.intro.intro.inr\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\nthis : Icc b a \u2286 s\n\u22a2 m \u2208 f' '' s",
185
+ "tactic": "rcases exists_hasDerivWithinAt_eq_of_lt_of_gt hab (fun x hx => (hf x <| this hx).mono this) hmb\n hma with\n \u27e8c, cmem, hc\u27e9"
186
+ },
187
+ {
188
+ "annotated_tactic": [
189
+ "\"exact \\u27e8c, this <| <a>Ioo_subset_Icc_self<\\/a> cmem, hc\\u27e9\"",
190
+ "[{\"full_name\":\"Set.Ioo_subset_Icc_self\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[532,9],\"def_end_pos\":[532,28]}]"
191
+ ],
192
+ "state_after": "no goals",
193
+ "state_before": "case hs.intro.intro.intro.intro.intro.inr.intro.intro\na\u271d b\u271d : \u211d\nf f' : \u211d \u2192 \u211d\ns : Set \u211d\nhs : s.OrdConnected\nhf : \u2200 x \u2208 s, HasDerivWithinAt f (f' x) s x\na : \u211d\nha : a \u2208 s\nb : \u211d\nhb : b \u2208 s\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab : b \u2264 a\nthis : Icc b a \u2286 s\nc : \u211d\ncmem : c \u2208 Ioo b a\nhc : f' c = m\n\u22a2 m \u2208 f' '' s",
194
+ "tactic": "exact \u27e8c, this <| Ioo_subset_Icc_self cmem, hc\u27e9"
195
+ }
196
+ ],
197
+ "url": "https://github.com/leanprover-community/mathlib4"
198
+ },
199
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
200
+ },
201
+ {
202
+ "error": "theorem has no selected tactic transitions",
203
+ "full_name": "exists_hasDerivWithinAt_eq_of_ge_of_le",
204
+ "id": "exists_hasDerivWithinAt_eq_of_ge_of_le",
205
+ "reason": "import_failure",
206
+ "selection": {
207
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
208
+ "end": [
209
+ 134,
210
+ 58
211
+ ],
212
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
213
+ "full_name": "exists_hasDerivWithinAt_eq_of_ge_of_le",
214
+ "start": [
215
+ 130,
216
+ 1
217
+ ],
218
+ "traced_tactics": [],
219
+ "url": "https://github.com/leanprover-community/mathlib4"
220
+ },
221
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
222
+ },
223
+ {
224
+ "error": "no internal trace for tactic 'exact (lt_asymm hma hmb).elim'",
225
+ "full_name": "exists_hasDerivWithinAt_eq_of_gt_of_lt",
226
+ "id": "exists_hasDerivWithinAt_eq_of_gt_of_lt",
227
+ "reason": "import_failure",
228
+ "selection": {
229
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
230
+ "end": [
231
+ 60,
232
+ 78
233
+ ],
234
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
235
+ "full_name": "exists_hasDerivWithinAt_eq_of_gt_of_lt",
236
+ "start": [
237
+ 28,
238
+ 1
239
+ ],
240
+ "traced_tactics": [
241
+ {
242
+ "annotated_tactic": [
243
+ "\"rcases hab.eq_or_lt with (rfl | hab')\"",
244
+ "[]"
245
+ ],
246
+ "state_after": "case inl\na : \u211d\nf f' : \u211d \u2192 \u211d\nm : \u211d\nhma : f' a < m\nhab : a \u2264 a\nhf : \u2200 x \u2208 Icc a a, HasDerivWithinAt f (f' x) (Icc a a) x\nhmb : m < f' a\n\u22a2 m \u2208 f' '' Ioo a a\n\ncase inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\n\u22a2 m \u2208 f' '' Ioo a b",
247
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\n\u22a2 m \u2208 f' '' Ioo a b",
248
+ "tactic": "rcases hab.eq_or_lt with (rfl | hab')"
249
+ },
250
+ {
251
+ "annotated_tactic": [
252
+ "\"set g : \\u211d \\u2192 \\u211d := fun x => f x - m * x\"",
253
+ "[]"
254
+ ],
255
+ "state_after": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\n\u22a2 m \u2208 f' '' Ioo a b",
256
+ "state_before": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\n\u22a2 m \u2208 f' '' Ioo a b",
257
+ "tactic": "set g : \u211d \u2192 \u211d := fun x => f x - m * x"
258
+ },
259
+ {
260
+ "annotated_tactic": [
261
+ "\"have hg : \\u2200 x \\u2208 <a>Icc<\\/a> a b, <a>HasDerivWithinAt<\\/a> g (f' x - m) (<a>Icc<\\/a> a b) x := by\\n intro x hx\\n simpa using (hf x hx).<a>sub<\\/a> ((<a>hasDerivWithinAt_id<\\/a> x _).<a>const_mul<\\/a> m)\"",
262
+ "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"HasDerivWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[117,5],\"def_end_pos\":[117,21]},{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"HasDerivWithinAt.sub\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Add.lean\",\"def_pos\":[302,16],\"def_end_pos\":[302,36]},{\"full_name\":\"hasDerivWithinAt_id\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[672,9],\"def_end_pos\":[672,28]},{\"full_name\":\"HasDerivWithinAt.const_mul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,35]}]"
263
+ ],
264
+ "state_after": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\n\u22a2 m \u2208 f' '' Ioo a b",
265
+ "state_before": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\n\u22a2 m \u2208 f' '' Ioo a b",
266
+ "tactic": "have hg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x := by\n intro x hx\n simpa using (hf x hx).sub ((hasDerivWithinAt_id x _).const_mul m)"
267
+ },
268
+ {
269
+ "annotated_tactic": [
270
+ "\"obtain \\u27e8c, cmem, hc\\u27e9 : \\u2203 c \\u2208 <a>Icc<\\/a> a b, <a>IsMinOn<\\/a> g (<a>Icc<\\/a> a b) c :=\\n isCompact_Icc.exists_isMinOn (<a>nonempty_Icc<\\/a>.2 <| hab) fun x hx => (hg x hx).<a>continuousWithinAt<\\/a>\"",
271
+ "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"IsMinOn\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Extr.lean\",\"def_pos\":[111,5],\"def_end_pos\":[111,12]},{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"Set.nonempty_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[266,9],\"def_end_pos\":[266,21]},{\"full_name\":\"HasDerivWithinAt.continuousWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[754,9],\"def_end_pos\":[754,44]}]"
272
+ ],
273
+ "state_after": "case inr.intro.intro\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\n\u22a2 m \u2208 f' '' Ioo a b",
274
+ "state_before": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\n\u22a2 m \u2208 f' '' Ioo a b",
275
+ "tactic": "obtain \u27e8c, cmem, hc\u27e9 : \u2203 c \u2208 Icc a b, IsMinOn g (Icc a b) c :=\n isCompact_Icc.exists_isMinOn (nonempty_Icc.2 <| hab) fun x hx => (hg x hx).continuousWithinAt"
276
+ },
277
+ {
278
+ "annotated_tactic": [
279
+ "\"use c, cmem'\"",
280
+ "[]"
281
+ ],
282
+ "state_after": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 f' c = m",
283
+ "state_before": "case inr.intro.intro\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 m \u2208 f' '' Ioo a b",
284
+ "tactic": "use c, cmem'"
285
+ },
286
+ {
287
+ "annotated_tactic": [
288
+ "\"rw [\\u2190 <a>sub_eq_zero<\\/a>]\"",
289
+ "[{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]}]"
290
+ ],
291
+ "state_after": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 f' c - m = 0",
292
+ "state_before": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 f' c = m",
293
+ "tactic": "rw [\u2190 sub_eq_zero]"
294
+ },
295
+ {
296
+ "annotated_tactic": [
297
+ "\"have : <a>Icc<\\/a> a b \\u2208 \\ud835\\udcdd c := by rwa [\\u2190 <a>mem_interior_iff_mem_nhds<\\/a>, <a>interior_Icc<\\/a>]\"",
298
+ "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"mem_interior_iff_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[852,9],\"def_end_pos\":[852,34]},{\"full_name\":\"interior_Icc\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/DenselyOrdered.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,21]}]"
299
+ ],
300
+ "state_after": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\nthis : Icc a b \u2208 \ud835\udcdd c\n\u22a2 f' c - m = 0",
301
+ "state_before": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 f' c - m = 0",
302
+ "tactic": "have : Icc a b \u2208 \ud835\udcdd c := by rwa [\u2190 mem_interior_iff_mem_nhds, interior_Icc]"
303
+ },
304
+ {
305
+ "annotated_tactic": [
306
+ "\"exact (hc.isLocalMin this).<a>hasDerivAt_eq_zero<\\/a> ((hg c cmem).<a>hasDerivAt<\\/a> this)\"",
307
+ "[{\"full_name\":\"IsLocalMin.hasDerivAt_eq_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/LocalExtr\\/Basic.lean\",\"def_pos\":[232,9],\"def_end_pos\":[232,38]},{\"full_name\":\"HasDerivWithinAt.hasDerivAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[423,9],\"def_end_pos\":[423,36]}]"
308
+ ],
309
+ "state_after": "no goals",
310
+ "state_before": "case right\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\nthis : Icc a b \u2208 \ud835\udcdd c\n\u22a2 f' c - m = 0",
311
+ "tactic": "exact (hc.isLocalMin this).hasDerivAt_eq_zero ((hg c cmem).hasDerivAt this)"
312
+ },
313
+ {
314
+ "annotated_tactic": [
315
+ "\"exact (<a>lt_asymm<\\/a> hma hmb).<a>elim<\\/a>\"",
316
+ "[{\"full_name\":\"lt_asymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[107,9],\"def_end_pos\":[107,17]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
317
+ ],
318
+ "state_after": "no goals",
319
+ "state_before": "case inl\na : \u211d\nf f' : \u211d \u2192 \u211d\nm : \u211d\nhma : f' a < m\nhab : a \u2264 a\nhf : \u2200 x \u2208 Icc a a, HasDerivWithinAt f (f' x) (Icc a a) x\nhmb : m < f' a\n\u22a2 m \u2208 f' '' Ioo a a",
320
+ "tactic": "exact (lt_asymm hma hmb).elim"
321
+ },
322
+ {
323
+ "annotated_tactic": [
324
+ "\"intro x hx\"",
325
+ "[]"
326
+ ],
327
+ "state_after": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 HasDerivWithinAt g (f' x - m) (Icc a b) x",
328
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\n\u22a2 \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x",
329
+ "tactic": "intro x hx"
330
+ },
331
+ {
332
+ "annotated_tactic": [
333
+ "\"simpa using (hf x hx).<a>sub<\\/a> ((<a>hasDerivWithinAt_id<\\/a> x _).<a>const_mul<\\/a> m)\"",
334
+ "[{\"full_name\":\"HasDerivWithinAt.sub\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Add.lean\",\"def_pos\":[302,16],\"def_end_pos\":[302,36]},{\"full_name\":\"hasDerivWithinAt_id\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[672,9],\"def_end_pos\":[672,28]},{\"full_name\":\"HasDerivWithinAt.const_mul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,35]}]"
335
+ ],
336
+ "state_after": "no goals",
337
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 HasDerivWithinAt g (f' x - m) (Icc a b) x",
338
+ "tactic": "simpa using (hf x hx).sub ((hasDerivWithinAt_id x _).const_mul m)"
339
+ },
340
+ {
341
+ "annotated_tactic": [
342
+ "\"rcases cmem.1.<a>eq_or_lt<\\/a> with (rfl | hac)\"",
343
+ "[{\"full_name\":\"LE.le.eq_or_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[395,7],\"def_end_pos\":[395,21]}]"
344
+ ],
345
+ "state_after": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\n\u22a2 a \u2208 Ioo a b\n\ncase inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\nhac : a < c\n\u22a2 c \u2208 Ioo a b",
346
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\n\u22a2 c \u2208 Ioo a b",
347
+ "tactic": "rcases cmem.1.eq_or_lt with (rfl | hac)"
348
+ },
349
+ {
350
+ "annotated_tactic": [
351
+ "\"rcases cmem.2.<a>eq_or_gt<\\/a> with (rfl | hcb)\"",
352
+ "[{\"full_name\":\"LE.le.eq_or_gt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[397,7],\"def_end_pos\":[397,21]}]"
353
+ ],
354
+ "state_after": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n\u22a2 b \u2208 Ioo a b\n\ncase inr.inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\nhac : a < c\nhcb : c < b\n\u22a2 c \u2208 Ioo a b",
355
+ "state_before": "case inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\nhac : a < c\n\u22a2 c \u2208 Ioo a b",
356
+ "tactic": "rcases cmem.2.eq_or_gt with (rfl | hcb)"
357
+ },
358
+ {
359
+ "annotated_tactic": [
360
+ "\"exact \\u27e8hac, hcb\\u27e9\"",
361
+ "[]"
362
+ ],
363
+ "state_after": "no goals",
364
+ "state_before": "case inr.inr\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\nhac : a < c\nhcb : c < b\n\u22a2 c \u2208 Ioo a b",
365
+ "tactic": "exact \u27e8hac, hcb\u27e9"
366
+ },
367
+ {
368
+ "annotated_tactic": [
369
+ "\"refine <a>absurd<\\/a> (<a>sub_nonneg<\\/a>.1 <| <a>nonneg_of_mul_nonneg_right<\\/a> ?_ (<a>sub_pos<\\/a>.2 hab'))\\n (<a>not_le_of_lt<\\/a> hma)\"",
370
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]},{\"full_name\":\"sub_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[720,30],\"def_end_pos\":[720,40]},{\"full_name\":\"nonneg_of_mul_nonneg_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[868,9],\"def_end_pos\":[868,35]},{\"full_name\":\"sub_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[886,30],\"def_end_pos\":[886,37]},{\"full_name\":\"not_le_of_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[337,9],\"def_end_pos\":[337,21]}]"
371
+ ],
372
+ "state_after": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\n\u22a2 0 \u2264 (b - a) * (f' a - m)",
373
+ "state_before": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\n\u22a2 a \u2208 Ioo a b",
374
+ "tactic": "refine absurd (sub_nonneg.1 <| nonneg_of_mul_nonneg_right ?_ (sub_pos.2 hab'))\n (not_le_of_lt hma)"
375
+ },
376
+ {
377
+ "annotated_tactic": [
378
+ "\"have : b - a \\u2208 <a>posTangentConeAt<\\/a> (<a>Icc<\\/a> a b) a :=\\n <a>mem_posTangentConeAt_of_segment_subset<\\/a> (<a>segment_eq_Icc<\\/a> hab \\u25b8 <a>Subset.refl<\\/a> _)\"",
379
+ "[{\"full_name\":\"posTangentConeAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/LocalExtr\\/Basic.lean\",\"def_pos\":[76,5],\"def_end_pos\":[76,21]},{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"mem_posTangentConeAt_of_segment_subset\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/LocalExtr\\/Basic.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,47]},{\"full_name\":\"segment_eq_Icc\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Segment.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,23]},{\"full_name\":\"Set.Subset.refl\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[331,9],\"def_end_pos\":[331,20]}]"
380
+ ],
381
+ "state_after": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\nthis : b - a \u2208 posTangentConeAt (Icc a b) a\n\u22a2 0 \u2264 (b - a) * (f' a - m)",
382
+ "state_before": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\n\u22a2 0 \u2264 (b - a) * (f' a - m)",
383
+ "tactic": "have : b - a \u2208 posTangentConeAt (Icc a b) a :=\n mem_posTangentConeAt_of_segment_subset (segment_eq_Icc hab \u25b8 Subset.refl _)"
384
+ },
385
+ {
386
+ "annotated_tactic": [
387
+ "\"simpa only [<a>ContinuousLinearMap.smulRight_apply<\\/a>, <a>ContinuousLinearMap.one_apply<\\/a>]\\n using hc.localize.hasFDerivWithinAt_nonneg (hg a (<a>left_mem_Icc<\\/a>.2 hab)) this\"",
388
+ "[{\"full_name\":\"ContinuousLinearMap.smulRight_apply\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[1222,9],\"def_end_pos\":[1222,24]},{\"full_name\":\"ContinuousLinearMap.one_apply\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[702,9],\"def_end_pos\":[702,18]},{\"full_name\":\"Set.left_mem_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[191,9],\"def_end_pos\":[191,21]}]"
389
+ ],
390
+ "state_after": "no goals",
391
+ "state_before": "case inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : a \u2208 Icc a b\nhc : IsMinOn g (Icc a b) a\nthis : b - a \u2208 posTangentConeAt (Icc a b) a\n\u22a2 0 \u2264 (b - a) * (f' a - m)",
392
+ "tactic": "simpa only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply]\n using hc.localize.hasFDerivWithinAt_nonneg (hg a (left_mem_Icc.2 hab)) this"
393
+ },
394
+ {
395
+ "annotated_tactic": [
396
+ "\"refine <a>absurd<\\/a> (<a>sub_nonpos<\\/a>.1 <| <a>nonpos_of_mul_nonneg_right<\\/a> ?_ (<a>sub_lt_zero<\\/a>.2 hab'))\\n (<a>not_le_of_lt<\\/a> hmb)\"",
397
+ "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]},{\"full_name\":\"sub_nonpos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[730,15],\"def_end_pos\":[730,25]},{\"full_name\":\"nonpos_of_mul_nonneg_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[955,9],\"def_end_pos\":[955,35]},{\"full_name\":\"sub_lt_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[906,7],\"def_end_pos\":[906,18]},{\"full_name\":\"not_le_of_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[337,9],\"def_end_pos\":[337,21]}]"
398
+ ],
399
+ "state_after": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n\u22a2 0 \u2264 (a - b) * (f' b - m)",
400
+ "state_before": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n\u22a2 b \u2208 Ioo a b",
401
+ "tactic": "refine absurd (sub_nonpos.1 <| nonpos_of_mul_nonneg_right ?_ (sub_lt_zero.2 hab'))\n (not_le_of_lt hmb)"
402
+ },
403
+ {
404
+ "annotated_tactic": [
405
+ "\"have : a - b \\u2208 <a>posTangentConeAt<\\/a> (<a>Icc<\\/a> a b) b :=\\n <a>mem_posTangentConeAt_of_segment_subset<\\/a> (by rw [<a>segment_symm<\\/a>, <a>segment_eq_Icc<\\/a> hab])\"",
406
+ "[{\"full_name\":\"posTangentConeAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/LocalExtr\\/Basic.lean\",\"def_pos\":[76,5],\"def_end_pos\":[76,21]},{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"mem_posTangentConeAt_of_segment_subset\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/LocalExtr\\/Basic.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,47]},{\"full_name\":\"segment_symm\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Segment.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,21]},{\"full_name\":\"segment_eq_Icc\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Segment.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,23]}]"
407
+ ],
408
+ "state_after": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\nthis : a - b \u2208 posTangentConeAt (Icc a b) b\n\u22a2 0 \u2264 (a - b) * (f' b - m)",
409
+ "state_before": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n\u22a2 0 \u2264 (a - b) * (f' b - m)",
410
+ "tactic": "have : a - b \u2208 posTangentConeAt (Icc a b) b :=\n mem_posTangentConeAt_of_segment_subset (by rw [segment_symm, segment_eq_Icc hab])"
411
+ },
412
+ {
413
+ "annotated_tactic": [
414
+ "\"simpa only [<a>ContinuousLinearMap.smulRight_apply<\\/a>, <a>ContinuousLinearMap.one_apply<\\/a>]\\n using hc.localize.hasFDerivWithinAt_nonneg (hg b (<a>right_mem_Icc<\\/a>.2 hab)) this\"",
415
+ "[{\"full_name\":\"ContinuousLinearMap.smulRight_apply\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[1222,9],\"def_end_pos\":[1222,24]},{\"full_name\":\"ContinuousLinearMap.one_apply\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[702,9],\"def_end_pos\":[702,18]},{\"full_name\":\"Set.right_mem_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[214,9],\"def_end_pos\":[214,22]}]"
416
+ ],
417
+ "state_after": "no goals",
418
+ "state_before": "case inr.inl\na b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\nthis : a - b \u2208 posTangentConeAt (Icc a b) b\n\u22a2 0 \u2264 (a - b) * (f' b - m)",
419
+ "tactic": "simpa only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply]\n using hc.localize.hasFDerivWithinAt_nonneg (hg b (right_mem_Icc.2 hab)) this"
420
+ },
421
+ {
422
+ "annotated_tactic": [
423
+ "\"rw [<a>segment_symm<\\/a>, <a>segment_eq_Icc<\\/a> hab]\"",
424
+ "[{\"full_name\":\"segment_symm\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Segment.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,21]},{\"full_name\":\"segment_eq_Icc\",\"def_path\":\"Mathlib\\/Analysis\\/Convex\\/Segment.lean\",\"def_pos\":[531,9],\"def_end_pos\":[531,23]}]"
425
+ ],
426
+ "state_after": "no goals",
427
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b \u2208 Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n\u22a2 segment \u211d b a \u2286 Icc a b",
428
+ "tactic": "rw [segment_symm, segment_eq_Icc hab]"
429
+ },
430
+ {
431
+ "annotated_tactic": [
432
+ "\"rwa [\\u2190 <a>mem_interior_iff_mem_nhds<\\/a>, <a>interior_Icc<\\/a>]\"",
433
+ "[{\"full_name\":\"mem_interior_iff_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[852,9],\"def_end_pos\":[852,34]},{\"full_name\":\"interior_Icc\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/DenselyOrdered.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,21]}]"
434
+ ],
435
+ "state_after": "no goals",
436
+ "state_before": "a b : \u211d\nf f' : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : \u2200 x \u2208 Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : \u211d\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : \u211d \u2192 \u211d := fun x => f x - m * x\nhg : \u2200 x \u2208 Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\nc : \u211d\ncmem : c \u2208 Icc a b\nhc : IsMinOn g (Icc a b) c\ncmem' : c \u2208 Ioo a b\n\u22a2 Icc a b \u2208 \ud835\udcdd c",
437
+ "tactic": "rwa [\u2190 mem_interior_iff_mem_nhds, interior_Icc]"
438
+ }
439
+ ],
440
+ "url": "https://github.com/leanprover-community/mathlib4"
441
+ },
442
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
443
+ },
444
+ {
445
+ "error": "theorem has no selected tactic transitions",
446
+ "full_name": "exists_hasDerivWithinAt_eq_of_le_of_ge",
447
+ "id": "exists_hasDerivWithinAt_eq_of_le_of_ge",
448
+ "reason": "import_failure",
449
+ "selection": {
450
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
451
+ "end": [
452
+ 143,
453
+ 58
454
+ ],
455
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
456
+ "full_name": "exists_hasDerivWithinAt_eq_of_le_of_ge",
457
+ "start": [
458
+ 139,
459
+ 1
460
+ ],
461
+ "traced_tactics": [],
462
+ "url": "https://github.com/leanprover-community/mathlib4"
463
+ },
464
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
465
+ },
466
+ {
467
+ "error": "theorem has no selected tactic transitions",
468
+ "full_name": "exists_hasDerivWithinAt_eq_of_lt_of_gt",
469
+ "id": "exists_hasDerivWithinAt_eq_of_lt_of_gt",
470
+ "reason": "import_failure",
471
+ "selection": {
472
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
473
+ "end": [
474
+ 71,
475
+ 30
476
+ ],
477
+ "file_path": "Mathlib/Analysis/Calculus/Darboux.lean",
478
+ "full_name": "exists_hasDerivWithinAt_eq_of_lt_of_gt",
479
+ "start": [
480
+ 65,
481
+ 1
482
+ ],
483
+ "traced_tactics": [],
484
+ "url": "https://github.com/leanprover-community/mathlib4"
485
+ },
486
+ "source_file": "Mathlib/Analysis/Calculus/Darboux.lean"
487
+ },
488
+ {
489
+ "error": "theorem has no selected tactic transitions",
490
+ "full_name": "HasDerivAt.add",
491
+ "id": "HasDerivAt.add",
492
+ "reason": "import_failure",
493
+ "selection": {
494
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
495
+ "end": [
496
+ 62,
497
+ 12
498
+ ],
499
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
500
+ "full_name": "HasDerivAt.add",
501
+ "start": [
502
+ 60,
503
+ 8
504
+ ],
505
+ "traced_tactics": [],
506
+ "url": "https://github.com/leanprover-community/mathlib4"
507
+ },
508
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
509
+ },
510
+ {
511
+ "error": "theorem has no selected tactic transitions",
512
+ "full_name": "HasDerivAt.add_const",
513
+ "id": "HasDerivAt.add_const",
514
+ "reason": "import_failure",
515
+ "selection": {
516
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
517
+ "end": [
518
+ 93,
519
+ 17
520
+ ],
521
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
522
+ "full_name": "HasDerivAt.add_const",
523
+ "start": [
524
+ 91,
525
+ 8
526
+ ],
527
+ "traced_tactics": [],
528
+ "url": "https://github.com/leanprover-community/mathlib4"
529
+ },
530
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
531
+ },
532
+ {
533
+ "error": "theorem has no selected tactic transitions",
534
+ "full_name": "HasDerivAt.const_add",
535
+ "id": "HasDerivAt.const_add",
536
+ "reason": "import_failure",
537
+ "selection": {
538
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
539
+ "end": [
540
+ 126,
541
+ 17
542
+ ],
543
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
544
+ "full_name": "HasDerivAt.const_add",
545
+ "start": [
546
+ 124,
547
+ 8
548
+ ],
549
+ "traced_tactics": [],
550
+ "url": "https://github.com/leanprover-community/mathlib4"
551
+ },
552
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
553
+ },
554
+ {
555
+ "error": "theorem has no selected tactic transitions",
556
+ "full_name": "HasDerivAt.const_sub",
557
+ "id": "HasDerivAt.const_sub",
558
+ "reason": "import_failure",
559
+ "selection": {
560
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
561
+ "end": [
562
+ 370,
563
+ 17
564
+ ],
565
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
566
+ "full_name": "HasDerivAt.const_sub",
567
+ "start": [
568
+ 368,
569
+ 8
570
+ ],
571
+ "traced_tactics": [],
572
+ "url": "https://github.com/leanprover-community/mathlib4"
573
+ },
574
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
575
+ },
576
+ {
577
+ "error": "theorem has no selected tactic transitions",
578
+ "full_name": "HasDerivAt.neg",
579
+ "id": "HasDerivAt.neg",
580
+ "reason": "import_failure",
581
+ "selection": {
582
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
583
+ "end": [
584
+ 199,
585
+ 8
586
+ ],
587
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
588
+ "full_name": "HasDerivAt.neg",
589
+ "start": [
590
+ 198,
591
+ 8
592
+ ],
593
+ "traced_tactics": [],
594
+ "url": "https://github.com/leanprover-community/mathlib4"
595
+ },
596
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
597
+ },
598
+ {
599
+ "error": "theorem has no selected tactic transitions",
600
+ "full_name": "HasDerivAt.sub",
601
+ "id": "HasDerivAt.sub",
602
+ "reason": "import_failure",
603
+ "selection": {
604
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
605
+ "end": [
606
+ 309,
607
+ 12
608
+ ],
609
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
610
+ "full_name": "HasDerivAt.sub",
611
+ "start": [
612
+ 307,
613
+ 8
614
+ ],
615
+ "traced_tactics": [],
616
+ "url": "https://github.com/leanprover-community/mathlib4"
617
+ },
618
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
619
+ },
620
+ {
621
+ "error": "theorem has no selected tactic transitions",
622
+ "full_name": "HasDerivAt.sub_const",
623
+ "id": "HasDerivAt.sub_const",
624
+ "reason": "import_failure",
625
+ "selection": {
626
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
627
+ "end": [
628
+ 341,
629
+ 17
630
+ ],
631
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
632
+ "full_name": "HasDerivAt.sub_const",
633
+ "start": [
634
+ 339,
635
+ 8
636
+ ],
637
+ "traced_tactics": [],
638
+ "url": "https://github.com/leanprover-community/mathlib4"
639
+ },
640
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
641
+ },
642
+ {
643
+ "error": "theorem has no selected tactic transitions",
644
+ "full_name": "HasDerivAt.sum",
645
+ "id": "HasDerivAt.sum",
646
+ "reason": "import_failure",
647
+ "selection": {
648
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
649
+ "end": [
650
+ 168,
651
+ 25
652
+ ],
653
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
654
+ "full_name": "HasDerivAt.sum",
655
+ "start": [
656
+ 166,
657
+ 1
658
+ ],
659
+ "traced_tactics": [],
660
+ "url": "https://github.com/leanprover-community/mathlib4"
661
+ },
662
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
663
+ },
664
+ {
665
+ "error": "theorem has no selected tactic transitions",
666
+ "full_name": "HasDerivAtFilter.add_const",
667
+ "id": "HasDerivAtFilter.add_const",
668
+ "reason": "import_failure",
669
+ "selection": {
670
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
671
+ "end": [
672
+ 83,
673
+ 54
674
+ ],
675
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
676
+ "full_name": "HasDerivAtFilter.add_const",
677
+ "start": [
678
+ 81,
679
+ 1
680
+ ],
681
+ "traced_tactics": [],
682
+ "url": "https://github.com/leanprover-community/mathlib4"
683
+ },
684
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
685
+ },
686
+ {
687
+ "error": "theorem has no selected tactic transitions",
688
+ "full_name": "HasDerivAtFilter.const_add",
689
+ "id": "HasDerivAtFilter.const_add",
690
+ "reason": "import_failure",
691
+ "selection": {
692
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
693
+ "end": [
694
+ 116,
695
+ 54
696
+ ],
697
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
698
+ "full_name": "HasDerivAtFilter.const_add",
699
+ "start": [
700
+ 114,
701
+ 1
702
+ ],
703
+ "traced_tactics": [],
704
+ "url": "https://github.com/leanprover-community/mathlib4"
705
+ },
706
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
707
+ },
708
+ {
709
+ "error": "theorem has no selected tactic transitions",
710
+ "full_name": "HasDerivWithinAt.add",
711
+ "id": "HasDerivWithinAt.add",
712
+ "reason": "import_failure",
713
+ "selection": {
714
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
715
+ "end": [
716
+ 57,
717
+ 12
718
+ ],
719
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
720
+ "full_name": "HasDerivWithinAt.add",
721
+ "start": [
722
+ 55,
723
+ 8
724
+ ],
725
+ "traced_tactics": [],
726
+ "url": "https://github.com/leanprover-community/mathlib4"
727
+ },
728
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
729
+ },
730
+ {
731
+ "error": "theorem has no selected tactic transitions",
732
+ "full_name": "HasDerivWithinAt.add_const",
733
+ "id": "HasDerivWithinAt.add_const",
734
+ "reason": "import_failure",
735
+ "selection": {
736
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
737
+ "end": [
738
+ 88,
739
+ 17
740
+ ],
741
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
742
+ "full_name": "HasDerivWithinAt.add_const",
743
+ "start": [
744
+ 86,
745
+ 8
746
+ ],
747
+ "traced_tactics": [],
748
+ "url": "https://github.com/leanprover-community/mathlib4"
749
+ },
750
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
751
+ },
752
+ {
753
+ "error": "theorem has no selected tactic transitions",
754
+ "full_name": "HasDerivWithinAt.const_add",
755
+ "id": "HasDerivWithinAt.const_add",
756
+ "reason": "import_failure",
757
+ "selection": {
758
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
759
+ "end": [
760
+ 121,
761
+ 17
762
+ ],
763
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
764
+ "full_name": "HasDerivWithinAt.const_add",
765
+ "start": [
766
+ 119,
767
+ 8
768
+ ],
769
+ "traced_tactics": [],
770
+ "url": "https://github.com/leanprover-community/mathlib4"
771
+ },
772
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
773
+ },
774
+ {
775
+ "error": "theorem has no selected tactic transitions",
776
+ "full_name": "HasDerivWithinAt.const_sub",
777
+ "id": "HasDerivWithinAt.const_sub",
778
+ "reason": "import_failure",
779
+ "selection": {
780
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
781
+ "end": [
782
+ 360,
783
+ 17
784
+ ],
785
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
786
+ "full_name": "HasDerivWithinAt.const_sub",
787
+ "start": [
788
+ 358,
789
+ 8
790
+ ],
791
+ "traced_tactics": [],
792
+ "url": "https://github.com/leanprover-community/mathlib4"
793
+ },
794
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
795
+ },
796
+ {
797
+ "error": "theorem has no selected tactic transitions",
798
+ "full_name": "HasDerivWithinAt.neg",
799
+ "id": "HasDerivWithinAt.neg",
800
+ "reason": "import_failure",
801
+ "selection": {
802
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
803
+ "end": [
804
+ 195,
805
+ 8
806
+ ],
807
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
808
+ "full_name": "HasDerivWithinAt.neg",
809
+ "start": [
810
+ 193,
811
+ 8
812
+ ],
813
+ "traced_tactics": [],
814
+ "url": "https://github.com/leanprover-community/mathlib4"
815
+ },
816
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
817
+ },
818
+ {
819
+ "error": "theorem has no selected tactic transitions",
820
+ "full_name": "HasDerivWithinAt.sub",
821
+ "id": "HasDerivWithinAt.sub",
822
+ "reason": "import_failure",
823
+ "selection": {
824
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
825
+ "end": [
826
+ 304,
827
+ 12
828
+ ],
829
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
830
+ "full_name": "HasDerivWithinAt.sub",
831
+ "start": [
832
+ 302,
833
+ 8
834
+ ],
835
+ "traced_tactics": [],
836
+ "url": "https://github.com/leanprover-community/mathlib4"
837
+ },
838
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
839
+ },
840
+ {
841
+ "error": "theorem has no selected tactic transitions",
842
+ "full_name": "HasDerivWithinAt.sub_const",
843
+ "id": "HasDerivWithinAt.sub_const",
844
+ "reason": "import_failure",
845
+ "selection": {
846
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
847
+ "end": [
848
+ 336,
849
+ 17
850
+ ],
851
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
852
+ "full_name": "HasDerivWithinAt.sub_const",
853
+ "start": [
854
+ 334,
855
+ 8
856
+ ],
857
+ "traced_tactics": [],
858
+ "url": "https://github.com/leanprover-community/mathlib4"
859
+ },
860
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
861
+ },
862
+ {
863
+ "error": "theorem has no selected tactic transitions",
864
+ "full_name": "HasDerivWithinAt.sum",
865
+ "id": "HasDerivWithinAt.sum",
866
+ "reason": "import_failure",
867
+ "selection": {
868
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
869
+ "end": [
870
+ 163,
871
+ 25
872
+ ],
873
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean",
874
+ "full_name": "HasDerivWithinAt.sum",
875
+ "start": [
876
+ 161,
877
+ 1
878
+ ],
879
+ "traced_tactics": [],
880
+ "url": "https://github.com/leanprover-community/mathlib4"
881
+ },
882
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Add.lean"
883
+ },
884
+ {
885
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+ },
1921
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
1922
+ },
1923
+ {
1924
+ "error": "theorem has no selected tactic transitions",
1925
+ "full_name": "hasDerivWithinAt_iff_tendsto",
1926
+ "id": "hasDerivWithinAt_iff_tendsto",
1927
+ "reason": "import_failure",
1928
+ "selection": {
1929
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1930
+ "end": [
1931
+ 281,
1932
+ 32
1933
+ ],
1934
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
1935
+ "full_name": "hasDerivWithinAt_iff_tendsto",
1936
+ "start": [
1937
+ 278,
1938
+ 1
1939
+ ],
1940
+ "traced_tactics": [],
1941
+ "url": "https://github.com/leanprover-community/mathlib4"
1942
+ },
1943
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
1944
+ },
1945
+ {
1946
+ "error": "theorem has no selected tactic transitions",
1947
+ "full_name": "hasDerivWithinAt_inter",
1948
+ "id": "hasDerivWithinAt_inter",
1949
+ "reason": "import_failure",
1950
+ "selection": {
1951
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1952
+ "end": [
1953
+ 415,
1954
+ 28
1955
+ ],
1956
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
1957
+ "full_name": "hasDerivWithinAt_inter",
1958
+ "start": [
1959
+ 413,
1960
+ 1
1961
+ ],
1962
+ "traced_tactics": [],
1963
+ "url": "https://github.com/leanprover-community/mathlib4"
1964
+ },
1965
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
1966
+ },
1967
+ {
1968
+ "error": "theorem has no selected tactic transitions",
1969
+ "full_name": "hasDerivWithinAt_inter'",
1970
+ "id": "hasDerivWithinAt_inter'",
1971
+ "reason": "import_failure",
1972
+ "selection": {
1973
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1974
+ "end": [
1975
+ 410,
1976
+ 29
1977
+ ],
1978
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
1979
+ "full_name": "hasDerivWithinAt_inter'",
1980
+ "start": [
1981
+ 408,
1982
+ 1
1983
+ ],
1984
+ "traced_tactics": [],
1985
+ "url": "https://github.com/leanprover-community/mathlib4"
1986
+ },
1987
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
1988
+ },
1989
+ {
1990
+ "error": "theorem has no selected tactic transitions",
1991
+ "full_name": "hasDerivWithinAt_univ",
1992
+ "id": "hasDerivWithinAt_univ",
1993
+ "reason": "import_failure",
1994
+ "selection": {
1995
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1996
+ "end": [
1997
+ 401,
1998
+ 25
1999
+ ],
2000
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2001
+ "full_name": "hasDerivWithinAt_univ",
2002
+ "start": [
2003
+ 400,
2004
+ 1
2005
+ ],
2006
+ "traced_tactics": [],
2007
+ "url": "https://github.com/leanprover-community/mathlib4"
2008
+ },
2009
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
2010
+ },
2011
+ {
2012
+ "error": "theorem has no selected tactic transitions",
2013
+ "full_name": "hasFDerivAt_iff_hasDerivAt",
2014
+ "id": "hasFDerivAt_iff_hasDerivAt",
2015
+ "reason": "import_failure",
2016
+ "selection": {
2017
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
2018
+ "end": [
2019
+ 194,
2020
+ 41
2021
+ ],
2022
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2023
+ "full_name": "hasFDerivAt_iff_hasDerivAt",
2024
+ "start": [
2025
+ 193,
2026
+ 1
2027
+ ],
2028
+ "traced_tactics": [],
2029
+ "url": "https://github.com/leanprover-community/mathlib4"
2030
+ },
2031
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
2032
+ },
2033
+ {
2034
+ "error": "theorem has no selected tactic transitions",
2035
+ "full_name": "hasStrictDerivAt_const",
2036
+ "id": "hasStrictDerivAt_const",
2037
+ "reason": "import_failure",
2038
+ "selection": {
2039
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
2040
+ "end": [
2041
+ 719,
2042
+ 49
2043
+ ],
2044
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2045
+ "full_name": "hasStrictDerivAt_const",
2046
+ "start": [
2047
+ 718,
2048
+ 1
2049
+ ],
2050
+ "traced_tactics": [],
2051
+ "url": "https://github.com/leanprover-community/mathlib4"
2052
+ },
2053
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
2054
+ },
2055
+ {
2056
+ "error": "theorem has no selected tactic transitions",
2057
+ "full_name": "hasStrictDerivAt_id",
2058
+ "id": "hasStrictDerivAt_id",
2059
+ "reason": "import_failure",
2060
+ "selection": {
2061
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
2062
+ "end": [
2063
+ 685,
2064
+ 44
2065
+ ],
2066
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2067
+ "full_name": "hasStrictDerivAt_id",
2068
+ "start": [
2069
+ 684,
2070
+ 1
2071
+ ],
2072
+ "traced_tactics": [],
2073
+ "url": "https://github.com/leanprover-community/mathlib4"
2074
+ },
2075
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Basic.lean"
2076
+ }
2077
+ ],
2078
+ "input_proofs": 131,
2079
+ "input_units": 2,
2080
+ "proofs": 37,
2081
+ "published_proofs": 36,
2082
+ "published_transitions": 49,
2083
+ "replay_failures": [],
2084
+ "transitions": 55,
2085
+ "validation_failures": [
2086
+ {
2087
+ "errors": [
2088
+ "internal transition boundary mismatch",
2089
+ "internal transition boundary mismatch"
2090
+ ],
2091
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2092
+ "proof_id": "mathlib:derivWithin_Ioi_eq_Ici",
2093
+ "provenance": {
2094
+ "selection": {
2095
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
2096
+ "end": [
2097
+ 569,
2098
+ 50
2099
+ ],
2100
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean",
2101
+ "full_name": "derivWithin_Ioi_eq_Ici",
2102
+ "start": [
2103
+ 561,
2104
+ 1
2105
+ ],
2106
+ "traced_tactics": [
2107
+ {
2108
+ "annotated_tactic": [
2109
+ "\"by_cases H : <a>DifferentiableWithinAt<\\/a> \\u211d f (<a>Ioi<\\/a> x) x\"",
2110
+ "[{\"full_name\":\"DifferentiableWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[171,5],\"def_end_pos\":[171,27]},{\"full_name\":\"Set.Ioi\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,8]}]"
2111
+ ],
2112
+ "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2113
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2114
+ "tactic": "by_cases H : DifferentiableWithinAt \u211d f (Ioi x) x"
2115
+ },
2116
+ {
2117
+ "annotated_tactic": [
2118
+ "\"have A := H.hasDerivWithinAt.Ici_of_Ioi\"",
2119
+ "[]"
2120
+ ],
2121
+ "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2122
+ "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2123
+ "tactic": "have A := H.hasDerivWithinAt.Ici_of_Ioi"
2124
+ },
2125
+ {
2126
+ "annotated_tactic": [
2127
+ "\"have B := (<a>differentiableWithinAt_Ioi_iff_Ici<\\/a>.1 H).<a>hasDerivWithinAt<\\/a>\"",
2128
+ "[{\"full_name\":\"differentiableWithinAt_Ioi_iff_Ici\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[554,9],\"def_end_pos\":[554,43]},{\"full_name\":\"DifferentiableWithinAt.hasDerivWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[428,9],\"def_end_pos\":[428,48]}]"
2129
+ ],
2130
+ "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\nB : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2131
+ "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2132
+ "tactic": "have B := (differentiableWithinAt_Ioi_iff_Ici.1 H).hasDerivWithinAt"
2133
+ },
2134
+ {
2135
+ "annotated_tactic": [
2136
+ "\"simpa using (<a>uniqueDiffOn_Ici<\\/a> x).<a>eq<\\/a> <a>left_mem_Ici<\\/a> A B\"",
2137
+ "[{\"full_name\":\"uniqueDiffOn_Ici\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/TangentCone.lean\",\"def_pos\":[389,9],\"def_end_pos\":[389,25]},{\"full_name\":\"UniqueDiffOn.eq\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[293,9],\"def_end_pos\":[293,24]},{\"full_name\":\"Set.left_mem_Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,21]}]"
2138
+ ],
2139
+ "state_after": "no goals",
2140
+ "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\nB : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2141
+ "tactic": "simpa using (uniqueDiffOn_Ici x).eq left_mem_Ici A B"
2142
+ },
2143
+ {
2144
+ "annotated_tactic": [
2145
+ "\"rw [<a>derivWithin_zero_of_not_differentiableWithinAt<\\/a> H,\\n <a>derivWithin_zero_of_not_differentiableWithinAt<\\/a>]\"",
2146
+ "[{\"full_name\":\"derivWithin_zero_of_not_differentiableWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,55]},{\"full_name\":\"derivWithin_zero_of_not_differentiableWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,55]}]"
2147
+ ],
2148
+ "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 \u00acDifferentiableWithinAt \u211d f (Ici x) x",
2149
+ "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
2150
+ "tactic": "rw [derivWithin_zero_of_not_differentiableWithinAt H,\n derivWithin_zero_of_not_differentiableWithinAt]"
2151
+ },
2152
+ {
2153
+ "annotated_tactic": [
2154
+ "\"rwa [<a>differentiableWithinAt_Ioi_iff_Ici<\\/a>] at H\"",
2155
+ "[{\"full_name\":\"differentiableWithinAt_Ioi_iff_Ici\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[554,9],\"def_end_pos\":[554,43]}]"
2156
+ ],
2157
+ "state_after": "no goals",
2158
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2159
+ "tactic": "rwa [differentiableWithinAt_Ioi_iff_Ici] at H"
2160
+ }
2161
+ ],
2162
+ "url": "https://github.com/leanprover-community/mathlib4"
2163
+ }
2164
+ },
2165
+ "theorem_name": "derivWithin_Ioi_eq_Ici"
2166
+ }
2167
+ ],
2168
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2169
+ "zero_transition_proofs": 0
2170
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2171
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2206
+ "mathlib:HasDerivAtFilter.isBigO_sub_rev",
2207
+ "mathlib:derivWithin_congr",
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2225
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735
+ {
736
+ "error": "theorem has no selected tactic transitions",
737
+ "full_name": "differentiableWithinAt_inv",
738
+ "id": "differentiableWithinAt_inv",
739
+ "reason": "import_failure",
740
+ "selection": {
741
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
742
+ "end": [
743
+ 80,
744
+ 60
745
+ ],
746
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
747
+ "full_name": "differentiableWithinAt_inv",
748
+ "start": [
749
+ 78,
750
+ 1
751
+ ],
752
+ "traced_tactics": [],
753
+ "url": "https://github.com/leanprover-community/mathlib4"
754
+ },
755
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
756
+ },
757
+ {
758
+ "error": "theorem has no selected tactic transitions",
759
+ "full_name": "hasDerivAt_inv",
760
+ "id": "hasDerivAt_inv",
761
+ "reason": "import_failure",
762
+ "selection": {
763
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
764
+ "end": [
765
+ 65,
766
+ 46
767
+ ],
768
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
769
+ "full_name": "hasDerivAt_inv",
770
+ "start": [
771
+ 64,
772
+ 1
773
+ ],
774
+ "traced_tactics": [],
775
+ "url": "https://github.com/leanprover-community/mathlib4"
776
+ },
777
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
778
+ },
779
+ {
780
+ "error": "theorem has no selected tactic transitions",
781
+ "full_name": "hasDerivWithinAt_inv",
782
+ "id": "hasDerivWithinAt_inv",
783
+ "reason": "import_failure",
784
+ "selection": {
785
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
786
+ "end": [
787
+ 70,
788
+ 46
789
+ ],
790
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
791
+ "full_name": "hasDerivWithinAt_inv",
792
+ "start": [
793
+ 68,
794
+ 1
795
+ ],
796
+ "traced_tactics": [],
797
+ "url": "https://github.com/leanprover-community/mathlib4"
798
+ },
799
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
800
+ },
801
+ {
802
+ "error": "theorem has no selected tactic transitions",
803
+ "full_name": "hasFDerivAt_inv",
804
+ "id": "hasFDerivAt_inv",
805
+ "reason": "import_failure",
806
+ "selection": {
807
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
808
+ "end": [
809
+ 106,
810
+ 27
811
+ ],
812
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
813
+ "full_name": "hasFDerivAt_inv",
814
+ "start": [
815
+ 104,
816
+ 1
817
+ ],
818
+ "traced_tactics": [],
819
+ "url": "https://github.com/leanprover-community/mathlib4"
820
+ },
821
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
822
+ },
823
+ {
824
+ "error": "theorem has no selected tactic transitions",
825
+ "full_name": "hasFDerivWithinAt_inv",
826
+ "id": "hasFDerivWithinAt_inv",
827
+ "reason": "import_failure",
828
+ "selection": {
829
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
830
+ "end": [
831
+ 111,
832
+ 48
833
+ ],
834
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
835
+ "full_name": "hasFDerivWithinAt_inv",
836
+ "start": [
837
+ 109,
838
+ 1
839
+ ],
840
+ "traced_tactics": [],
841
+ "url": "https://github.com/leanprover-community/mathlib4"
842
+ },
843
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
844
+ },
845
+ {
846
+ "error": "no internal trace for tactic \"suffices\\n (fun p : \ud835\udd5c \u00d7 \ud835\udd5c => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p =>\\n (p.1 - p.2) * 1 by\\n refine this.congr' ?_ (eventually_of_forall fun _ => mul_one _)\\n refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds \u27e8hx, hx\u27e9) ?_\\n rintro \u27e8y, z\u27e9 \u27e8hy, hz\u27e9\\n simp only [mem_setOf_eq] at hy hz\\n field_simp [hx, hy, hz]\\n ring\"",
847
+ "full_name": "hasStrictDerivAt_inv",
848
+ "id": "hasStrictDerivAt_inv",
849
+ "reason": "import_failure",
850
+ "selection": {
851
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
852
+ "end": [
853
+ 61,
854
+ 91
855
+ ],
856
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
857
+ "full_name": "hasStrictDerivAt_inv",
858
+ "start": [
859
+ 48,
860
+ 1
861
+ ],
862
+ "traced_tactics": [
863
+ {
864
+ "annotated_tactic": [
865
+ "\"suffices\\n (fun p : \\ud835\\udd5c \\u00d7 \\ud835\\udd5c => (p.1 - p.2) * ((x * x)\\u207b\\u00b9 - (p.1 * p.2)\\u207b\\u00b9)) =o[\\ud835\\udcdd (x, x)] fun p =>\\n (p.1 - p.2) * 1 by\\n refine this.congr' ?_ (<a>eventually_of_forall<\\/a> fun _ => <a>mul_one<\\/a> _)\\n refine <a>Eventually.mono<\\/a> ((isOpen_ne.prod <a>isOpen_ne<\\/a>).<a>mem_nhds<\\/a> \\u27e8hx, hx\\u27e9) ?_\\n rintro \\u27e8y, z\\u27e9 \\u27e8hy, hz\\u27e9\\n simp only [<a>mem_setOf_eq<\\/a>] at hy hz\\n -- hy : y \\u2260 0, hz : z \\u2260 0\\n field_simp [hx, hy, hz]\\n ring\"",
866
+ "[{\"full_name\":\"Filter.eventually_of_forall\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1129,9],\"def_end_pos\":[1129,29]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"Filter.Eventually.mono\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1158,9],\"def_end_pos\":[1158,24]},{\"full_name\":\"isOpen_ne\",\"def_path\":\"Mathlib\\/Topology\\/Separation.lean\",\"def_pos\":[563,9],\"def_end_pos\":[563,18]},{\"full_name\":\"IsOpen.mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,24]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
867
+ ],
868
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1",
869
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 HasStrictDerivAt Inv.inv (-(x ^ 2)\u207b\u00b9) x",
870
+ "tactic": "suffices\n (fun p : \ud835\udd5c \u00d7 \ud835\udd5c => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p =>\n (p.1 - p.2) * 1 by\n refine this.congr' ?_ (eventually_of_forall fun _ => mul_one _)\n refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds \u27e8hx, hx\u27e9) ?_\n rintro \u27e8y, z\u27e9 \u27e8hy, hz\u27e9\n simp only [mem_setOf_eq] at hy hz\n field_simp [hx, hy, hz]\n ring"
871
+ },
872
+ {
873
+ "annotated_tactic": [
874
+ "\"refine (<a>isBigO_refl<\\/a> (fun p : \\ud835\\udd5c \\u00d7 \\ud835\\udd5c => p.1 - p.2) _).<a>mul_isLittleO<\\/a> ((<a>isLittleO_one_iff<\\/a> \\ud835\\udd5c).2 ?_)\"",
875
+ "[{\"full_name\":\"Asymptotics.isBigO_refl\",\"def_path\":\"Mathlib\\/Analysis\\/Asymptotics\\/Asymptotics.lean\",\"def_pos\":[598,9],\"def_end_pos\":[598,20]},{\"full_name\":\"Asymptotics.IsBigO.mul_isLittleO\",\"def_path\":\"Mathlib\\/Analysis\\/Asymptotics\\/Asymptotics.lean\",\"def_pos\":[1634,9],\"def_end_pos\":[1634,29]},{\"full_name\":\"Asymptotics.isLittleO_one_iff\",\"def_path\":\"Mathlib\\/Analysis\\/Asymptotics\\/Asymptotics.lean\",\"def_pos\":[1362,9],\"def_end_pos\":[1362,26]}]"
876
+ ],
877
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => (x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9) (\ud835\udcdd (x, x)) (\ud835\udcdd 0)",
878
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1",
879
+ "tactic": "refine (isBigO_refl (fun p : \ud835\udd5c \u00d7 \ud835\udd5c => p.1 - p.2) _).mul_isLittleO ((isLittleO_one_iff \ud835\udd5c).2 ?_)"
880
+ },
881
+ {
882
+ "annotated_tactic": [
883
+ "\"rw [\\u2190 <a>sub_self<\\/a> (x * x)\\u207b\\u00b9]\"",
884
+ "[{\"full_name\":\"sub_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1003,30],\"def_end_pos\":[1003,38]}]"
885
+ ],
886
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => (x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9) (\ud835\udcdd (x, x)) (\ud835\udcdd ((x * x)\u207b\u00b9 - (x * x)\u207b\u00b9))",
887
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => (x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9) (\ud835\udcdd (x, x)) (\ud835\udcdd 0)",
888
+ "tactic": "rw [\u2190 sub_self (x * x)\u207b\u00b9]"
889
+ },
890
+ {
891
+ "annotated_tactic": [
892
+ "\"exact tendsto_const_nhds.sub ((continuous_mul.tendsto (x, x)).<a>inv\\u2080<\\/a> <| <a>mul_ne_zero<\\/a> hx hx)\"",
893
+ "[{\"full_name\":\"Filter.Tendsto.inv\\u2080\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/GroupWithZero.lean\",\"def_pos\":[114,9],\"def_end_pos\":[114,28]},{\"full_name\":\"mul_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,20]}]"
894
+ ],
895
+ "state_after": "no goals",
896
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\n\u22a2 Tendsto (fun p => (x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9) (\ud835\udcdd (x, x)) (\ud835\udcdd ((x * x)\u207b\u00b9 - (x * x)\u207b\u00b9))",
897
+ "tactic": "exact tendsto_const_nhds.sub ((continuous_mul.tendsto (x, x)).inv\u2080 <| mul_ne_zero hx hx)"
898
+ },
899
+ {
900
+ "annotated_tactic": [
901
+ "\"refine this.congr' ?_ (<a>eventually_of_forall<\\/a> fun _ => <a>mul_one<\\/a> _)\"",
902
+ "[{\"full_name\":\"Filter.eventually_of_forall\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1129,9],\"def_end_pos\":[1129,29]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
903
+ ],
904
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =\u1da0[\ud835\udcdd (x, x)] fun p =>\n p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)",
905
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\n\u22a2 HasStrictDerivAt Inv.inv (-(x ^ 2)\u207b\u00b9) x",
906
+ "tactic": "refine this.congr' ?_ (eventually_of_forall fun _ => mul_one _)"
907
+ },
908
+ {
909
+ "annotated_tactic": [
910
+ "\"refine <a>Eventually.mono<\\/a> ((isOpen_ne.prod <a>isOpen_ne<\\/a>).<a>mem_nhds<\\/a> \\u27e8hx, hx\\u27e9) ?_\"",
911
+ "[{\"full_name\":\"Filter.Eventually.mono\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1158,9],\"def_end_pos\":[1158,24]},{\"full_name\":\"isOpen_ne\",\"def_path\":\"Mathlib\\/Topology\\/Separation.lean\",\"def_pos\":[563,9],\"def_end_pos\":[563,18]},{\"full_name\":\"IsOpen.mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,24]}]"
912
+ ],
913
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\n\u22a2 \u2200 (x_1 : \ud835\udd5c \u00d7 \ud835\udd5c),\n ({y | y \u2260 0} \u00d7\u02e2 {y | y \u2260 0}) x_1 \u2192\n (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) x_1 =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) x_1",
914
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =\u1da0[\ud835\udcdd (x, x)] fun p =>\n p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)",
915
+ "tactic": "refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds \u27e8hx, hx\u27e9) ?_"
916
+ },
917
+ {
918
+ "annotated_tactic": [
919
+ "\"rintro \\u27e8y, z\\u27e9 \\u27e8hy, hz\\u27e9\"",
920
+ "[]"
921
+ ],
922
+ "state_after": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : (y, z).1 \u2208 {y | y \u2260 0}\nhz : (y, z).2 \u2208 {y | y \u2260 0}\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) (y, z) =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) (y, z)",
923
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\n\u22a2 \u2200 (x_1 : \ud835\udd5c \u00d7 \ud835\udd5c),\n ({y | y \u2260 0} \u00d7\u02e2 {y | y \u2260 0}) x_1 \u2192\n (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) x_1 =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) x_1",
924
+ "tactic": "rintro \u27e8y, z\u27e9 \u27e8hy, hz\u27e9"
925
+ },
926
+ {
927
+ "annotated_tactic": [
928
+ "\"simp only [<a>mem_setOf_eq<\\/a>] at hy hz\"",
929
+ "[{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
930
+ ],
931
+ "state_after": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : y \u2260 0\nhz : z \u2260 0\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) (y, z) =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) (y, z)",
932
+ "state_before": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : (y, z).1 \u2208 {y | y \u2260 0}\nhz : (y, z).2 \u2208 {y | y \u2260 0}\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) (y, z) =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) (y, z)",
933
+ "tactic": "simp only [mem_setOf_eq] at hy hz"
934
+ },
935
+ {
936
+ "annotated_tactic": [
937
+ "\"field_simp [hx, hy, hz]\"",
938
+ "[]"
939
+ ],
940
+ "state_after": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : y \u2260 0\nhz : z \u2260 0\n\u22a2 (y - z) * (y * z - x * x) * (y * z * (x ^ 2 * x ^ 2)) =\n ((z - y) * (x ^ 2 * x ^ 2) - y * z * (-(y * x ^ 2) + x ^ 2 * z)) * (x * x * (y * z))",
941
+ "state_before": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : y \u2260 0\nhz : z \u2260 0\n\u22a2 (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) (y, z) =\n (fun p => p.1\u207b\u00b9 - p.2\u207b\u00b9 - (smulRight 1 (-(x ^ 2)\u207b\u00b9)) (p.1 - p.2)) (y, z)",
942
+ "tactic": "field_simp [hx, hy, hz]"
943
+ },
944
+ {
945
+ "annotated_tactic": [
946
+ "\"ring\"",
947
+ "[]"
948
+ ],
949
+ "state_after": "no goals",
950
+ "state_before": "case mk.intro\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhx : x \u2260 0\nthis : (fun p => (p.1 - p.2) * ((x * x)\u207b\u00b9 - (p.1 * p.2)\u207b\u00b9)) =o[\ud835\udcdd (x, x)] fun p => (p.1 - p.2) * 1\ny z : \ud835\udd5c\nhy : y \u2260 0\nhz : z \u2260 0\n\u22a2 (y - z) * (y * z - x * x) * (y * z * (x ^ 2 * x ^ 2)) =\n ((z - y) * (x ^ 2 * x ^ 2) - y * z * (-(y * x ^ 2) + x ^ 2 * z)) * (x * x * (y * z))",
951
+ "tactic": "ring"
952
+ }
953
+ ],
954
+ "url": "https://github.com/leanprover-community/mathlib4"
955
+ },
956
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Inv.lean"
957
+ }
958
+ ],
959
+ "input_proofs": 83,
960
+ "input_units": 2,
961
+ "proofs": 44,
962
+ "published_proofs": 41,
963
+ "published_transitions": 82,
964
+ "replay_failures": [],
965
+ "transitions": 91,
966
+ "validation_failures": [
967
+ {
968
+ "errors": [
969
+ "internal transition boundary mismatch"
970
+ ],
971
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
972
+ "proof_id": "mathlib:HasDerivWithinAt.div",
973
+ "provenance": {
974
+ "selection": {
975
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
976
+ "end": [
977
+ 184,
978
+ 9
979
+ ],
980
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
981
+ "full_name": "HasDerivWithinAt.div",
982
+ "start": [
983
+ 178,
984
+ 1
985
+ ],
986
+ "traced_tactics": [
987
+ {
988
+ "annotated_tactic": [
989
+ "\"convert hc.mul ((<a>hasDerivAt_inv<\\/a> hx).<a>comp_hasDerivWithinAt<\\/a> x hd) using 1\"",
990
+ "[{\"full_name\":\"hasDerivAt_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Inv.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,23]},{\"full_name\":\"HasDerivAt.comp_hasDerivWithinAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Comp.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,41]}]"
991
+ ],
992
+ "state_after": "case h.e'_6\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 (fun y => c y / d y) = fun y => c y * ((fun y => y\u207b\u00b9) \u2218 d) y\n\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') / d x ^ 2 = c' * ((fun y => y\u207b\u00b9) \u2218 d) x + c x * (-(d x ^ 2)\u207b\u00b9 * d')",
993
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 HasDerivWithinAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) s x",
994
+ "tactic": "convert hc.mul ((hasDerivAt_inv hx).comp_hasDerivWithinAt x hd) using 1"
995
+ },
996
+ {
997
+ "annotated_tactic": [
998
+ "\"field_simp\"",
999
+ "[]"
1000
+ ],
1001
+ "state_after": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') * (d x * d x ^ 2) = (c' * d x ^ 2 + -(c x * d' * d x)) * d x ^ 2",
1002
+ "state_before": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') / d x ^ 2 = c' * ((fun y => y\u207b\u00b9) \u2218 d) x + c x * (-(d x ^ 2)\u207b\u00b9 * d')",
1003
+ "tactic": "field_simp"
1004
+ },
1005
+ {
1006
+ "annotated_tactic": [
1007
+ "\"ring\"",
1008
+ "[]"
1009
+ ],
1010
+ "state_after": "no goals",
1011
+ "state_before": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') * (d x * d x ^ 2) = (c' * d x ^ 2 + -(c x * d' * d x)) * d x ^ 2",
1012
+ "tactic": "ring"
1013
+ }
1014
+ ],
1015
+ "url": "https://github.com/leanprover-community/mathlib4"
1016
+ }
1017
+ },
1018
+ "theorem_name": "HasDerivWithinAt.div"
1019
+ },
1020
+ {
1021
+ "errors": [
1022
+ "internal transition boundary mismatch"
1023
+ ],
1024
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
1025
+ "proof_id": "mathlib:HasStrictDerivAt.div",
1026
+ "provenance": {
1027
+ "selection": {
1028
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1029
+ "end": [
1030
+ 192,
1031
+ 9
1032
+ ],
1033
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
1034
+ "full_name": "HasStrictDerivAt.div",
1035
+ "start": [
1036
+ 187,
1037
+ 1
1038
+ ],
1039
+ "traced_tactics": [
1040
+ {
1041
+ "annotated_tactic": [
1042
+ "\"convert hc.mul ((<a>hasStrictDerivAt_inv<\\/a> hx).<a>comp<\\/a> x hd) using 1\"",
1043
+ "[{\"full_name\":\"hasStrictDerivAt_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Inv.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,29]},{\"full_name\":\"HasStrictDerivAt.comp\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Comp.lean\",\"def_pos\":[267,9],\"def_end_pos\":[267,30]}]"
1044
+ ],
1045
+ "state_after": "case h.e'_6\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 (fun y => c y / d y) = fun y => c y * (Inv.inv \u2218 d) y\n\ncase h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') / d x ^ 2 = c' * (Inv.inv \u2218 d) x + c x * (-(d x ^ 2)\u207b\u00b9 * d')",
1046
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x",
1047
+ "tactic": "convert hc.mul ((hasStrictDerivAt_inv hx).comp x hd) using 1"
1048
+ },
1049
+ {
1050
+ "annotated_tactic": [
1051
+ "\"field_simp\"",
1052
+ "[]"
1053
+ ],
1054
+ "state_after": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') * (d x * d x ^ 2) = (c' * d x ^ 2 + -(c x * d' * d x)) * d x ^ 2",
1055
+ "state_before": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') / d x ^ 2 = c' * (Inv.inv \u2218 d) x + c x * (-(d x ^ 2)\u207b\u00b9 * d')",
1056
+ "tactic": "field_simp"
1057
+ },
1058
+ {
1059
+ "annotated_tactic": [
1060
+ "\"ring\"",
1061
+ "[]"
1062
+ ],
1063
+ "state_after": "no goals",
1064
+ "state_before": "case h.e'_7\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nc d : \ud835\udd5c \u2192 \ud835\udd5c'\nc' d' : \ud835\udd5c'\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nhx : d x \u2260 0\n\u22a2 (c' * d x - c x * d') * (d x * d x ^ 2) = (c' * d x ^ 2 + -(c x * d' * d x)) * d x ^ 2",
1065
+ "tactic": "ring"
1066
+ }
1067
+ ],
1068
+ "url": "https://github.com/leanprover-community/mathlib4"
1069
+ }
1070
+ },
1071
+ "theorem_name": "HasStrictDerivAt.div"
1072
+ },
1073
+ {
1074
+ "errors": [
1075
+ "internal transition boundary mismatch",
1076
+ "internal transition boundary mismatch"
1077
+ ],
1078
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
1079
+ "proof_id": "mathlib:deriv_inv",
1080
+ "provenance": {
1081
+ "selection": {
1082
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
1083
+ "end": [
1084
+ 90,
1085
+ 37
1086
+ ],
1087
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean",
1088
+ "full_name": "deriv_inv",
1089
+ "start": [
1090
+ 87,
1091
+ 1
1092
+ ],
1093
+ "traced_tactics": [
1094
+ {
1095
+ "annotated_tactic": [
1096
+ "\"rcases <a>eq_or_ne<\\/a> x 0 with (rfl | hne)\"",
1097
+ "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
1098
+ ],
1099
+ "state_after": "case inl\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u22a2 deriv (fun x => x\u207b\u00b9) 0 = -(0 ^ 2)\u207b\u00b9\n\ncase inr\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhne : x \u2260 0\n\u22a2 deriv (fun x => x\u207b\u00b9) x = -(x ^ 2)\u207b\u00b9",
1100
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u22a2 deriv (fun x => x\u207b\u00b9) x = -(x ^ 2)\u207b\u00b9",
1101
+ "tactic": "rcases eq_or_ne x 0 with (rfl | hne)"
1102
+ },
1103
+ {
1104
+ "annotated_tactic": [
1105
+ "\"simp [<a>deriv_zero_of_not_differentiableAt<\\/a> (<a>mt<\\/a> <a>differentiableAt_inv<\\/a>.1 (<a>not_not<\\/a>.2 <a>rfl<\\/a>))]\"",
1106
+ "[{\"full_name\":\"deriv_zero_of_not_differentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,43]},{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"differentiableAt_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Inv.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,29]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
1107
+ ],
1108
+ "state_after": "no goals",
1109
+ "state_before": "case inl\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\n\u22a2 deriv (fun x => x\u207b\u00b9) 0 = -(0 ^ 2)\u207b\u00b9",
1110
+ "tactic": "simp [deriv_zero_of_not_differentiableAt (mt differentiableAt_inv.1 (not_not.2 rfl))]"
1111
+ },
1112
+ {
1113
+ "annotated_tactic": [
1114
+ "\"exact (<a>hasDerivAt_inv<\\/a> hne).<a>deriv<\\/a>\"",
1115
+ "[{\"full_name\":\"hasDerivAt_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Inv.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,23]},{\"full_name\":\"HasDerivAt.deriv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[453,9],\"def_end_pos\":[453,25]}]"
1116
+ ],
1117
+ "state_after": "no goals",
1118
+ "state_before": "case inr\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhne : x \u2260 0\n\u22a2 deriv (fun x => x\u207b\u00b9) x = -(x ^ 2)\u207b\u00b9",
1119
+ "tactic": "exact (hasDerivAt_inv hne).deriv"
1120
+ }
1121
+ ],
1122
+ "url": "https://github.com/leanprover-community/mathlib4"
1123
+ }
1124
+ },
1125
+ "theorem_name": "deriv_inv"
1126
+ }
1127
+ ],
1128
+ "validation_rejected_proofs": 3,
1129
+ "zero_transition_proofs": 0
1130
+ },
1131
+ "shards": [
1132
+ {
1133
+ "path": "data/train/mathlib-000406-0000-1496255583be0aee.parquet",
1134
+ "sha256": "1496255583be0aee418cf810b407bd4d6137ae5a72ac8da6410b79102feb70b3",
1135
+ "bytes": 431556,
1136
+ "rows": 68,
1137
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
1138
+ },
1139
+ {
1140
+ "path": "data/dev/mathlib-000406-0000-c9ab9883e4d0abd5.parquet",
1141
+ "sha256": "c9ab9883e4d0abd5386ca98f4242e2b9ef6ba00c12ea394433771092c4684a23",
1142
+ "bytes": 105109,
1143
+ "rows": 7,
1144
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
1145
+ },
1146
+ {
1147
+ "path": "data/internal_test/mathlib-000406-0000-22b219e1f253af86.parquet",
1148
+ "sha256": "22b219e1f253af8693d98a1721242a37d34d77abf81ede13fb65812ae0bddad2",
1149
+ "bytes": 73435,
1150
+ "rows": 7,
1151
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
1152
+ }
1153
+ ],
1154
+ "proof_ids": [
1155
+ "mathlib:HasDerivAt.comp_hasDerivWithinAt_of_eq",
1156
+ "mathlib:HasDerivAt.comp_hasFDerivAt_of_eq",
1157
+ "mathlib:HasDerivAt.comp_hasFDerivWithinAt_of_eq",
1158
+ "mathlib:HasDerivAt.comp_of_eq",
1159
+ "mathlib:HasDerivAt.div",
1160
+ "mathlib:HasDerivAt.inv",
1161
+ "mathlib:HasDerivAt.scomp_hasDerivWithinAt_of_eq",
1162
+ "mathlib:HasDerivAt.scomp_of_eq",
1163
+ "mathlib:HasDerivAtFilter.comp",
1164
+ "mathlib:HasDerivAtFilter.comp_hasFDerivAtFilter",
1165
+ "mathlib:HasDerivAtFilter.comp_hasFDerivAtFilter_of_eq",
1166
+ "mathlib:HasDerivAtFilter.comp_of_eq",
1167
+ "mathlib:HasDerivAtFilter.iterate",
1168
+ "mathlib:HasDerivAtFilter.scomp",
1169
+ "mathlib:HasDerivAtFilter.scomp_of_eq",
1170
+ "mathlib:HasDerivWithinAt.comp",
1171
+ "mathlib:HasDerivWithinAt.comp_of_eq",
1172
+ "mathlib:HasDerivWithinAt.inv",
1173
+ "mathlib:HasDerivWithinAt.iterate",
1174
+ "mathlib:HasDerivWithinAt.scomp_hasDerivAt_of_eq",
1175
+ "mathlib:HasDerivWithinAt.scomp_of_eq",
1176
+ "mathlib:HasFDerivAt.comp_hasDerivAt_of_eq",
1177
+ "mathlib:HasFDerivAt.comp_hasDerivWithinAt_of_eq",
1178
+ "mathlib:HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq",
1179
+ "mathlib:HasStrictDerivAt.comp",
1180
+ "mathlib:HasStrictDerivAt.comp_hasStrictFDerivAt",
1181
+ "mathlib:HasStrictDerivAt.comp_hasStrictFDerivAt_of_eq",
1182
+ "mathlib:HasStrictDerivAt.comp_of_eq",
1183
+ "mathlib:HasStrictDerivAt.iterate",
1184
+ "mathlib:HasStrictDerivAt.scomp",
1185
+ "mathlib:HasStrictDerivAt.scomp_of_eq",
1186
+ "mathlib:HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq",
1187
+ "mathlib:deriv.comp_of_eq",
1188
+ "mathlib:deriv.scomp_of_eq",
1189
+ "mathlib:derivWithin.comp_of_eq",
1190
+ "mathlib:derivWithin.scomp_of_eq",
1191
+ "mathlib:derivWithin_inv",
1192
+ "mathlib:fderiv.comp_deriv_of_eq",
1193
+ "mathlib:fderivWithin.comp_derivWithin_of_eq",
1194
+ "mathlib:fderivWithin_inv",
1195
+ "mathlib:fderiv_inv"
1196
+ ],
1197
+ "proof_keys": [
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+ ]
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+ }
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352
+ "start": [
353
+ 169,
354
+ 1
355
+ ],
356
+ "traced_tactics": [],
357
+ "url": "https://github.com/leanprover-community/mathlib4"
358
+ },
359
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
360
+ },
361
+ {
362
+ "error": "theorem has no selected tactic transitions",
363
+ "full_name": "derivWithin_finset_prod",
364
+ "id": "derivWithin_finset_prod",
365
+ "reason": "import_failure",
366
+ "selection": {
367
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
368
+ "end": [
369
+ 359,
370
+ 87
371
+ ],
372
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
373
+ "full_name": "derivWithin_finset_prod",
374
+ "start": [
375
+ 355,
376
+ 1
377
+ ],
378
+ "traced_tactics": [],
379
+ "url": "https://github.com/leanprover-community/mathlib4"
380
+ },
381
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
382
+ },
383
+ {
384
+ "error": "theorem has no selected tactic transitions",
385
+ "full_name": "derivWithin_mul",
386
+ "id": "derivWithin_mul",
387
+ "reason": "import_failure",
388
+ "selection": {
389
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
390
+ "end": [
391
+ 233,
392
+ 64
393
+ ],
394
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
395
+ "full_name": "derivWithin_mul",
396
+ "start": [
397
+ 230,
398
+ 1
399
+ ],
400
+ "traced_tactics": [],
401
+ "url": "https://github.com/leanprover-community/mathlib4"
402
+ },
403
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
404
+ },
405
+ {
406
+ "error": "theorem has no selected tactic transitions",
407
+ "full_name": "derivWithin_mul_const",
408
+ "id": "derivWithin_mul_const",
409
+ "reason": "import_failure",
410
+ "selection": {
411
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
412
+ "end": [
413
+ 266,
414
+ 52
415
+ ],
416
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
417
+ "full_name": "derivWithin_mul_const",
418
+ "start": [
419
+ 264,
420
+ 1
421
+ ],
422
+ "traced_tactics": [],
423
+ "url": "https://github.com/leanprover-community/mathlib4"
424
+ },
425
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
426
+ },
427
+ {
428
+ "error": "theorem has no selected tactic transitions",
429
+ "full_name": "derivWithin_smul",
430
+ "id": "derivWithin_smul",
431
+ "reason": "import_failure",
432
+ "selection": {
433
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
434
+ "end": [
435
+ 106,
436
+ 65
437
+ ],
438
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
439
+ "full_name": "derivWithin_smul",
440
+ "start": [
441
+ 103,
442
+ 1
443
+ ],
444
+ "traced_tactics": [],
445
+ "url": "https://github.com/leanprover-community/mathlib4"
446
+ },
447
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
448
+ },
449
+ {
450
+ "error": "theorem has no selected tactic transitions",
451
+ "full_name": "derivWithin_smul_const",
452
+ "id": "derivWithin_smul_const",
453
+ "reason": "import_failure",
454
+ "selection": {
455
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
456
+ "end": [
457
+ 135,
458
+ 53
459
+ ],
460
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
461
+ "full_name": "derivWithin_smul_const",
462
+ "start": [
463
+ 132,
464
+ 1
465
+ ],
466
+ "traced_tactics": [],
467
+ "url": "https://github.com/leanprover-community/mathlib4"
468
+ },
469
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
470
+ },
471
+ {
472
+ "error": "theorem has no selected tactic transitions",
473
+ "full_name": "deriv_clm_apply",
474
+ "id": "deriv_clm_apply",
475
+ "reason": "import_failure",
476
+ "selection": {
477
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
478
+ "end": [
479
+ 510,
480
+ 48
481
+ ],
482
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
483
+ "full_name": "deriv_clm_apply",
484
+ "start": [
485
+ 508,
486
+ 1
487
+ ],
488
+ "traced_tactics": [],
489
+ "url": "https://github.com/leanprover-community/mathlib4"
490
+ },
491
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
492
+ },
493
+ {
494
+ "error": "theorem has no selected tactic transitions",
495
+ "full_name": "deriv_clm_comp",
496
+ "id": "deriv_clm_comp",
497
+ "reason": "import_failure",
498
+ "selection": {
499
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
500
+ "end": [
501
+ 477,
502
+ 47
503
+ ],
504
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
505
+ "full_name": "deriv_clm_comp",
506
+ "start": [
507
+ 475,
508
+ 1
509
+ ],
510
+ "traced_tactics": [],
511
+ "url": "https://github.com/leanprover-community/mathlib4"
512
+ },
513
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
514
+ },
515
+ {
516
+ "error": "theorem has no selected tactic transitions",
517
+ "full_name": "deriv_const_mul",
518
+ "id": "deriv_const_mul",
519
+ "reason": "import_failure",
520
+ "selection": {
521
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
522
+ "end": [
523
+ 315,
524
+ 36
525
+ ],
526
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
527
+ "full_name": "deriv_const_mul",
528
+ "start": [
529
+ 313,
530
+ 1
531
+ ],
532
+ "traced_tactics": [],
533
+ "url": "https://github.com/leanprover-community/mathlib4"
534
+ },
535
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
536
+ },
537
+ {
538
+ "error": "theorem has no selected tactic transitions",
539
+ "full_name": "deriv_const_mul_field'",
540
+ "id": "deriv_const_mul_field'",
541
+ "reason": "import_failure",
542
+ "selection": {
543
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
544
+ "end": [
545
+ 324,
546
+ 42
547
+ ],
548
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
549
+ "full_name": "deriv_const_mul_field'",
550
+ "start": [
551
+ 323,
552
+ 1
553
+ ],
554
+ "traced_tactics": [],
555
+ "url": "https://github.com/leanprover-community/mathlib4"
556
+ },
557
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
558
+ },
559
+ {
560
+ "error": "theorem has no selected tactic transitions",
561
+ "full_name": "deriv_const_smul",
562
+ "id": "deriv_const_smul",
563
+ "reason": "import_failure",
564
+ "selection": {
565
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
566
+ "end": [
567
+ 177,
568
+ 37
569
+ ],
570
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
571
+ "full_name": "deriv_const_smul",
572
+ "start": [
573
+ 175,
574
+ 1
575
+ ],
576
+ "traced_tactics": [],
577
+ "url": "https://github.com/leanprover-community/mathlib4"
578
+ },
579
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
580
+ },
581
+ {
582
+ "error": "no internal trace for tactic 'contrapose! hf'",
583
+ "full_name": "deriv_const_smul'",
584
+ "id": "deriv_const_smul'",
585
+ "reason": "import_failure",
586
+ "selection": {
587
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
588
+ "end": [
589
+ 194,
590
+ 98
591
+ ],
592
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
593
+ "full_name": "deriv_const_smul'",
594
+ "start": [
595
+ 180,
596
+ 1
597
+ ],
598
+ "traced_tactics": [
599
+ {
600
+ "annotated_tactic": [
601
+ "\"by_cases hf : <a>DifferentiableAt<\\/a> \\ud835\\udd5c f x\"",
602
+ "[{\"full_name\":\"DifferentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[178,5],\"def_end_pos\":[178,21]}]"
603
+ ],
604
+ "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : DifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
605
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
606
+ "tactic": "by_cases hf : DifferentiableAt \ud835\udd5c f x"
607
+ },
608
+ {
609
+ "annotated_tactic": [
610
+ "\"exact <a>deriv_const_smul<\\/a> c hf\"",
611
+ "[{\"full_name\":\"deriv_const_smul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[175,9],\"def_end_pos\":[175,25]}]"
612
+ ],
613
+ "state_after": "no goals",
614
+ "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : DifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
615
+ "tactic": "exact deriv_const_smul c hf"
616
+ },
617
+ {
618
+ "annotated_tactic": [
619
+ "\"rcases <a>eq_or_ne<\\/a> c 0 with rfl | hc\"",
620
+ "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
621
+ ],
622
+ "state_after": "case neg.inl\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nhf : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => 0 \u2022 f y) x = 0 \u2022 deriv f x\n\ncase neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\nhc : c \u2260 0\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
623
+ "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
624
+ "tactic": "rcases eq_or_ne c 0 with rfl | hc"
625
+ },
626
+ {
627
+ "annotated_tactic": [
628
+ "\"simp only [<a>zero_smul<\\/a>, <a>deriv_const'<\\/a>]\"",
629
+ "[{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]},{\"full_name\":\"deriv_const'\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[735,9],\"def_end_pos\":[735,21]}]"
630
+ ],
631
+ "state_after": "no goals",
632
+ "state_before": "case neg.inl\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nhf : \u00acDifferentiableAt \ud835\udd5c f x\n\u22a2 deriv (fun y => 0 \u2022 f y) x = 0 \u2022 deriv f x",
633
+ "tactic": "simp only [zero_smul, deriv_const']"
634
+ },
635
+ {
636
+ "annotated_tactic": [
637
+ "\"have H : \\u00ac<a>DifferentiableAt<\\/a> \\ud835\\udd5c (fun y \\u21a6 c \\u2022 f y) x := by\\n contrapose! hf\\n change <a>DifferentiableAt<\\/a> \\ud835\\udd5c (fun y \\u21a6 f y) x\\n conv => enter [2, y]; rw [\\u2190 <a>inv_smul_smul\\u2080<\\/a> hc (f y)]\\n exact <a>DifferentiableAt.const_smul<\\/a> hf c\\u207b\\u00b9\"",
638
+ "[{\"full_name\":\"DifferentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[178,5],\"def_end_pos\":[178,21]},{\"full_name\":\"DifferentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[178,5],\"def_end_pos\":[178,21]},{\"full_name\":\"inv_smul_smul\\u2080\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Group.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,23]},{\"full_name\":\"DifferentiableAt.const_smul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Add.lean\",\"def_pos\":[83,9],\"def_end_pos\":[83,36]}]"
639
+ ],
640
+ "state_after": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\nhc : c \u2260 0\nH : \u00acDifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
641
+ "state_before": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\nhc : c \u2260 0\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
642
+ "tactic": "have H : \u00acDifferentiableAt \ud835\udd5c (fun y \u21a6 c \u2022 f y) x := by\n contrapose! hf\n change DifferentiableAt \ud835\udd5c (fun y \u21a6 f y) x\n conv => enter [2, y]; rw [\u2190 inv_smul_smul\u2080 hc (f y)]\n exact DifferentiableAt.const_smul hf c\u207b\u00b9"
643
+ },
644
+ {
645
+ "annotated_tactic": [
646
+ "\"rw [<a>deriv_zero_of_not_differentiableAt<\\/a> hf, <a>deriv_zero_of_not_differentiableAt<\\/a> H, <a>smul_zero<\\/a>]\"",
647
+ "[{\"full_name\":\"deriv_zero_of_not_differentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,43]},{\"full_name\":\"deriv_zero_of_not_differentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,43]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
648
+ ],
649
+ "state_after": "no goals",
650
+ "state_before": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\nhc : c \u2260 0\nH : \u00acDifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 deriv (fun y => c \u2022 f y) x = c \u2022 deriv f x",
651
+ "tactic": "rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt H, smul_zero]"
652
+ },
653
+ {
654
+ "annotated_tactic": [
655
+ "\"contrapose! hf\"",
656
+ "[]"
657
+ ],
658
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c f x",
659
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : \u00acDifferentiableAt \ud835\udd5c f x\nhc : c \u2260 0\n\u22a2 \u00acDifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x",
660
+ "tactic": "contrapose! hf"
661
+ },
662
+ {
663
+ "annotated_tactic": [
664
+ "\"change <a>DifferentiableAt<\\/a> \\ud835\\udd5c (fun y \\u21a6 f y) x\"",
665
+ "[{\"full_name\":\"DifferentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[178,5],\"def_end_pos\":[178,21]}]"
666
+ ],
667
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => f y) x",
668
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c f x",
669
+ "tactic": "change DifferentiableAt \ud835\udd5c (fun y \u21a6 f y) x"
670
+ },
671
+ {
672
+ "annotated_tactic": [
673
+ "\"conv => enter [2, y]; rw [\\u2190 <a>inv_smul_smul\\u2080<\\/a> hc (f y)]\"",
674
+ "[{\"full_name\":\"inv_smul_smul\\u2080\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Group.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,23]}]"
675
+ ],
676
+ "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => c\u207b\u00b9 \u2022 c \u2022 f y) x",
677
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => f y) x",
678
+ "tactic": "conv => enter [2, y]; rw [\u2190 inv_smul_smul\u2080 hc (f y)]"
679
+ },
680
+ {
681
+ "annotated_tactic": [
682
+ "\"exact <a>DifferentiableAt.const_smul<\\/a> hf c\\u207b\\u00b9\"",
683
+ "[{\"full_name\":\"DifferentiableAt.const_smul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Add.lean\",\"def_pos\":[83,9],\"def_end_pos\":[83,36]}]"
684
+ ],
685
+ "state_after": "no goals",
686
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR\u271d : Type u_2\ninst\u271d\u2077 : Semiring R\u271d\ninst\u271d\u2076 : Module R\u271d F\ninst\u271d\u2075 : SMulCommClass \ud835\udd5c R\u271d F\ninst\u271d\u2074 : ContinuousConstSMul R\u271d F\nf : \ud835\udd5c \u2192 F\nx : \ud835\udd5c\nR : Type u_3\ninst\u271d\u00b3 : Field R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhc : c \u2260 0\nhf : DifferentiableAt \ud835\udd5c (fun y => c \u2022 f y) x\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => c\u207b\u00b9 \u2022 c \u2022 f y) x",
687
+ "tactic": "exact DifferentiableAt.const_smul hf c\u207b\u00b9"
688
+ }
689
+ ],
690
+ "url": "https://github.com/leanprover-community/mathlib4"
691
+ },
692
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
693
+ },
694
+ {
695
+ "error": "theorem has no selected tactic transitions",
696
+ "full_name": "deriv_finset_prod",
697
+ "id": "deriv_finset_prod",
698
+ "reason": "import_failure",
699
+ "selection": {
700
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
701
+ "end": [
702
+ 353,
703
+ 65
704
+ ],
705
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
706
+ "full_name": "deriv_finset_prod",
707
+ "start": [
708
+ 351,
709
+ 1
710
+ ],
711
+ "traced_tactics": [],
712
+ "url": "https://github.com/leanprover-community/mathlib4"
713
+ },
714
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
715
+ },
716
+ {
717
+ "error": "theorem has no selected tactic transitions",
718
+ "full_name": "deriv_mul",
719
+ "id": "deriv_mul",
720
+ "reason": "import_failure",
721
+ "selection": {
722
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
723
+ "end": [
724
+ 239,
725
+ 42
726
+ ],
727
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
728
+ "full_name": "deriv_mul",
729
+ "start": [
730
+ 237,
731
+ 1
732
+ ],
733
+ "traced_tactics": [],
734
+ "url": "https://github.com/leanprover-community/mathlib4"
735
+ },
736
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
737
+ },
738
+ {
739
+ "error": "theorem has no selected tactic transitions",
740
+ "full_name": "deriv_mul_const_field'",
741
+ "id": "deriv_mul_const_field'",
742
+ "reason": "import_failure",
743
+ "selection": {
744
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
745
+ "end": [
746
+ 286,
747
+ 42
748
+ ],
749
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
750
+ "full_name": "deriv_mul_const_field'",
751
+ "start": [
752
+ 285,
753
+ 1
754
+ ],
755
+ "traced_tactics": [],
756
+ "url": "https://github.com/leanprover-community/mathlib4"
757
+ },
758
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
759
+ },
760
+ {
761
+ "error": "theorem has no selected tactic transitions",
762
+ "full_name": "deriv_smul",
763
+ "id": "deriv_smul",
764
+ "reason": "import_failure",
765
+ "selection": {
766
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
767
+ "end": [
768
+ 111,
769
+ 43
770
+ ],
771
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
772
+ "full_name": "deriv_smul",
773
+ "start": [
774
+ 109,
775
+ 1
776
+ ],
777
+ "traced_tactics": [],
778
+ "url": "https://github.com/leanprover-community/mathlib4"
779
+ },
780
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
781
+ },
782
+ {
783
+ "error": "theorem has no selected tactic transitions",
784
+ "full_name": "deriv_smul_const",
785
+ "id": "deriv_smul_const",
786
+ "reason": "import_failure",
787
+ "selection": {
788
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
789
+ "end": [
790
+ 140,
791
+ 37
792
+ ],
793
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
794
+ "full_name": "deriv_smul_const",
795
+ "start": [
796
+ 138,
797
+ 1
798
+ ],
799
+ "traced_tactics": [],
800
+ "url": "https://github.com/leanprover-community/mathlib4"
801
+ },
802
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Mul.lean"
803
+ },
804
+ {
805
+ "error": "theorem has no selected tactic transitions",
806
+ "full_name": "deriv_single",
807
+ "id": "deriv_single",
808
+ "reason": "import_failure",
809
+ "selection": {
810
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
811
+ "end": [
812
+ 34,
813
+ 21
814
+ ],
815
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pi.lean",
816
+ "full_name": "deriv_single",
817
+ "start": [
818
+ 32,
819
+ 1
820
+ ],
821
+ "traced_tactics": [],
822
+ "url": "https://github.com/leanprover-community/mathlib4"
823
+ },
824
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pi.lean"
825
+ },
826
+ {
827
+ "error": "theorem has no selected tactic transitions",
828
+ "full_name": "deriv_update",
829
+ "id": "deriv_update",
830
+ "reason": "import_failure",
831
+ "selection": {
832
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
833
+ "end": [
834
+ 30,
835
+ 34
836
+ ],
837
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pi.lean",
838
+ "full_name": "deriv_update",
839
+ "start": [
840
+ 28,
841
+ 1
842
+ ],
843
+ "traced_tactics": [],
844
+ "url": "https://github.com/leanprover-community/mathlib4"
845
+ },
846
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pi.lean"
847
+ },
848
+ {
849
+ "error": "theorem has no selected tactic transitions",
850
+ "full_name": "hasDerivAt_single",
851
+ "id": "hasDerivAt_single",
852
+ "reason": "import_failure",
853
+ "selection": {
854
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
855
+ "end": [
856
+ 26,
857
+ 26
858
+ ],
859
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pi.lean",
860
+ "full_name": "hasDerivAt_single",
861
+ "start": [
862
+ 24,
863
+ 1
864
+ ],
865
+ "traced_tactics": [],
866
+ "url": "https://github.com/leanprover-community/mathlib4"
867
+ },
868
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pi.lean"
869
+ }
870
+ ],
871
+ "input_proofs": 71,
872
+ "input_units": 2,
873
+ "proofs": 36,
874
+ "published_proofs": 34,
875
+ "published_transitions": 52,
876
+ "replay_failures": [],
877
+ "transitions": 65,
878
+ "validation_failures": [
879
+ {
880
+ "errors": [
881
+ "internal transition boundary mismatch",
882
+ "internal transition boundary mismatch",
883
+ "internal transition boundary mismatch",
884
+ "internal transition boundary mismatch"
885
+ ],
886
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
887
+ "proof_id": "mathlib:deriv_mul_const_field",
888
+ "provenance": {
889
+ "selection": {
890
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
891
+ "end": [
892
+ 281,
893
+ 66
894
+ ],
895
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean",
896
+ "full_name": "deriv_mul_const_field",
897
+ "start": [
898
+ 274,
899
+ 1
900
+ ],
901
+ "traced_tactics": [
902
+ {
903
+ "annotated_tactic": [
904
+ "\"by_cases hu : <a>DifferentiableAt<\\/a> \\ud835\\udd5c u x\"",
905
+ "[{\"full_name\":\"DifferentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[178,5],\"def_end_pos\":[178,21]}]"
906
+ ],
907
+ "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : DifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v",
908
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v",
909
+ "tactic": "by_cases hu : DifferentiableAt \ud835\udd5c u x"
910
+ },
911
+ {
912
+ "annotated_tactic": [
913
+ "\"exact <a>deriv_mul_const<\\/a> hu v\"",
914
+ "[{\"full_name\":\"deriv_mul_const\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,24]}]"
915
+ ],
916
+ "state_after": "no goals",
917
+ "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : DifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v",
918
+ "tactic": "exact deriv_mul_const hu v"
919
+ },
920
+ {
921
+ "annotated_tactic": [
922
+ "\"rw [<a>deriv_zero_of_not_differentiableAt<\\/a> hu, <a>zero_mul<\\/a>]\"",
923
+ "[{\"full_name\":\"deriv_zero_of_not_differentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,43]},{\"full_name\":\"MulZeroClass.zero_mul\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,11]}]"
924
+ ],
925
+ "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = 0",
926
+ "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = deriv u x * v",
927
+ "tactic": "rw [deriv_zero_of_not_differentiableAt hu, zero_mul]"
928
+ },
929
+ {
930
+ "annotated_tactic": [
931
+ "\"rcases <a>eq_or_ne<\\/a> v 0 with (rfl | hd)\"",
932
+ "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
933
+ ],
934
+ "state_after": "case neg.inl\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * 0) x = 0\n\ncase neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\n\u22a2 deriv (fun y => u y * v) x = 0",
935
+ "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * v) x = 0",
936
+ "tactic": "rcases eq_or_ne v 0 with (rfl | hd)"
937
+ },
938
+ {
939
+ "annotated_tactic": [
940
+ "\"simp only [<a>mul_zero<\\/a>, <a>deriv_const<\\/a>]\"",
941
+ "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"deriv_const\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[730,9],\"def_end_pos\":[730,20]}]"
942
+ ],
943
+ "state_after": "no goals",
944
+ "state_before": "case neg.inl\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v : \ud835\udd5c \u2192 \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\n\u22a2 deriv (fun y => u y * 0) x = 0",
945
+ "tactic": "simp only [mul_zero, deriv_const]"
946
+ },
947
+ {
948
+ "annotated_tactic": [
949
+ "\"refine <a>deriv_zero_of_not_differentiableAt<\\/a> (<a>mt<\\/a> (fun H => ?_) hu)\"",
950
+ "[{\"full_name\":\"deriv_zero_of_not_differentiableAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,43]},{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]}]"
951
+ ],
952
+ "state_after": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\nH : DifferentiableAt \ud835\udd5c (fun y => u y * v) x\n\u22a2 DifferentiableAt \ud835\udd5c u x",
953
+ "state_before": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\n\u22a2 deriv (fun y => u y * v) x = 0",
954
+ "tactic": "refine deriv_zero_of_not_differentiableAt (mt (fun H => ?_) hu)"
955
+ },
956
+ {
957
+ "annotated_tactic": [
958
+ "\"simpa only [<a>mul_inv_cancel_right\\u2080<\\/a> hd] using H.mul_const v\\u207b\\u00b9\"",
959
+ "[{\"full_name\":\"mul_inv_cancel_right\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[252,9],\"def_end_pos\":[252,30]}]"
960
+ ],
961
+ "state_after": "no goals",
962
+ "state_before": "case neg.inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nG : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_2\n\ud835\udd38 : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38\nc d : \ud835\udd5c \u2192 \ud835\udd38\nc' d' : \ud835\udd38\nu v\u271d : \ud835\udd5c \u2192 \ud835\udd5c'\nv : \ud835\udd5c'\nhu : \u00acDifferentiableAt \ud835\udd5c u x\nhd : v \u2260 0\nH : DifferentiableAt \ud835\udd5c (fun y => u y * v) x\n\u22a2 DifferentiableAt \ud835\udd5c u x",
963
+ "tactic": "simpa only [mul_inv_cancel_right\u2080 hd] using H.mul_const v\u207b\u00b9"
964
+ }
965
+ ],
966
+ "url": "https://github.com/leanprover-community/mathlib4"
967
+ }
968
+ },
969
+ "theorem_name": "deriv_mul_const_field"
970
+ },
971
+ {
972
+ "errors": [
973
+ "internal transition boundary mismatch",
974
+ "internal transition boundary mismatch"
975
+ ],
976
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pi.lean",
977
+ "proof_id": "mathlib:hasDerivAt_update",
978
+ "provenance": {
979
+ "selection": {
980
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
981
+ "end": [
982
+ 22,
983
+ 32
984
+ ],
985
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pi.lean",
986
+ "full_name": "hasDerivAt_update",
987
+ "start": [
988
+ 15,
989
+ 1
990
+ ],
991
+ "traced_tactics": [
992
+ {
993
+ "annotated_tactic": [
994
+ "\"convert (<a>hasFDerivAt_update<\\/a> x y).<a>hasDerivAt<\\/a>\"",
995
+ "[{\"full_name\":\"hasFDerivAt_update\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Pi.lean\",\"def_pos\":[17,9],\"def_end_pos\":[17,27]},{\"full_name\":\"HasFDerivAt.hasDerivAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[197,9],\"def_end_pos\":[197,31]}]"
996
+ ],
997
+ "state_after": "case h.e'_7.h.e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny : \ud835\udd5c\n\u22a2 Pi.single i = \u21d1(ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c)))",
998
+ "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny : \ud835\udd5c\n\u22a2 HasDerivAt (Function.update x i) (Pi.single i 1) y",
999
+ "tactic": "convert (hasFDerivAt_update x y).hasDerivAt"
1000
+ },
1001
+ {
1002
+ "annotated_tactic": [
1003
+ "\"ext z j\"",
1004
+ "[]"
1005
+ ],
1006
+ "state_after": "case h.e'_7.h.e.h.h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\n\u22a2 Pi.single i z j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1007
+ "state_before": "case h.e'_7.h.e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny : \ud835\udd5c\n\u22a2 Pi.single i = \u21d1(ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c)))",
1008
+ "tactic": "ext z j"
1009
+ },
1010
+ {
1011
+ "annotated_tactic": [
1012
+ "\"rw [<a>Pi.single<\\/a>, <a>Function.update_apply<\\/a>]\"",
1013
+ "[{\"full_name\":\"Pi.single\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[348,3],\"def_end_pos\":[348,14]},{\"full_name\":\"Function.update_apply\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[568,9],\"def_end_pos\":[568,21]}]"
1014
+ ],
1015
+ "state_after": "case h.e'_7.h.e.h.h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\n\u22a2 (if j = i then z else 0 j) = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1016
+ "state_before": "case h.e'_7.h.e.h.h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\n\u22a2 Pi.single i z j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1017
+ "tactic": "rw [Pi.single, Function.update_apply]"
1018
+ },
1019
+ {
1020
+ "annotated_tactic": [
1021
+ "\"split_ifs with h\"",
1022
+ "[]"
1023
+ ],
1024
+ "state_after": "case pos\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\nh : j = i\n\u22a2 z = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j\n\ncase neg\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\nh : \u00acj = i\n\u22a2 0 j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1025
+ "state_before": "case h.e'_7.h.e.h.h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\n\u22a2 (if j = i then z else 0 j) = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1026
+ "tactic": "split_ifs with h"
1027
+ },
1028
+ {
1029
+ "annotated_tactic": [
1030
+ "\"simp [h]\"",
1031
+ "[]"
1032
+ ],
1033
+ "state_after": "no goals",
1034
+ "state_before": "case pos\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\nh : j = i\n\u22a2 z = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1035
+ "tactic": "simp [h]"
1036
+ },
1037
+ {
1038
+ "annotated_tactic": [
1039
+ "\"simp [<a>Pi.single_eq_of_ne<\\/a> h]\"",
1040
+ "[{\"full_name\":\"Pi.single_eq_of_ne\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[360,3],\"def_end_pos\":[360,14]}]"
1041
+ ],
1042
+ "state_after": "no goals",
1043
+ "state_before": "case neg\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nx : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\ny z : \ud835\udd5c\nj : \u03b9\nh : \u00acj = i\n\u22a2 0 j = (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id \ud835\udd5c \ud835\udd5c))) z j",
1044
+ "tactic": "simp [Pi.single_eq_of_ne h]"
1045
+ }
1046
+ ],
1047
+ "url": "https://github.com/leanprover-community/mathlib4"
1048
+ }
1049
+ },
1050
+ "theorem_name": "hasDerivAt_update"
1051
+ }
1052
+ ],
1053
+ "validation_rejected_proofs": 2,
1054
+ "zero_transition_proofs": 0
1055
+ },
1056
+ "shards": [
1057
+ {
1058
+ "path": "data/train/mathlib-000408-0000-3678753c0090868d.parquet",
1059
+ "sha256": "3678753c0090868d5f1b24a611f1b4d314f773c2aa02090a8bd2ab459bafe664",
1060
+ "bytes": 514981,
1061
+ "rows": 45,
1062
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
1063
+ },
1064
+ {
1065
+ "path": "data/internal_test/mathlib-000408-0000-4c43fef10d67d745.parquet",
1066
+ "sha256": "4c43fef10d67d745fa0c674d450ad63a74c26563c87c7e61a62cbcd241fd249b",
1067
+ "bytes": 106315,
1068
+ "rows": 7,
1069
+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
1070
+ }
1071
+ ],
1072
+ "proof_ids": [
1073
+ "mathlib:ContinuousLinearMap.hasDerivAt_of_bilinear",
1074
+ "mathlib:ContinuousLinearMap.hasDerivWithinAt_of_bilinear",
1075
+ "mathlib:ContinuousLinearMap.hasStrictDerivAt_of_bilinear",
1076
+ "mathlib:HasDerivAt.clm_apply",
1077
+ "mathlib:HasDerivAt.clm_comp",
1078
+ "mathlib:HasDerivAt.const_mul",
1079
+ "mathlib:HasDerivAt.div_const",
1080
+ "mathlib:HasDerivAt.finset_prod",
1081
+ "mathlib:HasDerivAt.mul",
1082
+ "mathlib:HasDerivAt.smul",
1083
+ "mathlib:HasDerivAt.smul_const",
1084
+ "mathlib:HasDerivAtFilter.const_smul",
1085
+ "mathlib:HasDerivWithinAt.clm_apply",
1086
+ "mathlib:HasDerivWithinAt.clm_comp",
1087
+ "mathlib:HasDerivWithinAt.const_mul",
1088
+ "mathlib:HasDerivWithinAt.div_const",
1089
+ "mathlib:HasDerivWithinAt.finset_prod",
1090
+ "mathlib:HasDerivWithinAt.mul",
1091
+ "mathlib:HasDerivWithinAt.mul_const",
1092
+ "mathlib:HasDerivWithinAt.smul",
1093
+ "mathlib:HasDerivWithinAt.smul_const",
1094
+ "mathlib:HasStrictDerivAt.clm_apply",
1095
+ "mathlib:HasStrictDerivAt.clm_comp",
1096
+ "mathlib:HasStrictDerivAt.const_mul",
1097
+ "mathlib:HasStrictDerivAt.const_smul",
1098
+ "mathlib:HasStrictDerivAt.div_const",
1099
+ "mathlib:HasStrictDerivAt.finset_prod",
1100
+ "mathlib:HasStrictDerivAt.mul",
1101
+ "mathlib:HasStrictDerivAt.mul_const",
1102
+ "mathlib:HasStrictDerivAt.smul",
1103
+ "mathlib:HasStrictDerivAt.smul_const",
1104
+ "mathlib:derivWithin_div_const",
1105
+ "mathlib:deriv_const_mul_field",
1106
+ "mathlib:deriv_div_const"
1107
+ ],
1108
+ "proof_keys": [
1109
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"ContinuousLinearMap.hasDerivAt_of_bilinear\"]",
1110
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"ContinuousLinearMap.hasDerivWithinAt_of_bilinear\"]",
1111
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"ContinuousLinearMap.hasStrictDerivAt_of_bilinear\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.clm_apply\"]",
1113
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.clm_comp\"]",
1114
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.const_mul\"]",
1115
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.div_const\"]",
1116
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.finset_prod\"]",
1117
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.mul\"]",
1118
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.smul\"]",
1119
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAt.smul_const\"]",
1120
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivAtFilter.const_smul\"]",
1121
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.clm_apply\"]",
1122
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.clm_comp\"]",
1123
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.const_mul\"]",
1124
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.div_const\"]",
1125
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.finset_prod\"]",
1126
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.mul\"]",
1127
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.mul_const\"]",
1128
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.smul\"]",
1129
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasDerivWithinAt.smul_const\"]",
1130
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.clm_apply\"]",
1131
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.clm_comp\"]",
1132
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.const_mul\"]",
1133
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.const_smul\"]",
1134
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.div_const\"]",
1135
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.finset_prod\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.mul\"]",
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+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.mul_const\"]",
1138
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.smul\"]",
1139
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"HasStrictDerivAt.smul_const\"]",
1140
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"derivWithin_div_const\"]",
1141
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"deriv_const_mul_field\"]",
1142
+ "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Calculus/Deriv/Mul.lean\",null,null,\"deriv_div_const\"]"
1143
+ ]
1144
+ }
manifests/batches/mathlib/000409.json ADDED
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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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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
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+ },
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+ {
76
+ "error": "theorem has no selected tactic transitions",
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+ "full_name": "Polynomial.differentiableAt",
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+ "id": "Polynomial.differentiableAt",
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
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+ },
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+ {
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+ "error": "theorem has no selected tactic transitions",
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+ "id": "Polynomial.differentiableAt_aeval",
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+ "reason": "import_failure",
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+ "url": "https://github.com/leanprover-community/mathlib4"
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+ },
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+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.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": "Polynomial.differentiableOn",
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+ "id": "Polynomial.differentiableOn",
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+ "reason": "import_failure",
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+ "selection": {
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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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+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.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": "Polynomial.differentiableOn_aeval",
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+ "id": "Polynomial.differentiableOn_aeval",
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+ "reason": "import_failure",
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+ "selection": {
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+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
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+ "end": [
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+ ],
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+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
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+ "full_name": "Polynomial.differentiableOn_aeval",
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+ "start": [
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+ ],
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
160
+ },
161
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
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+ },
163
+ {
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+ "error": "theorem has no selected tactic transitions",
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+ "full_name": "Polynomial.differentiableWithinAt",
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+ "id": "Polynomial.differentiableWithinAt",
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+ "reason": "import_failure",
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+ ],
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
182
+ },
183
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
184
+ },
185
+ {
186
+ "error": "theorem has no selected tactic transitions",
187
+ "full_name": "Polynomial.differentiableWithinAt_aeval",
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+ "id": "Polynomial.differentiableWithinAt_aeval",
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+ "reason": "import_failure",
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+ "selection": {
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+ ],
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+ "start": [
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+ ],
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+ "traced_tactics": [],
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+ "url": "https://github.com/leanprover-community/mathlib4"
204
+ },
205
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
206
+ },
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+ {
208
+ "error": "theorem has no selected tactic transitions",
209
+ "full_name": "Polynomial.differentiable_aeval",
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+ "id": "Polynomial.differentiable_aeval",
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+ "reason": "import_failure",
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+ "selection": {
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+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
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+ "full_name": "Polynomial.differentiable_aeval",
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+ "start": [
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+ "traced_tactics": [],
225
+ "url": "https://github.com/leanprover-community/mathlib4"
226
+ },
227
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
228
+ },
229
+ {
230
+ "error": "theorem has no selected tactic transitions",
231
+ "full_name": "Polynomial.fderiv",
232
+ "id": "Polynomial.fderiv",
233
+ "reason": "import_failure",
234
+ "selection": {
235
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
236
+ "end": [
237
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+ ],
240
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
241
+ "full_name": "Polynomial.fderiv",
242
+ "start": [
243
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244
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245
+ ],
246
+ "traced_tactics": [],
247
+ "url": "https://github.com/leanprover-community/mathlib4"
248
+ },
249
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
250
+ },
251
+ {
252
+ "error": "theorem has no selected tactic transitions",
253
+ "full_name": "Polynomial.fderivWithin",
254
+ "id": "Polynomial.fderivWithin",
255
+ "reason": "import_failure",
256
+ "selection": {
257
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
258
+ "end": [
259
+ 181,
260
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261
+ ],
262
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
263
+ "full_name": "Polynomial.fderivWithin",
264
+ "start": [
265
+ 179,
266
+ 11
267
+ ],
268
+ "traced_tactics": [],
269
+ "url": "https://github.com/leanprover-community/mathlib4"
270
+ },
271
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
272
+ },
273
+ {
274
+ "error": "theorem has no selected tactic transitions",
275
+ "full_name": "Polynomial.fderivWithin_aeval",
276
+ "id": "Polynomial.fderivWithin_aeval",
277
+ "reason": "import_failure",
278
+ "selection": {
279
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
280
+ "end": [
281
+ 186,
282
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283
+ ],
284
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
285
+ "full_name": "Polynomial.fderivWithin_aeval",
286
+ "start": [
287
+ 184,
288
+ 11
289
+ ],
290
+ "traced_tactics": [],
291
+ "url": "https://github.com/leanprover-community/mathlib4"
292
+ },
293
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
294
+ },
295
+ {
296
+ "error": "theorem has no selected tactic transitions",
297
+ "full_name": "Polynomial.fderiv_aeval",
298
+ "id": "Polynomial.fderiv_aeval",
299
+ "reason": "import_failure",
300
+ "selection": {
301
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
302
+ "end": [
303
+ 176,
304
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305
+ ],
306
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
307
+ "full_name": "Polynomial.fderiv_aeval",
308
+ "start": [
309
+ 174,
310
+ 11
311
+ ],
312
+ "traced_tactics": [],
313
+ "url": "https://github.com/leanprover-community/mathlib4"
314
+ },
315
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
316
+ },
317
+ {
318
+ "error": "theorem has no selected tactic transitions",
319
+ "full_name": "Polynomial.hasDerivAt",
320
+ "id": "Polynomial.hasDerivAt",
321
+ "reason": "import_failure",
322
+ "selection": {
323
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
324
+ "end": [
325
+ 74,
326
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327
+ ],
328
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
329
+ "full_name": "Polynomial.hasDerivAt",
330
+ "start": [
331
+ 73,
332
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333
+ ],
334
+ "traced_tactics": [],
335
+ "url": "https://github.com/leanprover-community/mathlib4"
336
+ },
337
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
338
+ },
339
+ {
340
+ "error": "theorem has no selected tactic transitions",
341
+ "full_name": "Polynomial.hasDerivAt_aeval",
342
+ "id": "Polynomial.hasDerivAt_aeval",
343
+ "reason": "import_failure",
344
+ "selection": {
345
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
346
+ "end": [
347
+ 79,
348
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349
+ ],
350
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
351
+ "full_name": "Polynomial.hasDerivAt_aeval",
352
+ "start": [
353
+ 77,
354
+ 11
355
+ ],
356
+ "traced_tactics": [],
357
+ "url": "https://github.com/leanprover-community/mathlib4"
358
+ },
359
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
360
+ },
361
+ {
362
+ "error": "theorem has no selected tactic transitions",
363
+ "full_name": "Polynomial.hasDerivWithinAt",
364
+ "id": "Polynomial.hasDerivWithinAt",
365
+ "reason": "import_failure",
366
+ "selection": {
367
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
368
+ "end": [
369
+ 84,
370
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371
+ ],
372
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
373
+ "full_name": "Polynomial.hasDerivWithinAt",
374
+ "start": [
375
+ 82,
376
+ 11
377
+ ],
378
+ "traced_tactics": [],
379
+ "url": "https://github.com/leanprover-community/mathlib4"
380
+ },
381
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
382
+ },
383
+ {
384
+ "error": "theorem has no selected tactic transitions",
385
+ "full_name": "Polynomial.hasDerivWithinAt_aeval",
386
+ "id": "Polynomial.hasDerivWithinAt_aeval",
387
+ "reason": "import_failure",
388
+ "selection": {
389
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
390
+ "end": [
391
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392
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393
+ ],
394
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
395
+ "full_name": "Polynomial.hasDerivWithinAt_aeval",
396
+ "start": [
397
+ 87,
398
+ 11
399
+ ],
400
+ "traced_tactics": [],
401
+ "url": "https://github.com/leanprover-community/mathlib4"
402
+ },
403
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
404
+ },
405
+ {
406
+ "error": "theorem has no selected tactic transitions",
407
+ "full_name": "Polynomial.hasFDerivAt",
408
+ "id": "Polynomial.hasFDerivAt",
409
+ "reason": "import_failure",
410
+ "selection": {
411
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
412
+ "end": [
413
+ 148,
414
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415
+ ],
416
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
417
+ "full_name": "Polynomial.hasFDerivAt",
418
+ "start": [
419
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420
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421
+ ],
422
+ "traced_tactics": [],
423
+ "url": "https://github.com/leanprover-community/mathlib4"
424
+ },
425
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
426
+ },
427
+ {
428
+ "error": "theorem has no selected tactic transitions",
429
+ "full_name": "Polynomial.hasFDerivAt_aeval",
430
+ "id": "Polynomial.hasFDerivAt_aeval",
431
+ "reason": "import_failure",
432
+ "selection": {
433
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
434
+ "end": [
435
+ 153,
436
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437
+ ],
438
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
439
+ "full_name": "Polynomial.hasFDerivAt_aeval",
440
+ "start": [
441
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442
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443
+ ],
444
+ "traced_tactics": [],
445
+ "url": "https://github.com/leanprover-community/mathlib4"
446
+ },
447
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
448
+ },
449
+ {
450
+ "error": "theorem has no selected tactic transitions",
451
+ "full_name": "Polynomial.hasFDerivWithinAt",
452
+ "id": "Polynomial.hasFDerivWithinAt",
453
+ "reason": "import_failure",
454
+ "selection": {
455
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
456
+ "end": [
457
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458
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459
+ ],
460
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
461
+ "full_name": "Polynomial.hasFDerivWithinAt",
462
+ "start": [
463
+ 156,
464
+ 11
465
+ ],
466
+ "traced_tactics": [],
467
+ "url": "https://github.com/leanprover-community/mathlib4"
468
+ },
469
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
470
+ },
471
+ {
472
+ "error": "theorem has no selected tactic transitions",
473
+ "full_name": "Polynomial.hasFDerivWithinAt_aeval",
474
+ "id": "Polynomial.hasFDerivWithinAt_aeval",
475
+ "reason": "import_failure",
476
+ "selection": {
477
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
478
+ "end": [
479
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480
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481
+ ],
482
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
483
+ "full_name": "Polynomial.hasFDerivWithinAt_aeval",
484
+ "start": [
485
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486
+ 11
487
+ ],
488
+ "traced_tactics": [],
489
+ "url": "https://github.com/leanprover-community/mathlib4"
490
+ },
491
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
492
+ },
493
+ {
494
+ "error": "no internal trace for tactic 'simpa using hp.add hq'",
495
+ "full_name": "Polynomial.hasStrictDerivAt",
496
+ "id": "Polynomial.hasStrictDerivAt",
497
+ "reason": "import_failure",
498
+ "selection": {
499
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
500
+ "end": [
501
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502
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503
+ ],
504
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean",
505
+ "full_name": "Polynomial.hasStrictDerivAt",
506
+ "start": [
507
+ 59,
508
+ 11
509
+ ],
510
+ "traced_tactics": [
511
+ {
512
+ "annotated_tactic": [
513
+ "\"induction p using <a>Polynomial.induction_on'<\\/a> with\\n | h_add p q hp hq => simpa using hp.add hq\\n | h_monomial n a => simpa [<a>mul_assoc<\\/a>] using (<a>hasStrictDerivAt_pow<\\/a> n x).<a>const_mul<\\/a> a\"",
514
+ "[{\"full_name\":\"Polynomial.induction_on'\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Induction.lean\",\"def_pos\":[63,19],\"def_end_pos\":[63,32]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"hasStrictDerivAt_pow\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Pow.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,29]},{\"full_name\":\"HasStrictDerivAt.const_mul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[301,9],\"def_end_pos\":[301,35]}]"
515
+ ],
516
+ "state_after": "no goals",
517
+ "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np : \ud835\udd5c[X]\nq : R[X]\nx : \ud835\udd5c\n\u22a2 HasStrictDerivAt (fun x => eval x p) (eval x (derivative p)) x",
518
+ "tactic": "induction p using Polynomial.induction_on' with\n| h_add p q hp hq => simpa using hp.add hq\n| h_monomial n a => simpa [mul_assoc] using (hasStrictDerivAt_pow n x).const_mul a"
519
+ },
520
+ {
521
+ "annotated_tactic": [
522
+ "\"simpa using hp.add hq\"",
523
+ "[]"
524
+ ],
525
+ "state_after": "no goals",
526
+ "state_before": "case h_add\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np\u271d : \ud835\udd5c[X]\nq\u271d : R[X]\nx : \ud835\udd5c\np q : \ud835\udd5c[X]\nhp : HasStrictDerivAt (fun x => eval x p) (eval x (derivative p)) x\nhq : HasStrictDerivAt (fun x => eval x q) (eval x (derivative q)) x\n\u22a2 HasStrictDerivAt (fun x => eval x (p + q)) (eval x (derivative (p + q))) x",
527
+ "tactic": "simpa using hp.add hq"
528
+ },
529
+ {
530
+ "annotated_tactic": [
531
+ "\"simpa [<a>mul_assoc<\\/a>] using (<a>hasStrictDerivAt_pow<\\/a> n x).<a>const_mul<\\/a> a\"",
532
+ "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"hasStrictDerivAt_pow\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Pow.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,29]},{\"full_name\":\"HasStrictDerivAt.const_mul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[301,9],\"def_end_pos\":[301,35]}]"
533
+ ],
534
+ "state_after": "no goals",
535
+ "state_before": "case h_monomial\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np : \ud835\udd5c[X]\nq : R[X]\nx : \ud835\udd5c\nn : \u2115\na : \ud835\udd5c\n\u22a2 HasStrictDerivAt (fun x => eval x ((monomial n) a)) (eval x (derivative ((monomial n) a))) x",
536
+ "tactic": "simpa [mul_assoc] using (hasStrictDerivAt_pow n x).const_mul a"
537
+ }
538
+ ],
539
+ "url": "https://github.com/leanprover-community/mathlib4"
540
+ },
541
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean"
542
+ },
543
+ {
544
+ "error": "theorem has no selected tactic transitions",
545
+ "full_name": "HasDerivWithinAt.pow",
546
+ "id": "HasDerivWithinAt.pow",
547
+ "reason": "import_failure",
548
+ "selection": {
549
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
550
+ "end": [
551
+ 96,
552
+ 54
553
+ ],
554
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
555
+ "full_name": "HasDerivWithinAt.pow",
556
+ "start": [
557
+ 94,
558
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559
+ ],
560
+ "traced_tactics": [],
561
+ "url": "https://github.com/leanprover-community/mathlib4"
562
+ },
563
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pow.lean"
564
+ },
565
+ {
566
+ "error": "theorem has no selected tactic transitions",
567
+ "full_name": "derivWithin_pow",
568
+ "id": "derivWithin_pow",
569
+ "reason": "import_failure",
570
+ "selection": {
571
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
572
+ "end": [
573
+ 91,
574
+ 47
575
+ ],
576
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
577
+ "full_name": "derivWithin_pow",
578
+ "start": [
579
+ 89,
580
+ 1
581
+ ],
582
+ "traced_tactics": [],
583
+ "url": "https://github.com/leanprover-community/mathlib4"
584
+ },
585
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pow.lean"
586
+ },
587
+ {
588
+ "error": "theorem has no selected tactic transitions",
589
+ "full_name": "derivWithin_pow'",
590
+ "id": "derivWithin_pow'",
591
+ "reason": "import_failure",
592
+ "selection": {
593
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
594
+ "end": [
595
+ 107,
596
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597
+ ],
598
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
599
+ "full_name": "derivWithin_pow'",
600
+ "start": [
601
+ 105,
602
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603
+ ],
604
+ "traced_tactics": [],
605
+ "url": "https://github.com/leanprover-community/mathlib4"
606
+ },
607
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pow.lean"
608
+ },
609
+ {
610
+ "error": "theorem has no selected tactic transitions",
611
+ "full_name": "deriv_pow",
612
+ "id": "deriv_pow",
613
+ "reason": "import_failure",
614
+ "selection": {
615
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
616
+ "end": [
617
+ 81,
618
+ 29
619
+ ],
620
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
621
+ "full_name": "deriv_pow",
622
+ "start": [
623
+ 80,
624
+ 1
625
+ ],
626
+ "traced_tactics": [],
627
+ "url": "https://github.com/leanprover-community/mathlib4"
628
+ },
629
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pow.lean"
630
+ },
631
+ {
632
+ "error": "theorem has no selected tactic transitions",
633
+ "full_name": "deriv_pow'",
634
+ "id": "deriv_pow'",
635
+ "reason": "import_failure",
636
+ "selection": {
637
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
638
+ "end": [
639
+ 86,
640
+ 30
641
+ ],
642
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
643
+ "full_name": "deriv_pow'",
644
+ "start": [
645
+ 85,
646
+ 1
647
+ ],
648
+ "traced_tactics": [],
649
+ "url": "https://github.com/leanprover-community/mathlib4"
650
+ },
651
+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Pow.lean"
652
+ },
653
+ {
654
+ "error": "theorem has no selected tactic transitions",
655
+ "full_name": "deriv_pow''",
656
+ "id": "deriv_pow''",
657
+ "reason": "import_failure",
658
+ "selection": {
659
+ "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
660
+ "end": [
661
+ 113,
662
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663
+ ],
664
+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean",
665
+ "full_name": "deriv_pow''",
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+ },
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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": "differentiableAt_pow",
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+ "tactic": "simpa using hasStrictDerivAt_id x"
832
+ },
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+ ],
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+ "state_after": "no goals",
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841
+ }
842
+ ],
843
+ "url": "https://github.com/leanprover-community/mathlib4"
844
+ }
845
+ },
846
+ "theorem_name": "hasStrictDerivAt_pow"
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+ }
848
+ ],
849
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+ "mathlib:Polynomial.derivWithin_aeval",
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+ "mathlib:Polynomial.hasStrictDerivAt_aeval"
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+ ],
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872
+ ]
873
+ }
manifests/batches/mathlib/000410.json ADDED
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+ {
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+ "error": "theorem has no selected tactic transitions",
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+ },
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+ "source_file": "Mathlib/Analysis/Calculus/Deriv/Shift.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": "deriv_comp_const_add",
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+ "id": "deriv_comp_const_add",
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+ "file_path": "Mathlib/Analysis/Calculus/Deriv/Shift.lean",
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+ "tactic": "by_cases h : DifferentiableAt \ud835\udd5c f (-a)"
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+ },
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+ {
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+ "annotated_tactic": [
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+ "\"simpa only [<a>deriv_neg<\\/a>, <a>neg_one_smul<\\/a>] using <a>deriv.scomp<\\/a> a h (<a>differentiable_neg<\\/a> _)\"",
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+ "tactic": "simpa only [deriv_neg, neg_one_smul] using deriv.scomp a h (differentiable_neg _)"
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+ },
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+ {
300
+ "annotated_tactic": [
301
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+ ],
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+ "state_after": "no goals",
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+ "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \ud835\udd5c \u2192 F\na : \ud835\udd5c\nh : \u00acDifferentiableAt \ud835\udd5c f (-a)\n\u22a2 deriv (fun x => f (-x)) a = -deriv f (-a)",
306
+ "tactic": "rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_comp_neg_iff.mpr h),\n deriv_zero_of_not_differentiableAt h, neg_zero]"
307
+ }
308
+ ],
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+ "url": "https://github.com/leanprover-community/mathlib4"
310
+ }
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+ "path": "data/train/numina-001170-0000-063f62282b7ac664.parquet",
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+ "sha256": "063f62282b7ac66457f529c7da0a4b9d9f3329d253722b62508d9c65d9f0eadf",
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+ "bytes": 648871,
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+ "rows": 39,
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+ "revision": "2bac0d6fed5bbac4cc041479575c727f538d876a"
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+ }
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+ ],
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+ "proof_ids": [
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+ "numina:020fd724-df38-5489-9dcc-77bff1078221",
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+ "numina:0a890f73-d69a-50c8-9be1-5a543a6a522b",
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+ "numina:6c638ffe-07e3-5a23-a022-4cdb727b6c84",
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+ "numina:bed6b02e-28c5-5557-af81-86c764867aa2",
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+ "numina:dd77b7c3-1d93-5a3d-9703-b2dc6817ab56"
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+ ],
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+ "proof_keys": [
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_005850.lean\",\"d1ac5dc8bd2bb56ea6b7f8cdd1ffcf268bd0da312f3acc750874a80f26758aba\",0,\"inequalities_260657\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_005851.lean\",\"fba8783180275cd3ba5d757020b14538c03eae45d73c22086f7b0d5f30704d9d\",1,\"algebra_144492\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_005852.lean\",\"a37ff94ccbb5af6076ea49c4ef6507741210225c16d4c89dafe7b3ce3eea104f\",0,\"algebra_135834\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_005853.lean\",\"41591e18a0ccab5626e450027d9f4cdd77d5d292ba63ff62e233bed8ebc2f9ed\",2,\"number_theory_137791\"]",
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+ "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_005854.lean\",\"fe5f6487aa111cec184a593bfdac1dc7cdd55bdaa4d4297341194e1768bf2ff0\",0,\"algebra_272513\"]"
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+ ]
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+ }