Record 21 verified HPC batch manifests

manifests/batches/mathlib/000560.json

@@ -0,0 +1,651 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000560",
+  "batch_fingerprint": "85896b3adc46dc74782911823721689f0c3da660b88ba63a6106e45381a96059",
+  "evidence": {
+    "batch_fingerprint": "85896b3adc46dc74782911823721689f0c3da660b88ba63a6106e45381a96059",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.eventually_ne",
+        "id": "Complex.IsExpCmpFilter.eventually_ne",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            92,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.eventually_ne",
+          "start": [
+            91,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "Complex.IsExpCmpFilter.isLittleO_log_abs_re",
+        "id": "Complex.IsExpCmpFilter.isLittleO_log_abs_re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            45
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.isLittleO_log_abs_re",
+          "start": [
+            127,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have h2 : 0 < \\u221a2 := by simp\"",
+                "[]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "tactic": "have h2 : 0 < \u221a2 := by simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hz' : 1 \\u2264 <a>abs<\\/a> z := hz.trans (<a>re_le_abs<\\/a> z)\"",
+                "[{\"full_name\":\"Complex.abs\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[56,19],\"def_end_pos\":[56,37]},{\"full_name\":\"Complex.re_le_abs\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,18]}]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "tactic": "have hz' : 1 \u2264 abs z := hz.trans (re_le_abs z)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hm\\u2080 : 0 < <a>max<\\/a> z.re |z.im| := <a>lt_max_iff<\\/a>.2 (<a>Or.inl<\\/a> <| one_pos.trans_le hz)\"",
+                "[{\"full_name\":\"Max.max\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1129,3],\"def_end_pos\":[1129,6]},{\"full_name\":\"lt_max_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[58,9],\"def_end_pos\":[58,19]},{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "tactic": "have hm\u2080 : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>one_mul<\\/a>, <a>Real.norm_eq_abs<\\/a>, <a>_root_.abs_of_nonneg<\\/a> (<a>Real.log_nonneg<\\/a> hz')]\"",
+                "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"Real.norm_eq_abs\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1434,9],\"def_end_pos\":[1434,20]},{\"full_name\":\"abs_of_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[95,3],\"def_end_pos\":[95,14]},{\"full_name\":\"Real.log_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[203,9],\"def_end_pos\":[203,19]}]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (abs z) \u2264 \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u2016Real.log (abs z)\u2016 \u2264 1 * \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "tactic": "rw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> ?_ (<a>le_abs_self<\\/a> _)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"le_abs_self\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[73,3],\"def_end_pos\":[73,14]}]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (abs z) \u2264 Real.log \u221a2 + Real.log (max z.re |z.im|)",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (abs z) \u2264 \u2016Real.log \u221a2 + Real.log (max z.re |z.im|)\u2016",
+              "tactic": "refine le_trans ?_ (le_abs_self _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Real.log_mul<\\/a>, <a>Real.log_le_log_iff<\\/a>, \\u2190 <a>_root_.abs_of_nonneg<\\/a> (<a>le_trans<\\/a> <a>zero_le_one<\\/a> hz)]\"",
+                "[{\"full_name\":\"Real.log_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,16]},{\"full_name\":\"Real.log_le_log_iff\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,23]},{\"full_name\":\"abs_of_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[95,3],\"def_end_pos\":[95,14]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"zero_le_one\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/ZeroLEOne.lean\",\"def_pos\":[26,15],\"def_end_pos\":[26,26]}]"
+              ],
+              "state_after": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 abs z \u2264 \u221a2 * max |z.re| |z.im|\n\ncase h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < abs z\n\ncase h\u2081\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < \u221a2 * max z.re |z.im|\n\ncase hx\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u221a2 \u2260 0\n\ncase hy\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 max z.re |z.im| \u2260 0",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 Real.log (abs z) \u2264 Real.log \u221a2 + Real.log (max z.re |z.im|)",
+              "tactic": "rw [\u2190 Real.log_mul, Real.log_le_log_iff, \u2190 _root_.abs_of_nonneg (le_trans zero_le_one hz)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exacts [<a>abs_le_sqrt_two_mul_max<\\/a> z, one_pos.trans_le hz', <a>mul_pos<\\/a> h2 hm\\u2080, h2.ne', hm\\u2080.ne']\"",
+                "[{\"full_name\":\"Complex.abs_le_sqrt_two_mul_max\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[213,9],\"def_end_pos\":[213,32]},{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 abs z \u2264 \u221a2 * max |z.re| |z.im|\n\ncase h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < abs z\n\ncase h\u2081\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 0 < \u221a2 * max z.re |z.im|\n\ncase hx\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 \u221a2 \u2260 0\n\ncase hy\nl : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\nh2 : 0 < \u221a2\nhz' : 1 \u2264 abs z\nhm\u2080 : 0 < max z.re |z.im|\n\u22a2 max z.re |z.im| \u2260 0",
+              "tactic": "exacts [abs_le_sqrt_two_mul_max z, one_pos.trans_le hz', mul_pos h2 hm\u2080, h2.ne', hm\u2080.ne']"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nz : \u2102\nhz : 1 \u2264 z.re\n\u22a2 0 < \u221a2",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [<a>isLittleO_iff_nat_mul_le'<\\/a>.1 hl.isLittleO_log_re_re n,\\n            hl.abs_im_pow_eventuallyLE_exp_re n,\\n            hl.tendsto_re.eventually_gt_atTop 1] with z hre him h\\u2081\"",
+                "[{\"full_name\":\"Asymptotics.isLittleO_iff_nat_mul_le'\",\"def_path\":\"Mathlib\\/Analysis\\/Asymptotics\\/Asymptotics.lean\",\"def_pos\":[300,9],\"def_end_pos\":[300,34]}]"
+              ],
+              "state_after": "case h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u2102) in l, \u2191n * \u2016Real.log (max x.re |x.im|)\u2016 \u2264 \u2016x.re\u2016",
+              "tactic": "filter_upwards [isLittleO_iff_nat_mul_le'.1 hl.isLittleO_log_re_re n,\n  hl.abs_im_pow_eventuallyLE_exp_re n,\n  hl.tendsto_re.eventually_gt_atTop 1] with z hre him h\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>le_total<\\/a> |z.im| z.re with hle | hle\"",
+                "[{\"full_name\":\"le_total\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,17]}]"
+              ],
+              "state_after": "case h.inl\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : |z.im| \u2264 z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016\n\ncase h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "state_before": "case h\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "tactic": "rcases le_total |z.im| z.re with hle | hle"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>max_eq_left<\\/a> hle]\"",
+                "[{\"full_name\":\"max_eq_left\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[133,9],\"def_end_pos\":[133,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.inl\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : |z.im| \u2264 z.re\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "tactic": "rwa [max_eq_left hle]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have H : 1 < |z.im| := h\\u2081.trans_le hle\"",
+                "[]"
+              ],
+              "state_after": "case h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "state_before": "case h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "tactic": "have H : 1 < |z.im| := h\u2081.trans_le hle"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_cast at *\"",
+                "[]"
+              ],
+              "state_after": "case h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nhle : z.re \u2264 |z.im|\nh\u2081 : 1 < z.re\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "state_before": "case h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nh\u2081 : 1 < z.re\nhle : z.re \u2264 |z.im|\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "tactic": "norm_cast at *"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>max_eq_right<\\/a> hle, <a>Real.norm_eq_abs<\\/a>, <a>Real.norm_eq_abs<\\/a>, <a>abs_of_pos<\\/a> (<a>Real.log_pos<\\/a> H),\\n              \\u2190 <a>Real.log_pow<\\/a>, <a>Real.log_le_iff_le_exp<\\/a> (<a>pow_pos<\\/a> (one_pos.trans H) _),\\n              <a>abs_of_pos<\\/a> (one_pos.trans h\\u2081)]\"",
+                "[{\"full_name\":\"max_eq_right\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,21]},{\"full_name\":\"Real.norm_eq_abs\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1434,9],\"def_end_pos\":[1434,20]},{\"full_name\":\"Real.norm_eq_abs\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1434,9],\"def_end_pos\":[1434,20]},{\"full_name\":\"abs_of_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[101,3],\"def_end_pos\":[101,14]},{\"full_name\":\"Real.log_pos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[176,9],\"def_end_pos\":[176,16]},{\"full_name\":\"Real.log_pow\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[300,9],\"def_end_pos\":[300,16]},{\"full_name\":\"Real.log_le_iff_le_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,26]},{\"full_name\":\"pow_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[552,9],\"def_end_pos\":[552,16]},{\"full_name\":\"abs_of_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[101,3],\"def_end_pos\":[101,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.inr\nl : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\nz : \u2102\nhre : \u2191n * \u2016Real.log z.re\u2016 \u2264 \u2016z.re\u2016\nhim : |z.im| ^ n \u2264 Real.exp z.re\nhle : z.re \u2264 |z.im|\nh\u2081 : 1 < z.re\nH : 1 < |z.im|\n\u22a2 \u2191n * \u2016Real.log (max z.re |z.im|)\u2016 \u2264 \u2016z.re\u2016",
+              "tactic": "rwa [max_eq_right hle, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos (Real.log_pos H),\n  \u2190 Real.log_pow, Real.log_le_iff_le_exp (pow_pos (one_pos.trans H) _),\n  abs_of_pos (one_pos.trans h\u2081)]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.isLittleO_log_re_re",
+        "id": "Complex.IsExpCmpFilter.isLittleO_log_re_re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            57
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.isLittleO_log_re_re",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.of_boundedUnder_im",
+        "id": "Complex.IsExpCmpFilter.of_boundedUnder_im",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            73
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.of_boundedUnder_im",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.of_isBigO_im_re_pow",
+        "id": "Complex.IsExpCmpFilter.of_isBigO_im_re_pow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.of_isBigO_im_re_pow",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.tendsto_abs",
+        "id": "Complex.IsExpCmpFilter.tendsto_abs",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            53
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.tendsto_abs",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.IsExpCmpFilter.tendsto_abs_re",
+        "id": "Complex.IsExpCmpFilter.tendsto_abs_re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            45
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+          "full_name": "Complex.IsExpCmpFilter.tendsto_abs_re",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AnalyticAt.cexp",
+        "id": "AnalyticAt.cexp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            33,
+            26
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticAt.cexp",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AnalyticAt.clog",
+        "id": "AnalyticAt.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            30
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticAt.clog",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'filter_upwards [(fa.continuousAt.eventually_ne (slitPlane_ne_zero m))]'",
+        "full_name": "AnalyticAt.cpow",
+        "id": "AnalyticAt.cpow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            34
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticAt.cpow",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have e : (fun z \\u21a6 f z ^ g z) =\\u1da0[\\ud835\\udcdd x] fun z \\u21a6 <a>exp<\\/a> (<a>log<\\/a> (f z) * g z) := by\\n    filter_upwards [(fa.continuousAt.eventually_ne (<a>slitPlane_ne_zero<\\/a> m))]\\n    intro z fz\\n    simp only [fz, <a>cpow_def<\\/a>, <a>if_false<\\/a>]\"",
+                "[{\"full_name\":\"Complex.exp\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[55,5],\"def_end_pos\":[55,8]},{\"full_name\":\"Complex.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[29,19],\"def_end_pos\":[29,22]},{\"full_name\":\"Complex.slitPlane_ne_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Complex\\/Basic.lean\",\"def_pos\":[748,7],\"def_end_pos\":[748,24]},{\"full_name\":\"Complex.cpow_def\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Pow\\/Complex.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,17]},{\"full_name\":\"if_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[26,17],\"def_end_pos\":[26,25]}]"
+              ],
+              "state_after": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\ne : (fun z => f z ^ g z) =\u1da0[\ud835\udcdd x] fun z => cexp (log (f z) * g z)\n\u22a2 AnalyticAt \u2102 (fun z => f z ^ g z) x",
+              "state_before": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\n\u22a2 AnalyticAt \u2102 (fun z => f z ^ g z) x",
+              "tactic": "have e : (fun z \u21a6 f z ^ g z) =\u1da0[\ud835\udcdd x] fun z \u21a6 exp (log (f z) * g z) := by\n  filter_upwards [(fa.continuousAt.eventually_ne (slitPlane_ne_zero m))]\n  intro z fz\n  simp only [fz, cpow_def, if_false]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>analyticAt_congr<\\/a> e]\"",
+                "[{\"full_name\":\"analyticAt_congr\",\"def_path\":\"Mathlib\\/Analysis\\/Analytic\\/Basic.lean\",\"def_pos\":[553,9],\"def_end_pos\":[553,25]}]"
+              ],
+              "state_after": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\ne : (fun z => f z ^ g z) =\u1da0[\ud835\udcdd x] fun z => cexp (log (f z) * g z)\n\u22a2 AnalyticAt \u2102 (fun z => cexp (log (f z) * g z)) x",
+              "state_before": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\ne : (fun z => f z ^ g z) =\u1da0[\ud835\udcdd x] fun z => cexp (log (f z) * g z)\n\u22a2 AnalyticAt \u2102 (fun z => f z ^ g z) x",
+              "tactic": "rw [analyticAt_congr e]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact ((fa.clog m).<a>mul<\\/a> ga).<a>cexp<\\/a>\"",
+                "[{\"full_name\":\"AnalyticAt.mul\",\"def_path\":\"Mathlib\\/Analysis\\/Analytic\\/Constructions.lean\",\"def_pos\":[172,7],\"def_end_pos\":[172,21]},{\"full_name\":\"AnalyticAt.cexp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Analytic.lean\",\"def_pos\":[32,9],\"def_end_pos\":[32,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\ne : (fun z => f z ^ g z) =\u1da0[\ud835\udcdd x] fun z => cexp (log (f z) * g z)\n\u22a2 AnalyticAt \u2102 (fun z => cexp (log (f z) * g z)) x",
+              "tactic": "exact ((fa.clog m).mul ga).cexp"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [(fa.continuousAt.eventually_ne (<a>slitPlane_ne_zero<\\/a> m))]\"",
+                "[{\"full_name\":\"Complex.slitPlane_ne_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Complex\\/Basic.lean\",\"def_pos\":[748,7],\"def_end_pos\":[748,24]}]"
+              ],
+              "state_after": "case h\nE : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\n\u22a2 \u2200 (a : E), f a \u2260 0 \u2192 f a ^ g a = cexp (log (f a) * g a)",
+              "state_before": "E : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\n\u22a2 (fun z => f z ^ g z) =\u1da0[\ud835\udcdd x] fun z => cexp (log (f z) * g z)",
+              "tactic": "filter_upwards [(fa.continuousAt.eventually_ne (slitPlane_ne_zero m))]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro z fz\"",
+                "[]"
+              ],
+              "state_after": "case h\nE : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz\u271d : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\nz : E\nfz : f z \u2260 0\n\u22a2 f z ^ g z = cexp (log (f z) * g z)",
+              "state_before": "case h\nE : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\n\u22a2 \u2200 (a : E), f a \u2260 0 \u2192 f a ^ g a = cexp (log (f a) * g a)",
+              "tactic": "intro z fz"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [fz, <a>cpow_def<\\/a>, <a>if_false<\\/a>]\"",
+                "[{\"full_name\":\"Complex.cpow_def\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Pow\\/Complex.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,17]},{\"full_name\":\"if_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[26,17],\"def_end_pos\":[26,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\nE : Type\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf g : E \u2192 \u2102\nz\u271d : \u2102\nx : E\ns : Set E\nfa : AnalyticAt \u2102 f x\nga : AnalyticAt \u2102 g x\nm : f x \u2208 slitPlane\nz : E\nfz : f z \u2260 0\n\u22a2 f z ^ g z = cexp (log (f z) * g z)",
+              "tactic": "simp only [fz, cpow_def, if_false]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AnalyticOn.cexp",
+        "id": "AnalyticOn.cexp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            42
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticOn.cexp",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AnalyticOn.clog",
+        "id": "AnalyticOn.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            54,
+            52
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticOn.clog",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AnalyticOn.cpow",
+        "id": "AnalyticOn.cpow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "AnalyticOn.cpow",
+          "start": [
+            67,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "analyticAt_cexp",
+        "id": "analyticAt_cexp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            29,
+            33
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean",
+          "full_name": "analyticAt_cexp",
+          "start": [
+            28,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean"
+      }
+    ],
+    "input_proofs": 26,
+    "input_units": 2,
+    "proofs": 12,
+    "published_proofs": 10,
+    "published_transitions": 16,
+    "replay_failures": [],
+    "transitions": 22,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+        "proof_id": "mathlib:Complex.IsExpCmpFilter.isLittleO_cpow_mul_exp",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              195,
+              16
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+            "full_name": "Complex.IsExpCmpFilter.isLittleO_cpow_mul_exp",
+            "start": [
+              183,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp only\"",
+                  "[]"
+                ],
+                "state_after": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 z : \u2102\nhz : z \u2260 0\n\u22a2 z ^ a\u2081 * cexp (\u2191b\u2081 * z) = z ^ a\u2082 * cexp (\u2191b\u2081 * z) * z ^ (a\u2081 - a\u2082)",
+                "state_before": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 z : \u2102\nhz : z \u2260 0\n\u22a2 (fun z => z ^ a\u2081 * cexp (\u2191b\u2081 * z)) z = (fun z => z ^ a\u2082 * cexp (\u2191b\u2081 * z) * z ^ (a\u2081 - a\u2082)) z",
+                "tactic": "simp only"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>mul_right_comm<\\/a>, \\u2190 <a>cpow_add<\\/a> _ _ hz, <a>add_sub_cancel<\\/a>]\"",
+                  "[{\"full_name\":\"mul_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,23]},{\"full_name\":\"Complex.cpow_add\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Pow\\/Complex.lean\",\"def_pos\":[91,9],\"def_end_pos\":[91,17]},{\"full_name\":\"add_sub_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 z : \u2102\nhz : z \u2260 0\n\u22a2 z ^ a\u2081 * cexp (\u2191b\u2081 * z) = z ^ a\u2082 * cexp (\u2191b\u2081 * z) * z ^ (a\u2081 - a\u2082)",
+                "tactic": "rw [mul_right_comm, \u2190 cpow_add _ _ hz, add_sub_cancel]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>ofReal_sub<\\/a>, <a>sub_mul<\\/a>, <a>mul_assoc<\\/a>, \\u2190 <a>exp_add<\\/a>, <a>add_sub_cancel<\\/a>]\"",
+                  "[{\"full_name\":\"Complex.ofReal_sub\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[789,9],\"def_end_pos\":[789,19]},{\"full_name\":\"sub_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[401,7],\"def_end_pos\":[401,14]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Complex.exp_add\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,16]},{\"full_name\":\"add_sub_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]}]"
+                ],
+                "state_after": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 : \u2102\n\u22a2 (fun z => z ^ a\u2082 * cexp (\u2191b\u2082 * z)) =\u1da0[l] fun z => z ^ a\u2082 * cexp (\u2191b\u2082 * z)",
+                "state_before": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 : \u2102\n\u22a2 (fun z => z ^ a\u2082 * cexp (\u2191b\u2081 * z) * cexp (\u2191(b\u2082 - b\u2081) * z)) =\u1da0[l] fun z => z ^ a\u2082 * cexp (\u2191b\u2082 * z)",
+                "tactic": "simp only [ofReal_sub, sub_mul, mul_assoc, \u2190 exp_add, add_sub_cancel]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"norm_cast\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "l : Filter \u2102\nb\u2081 b\u2082 : \u211d\nhl : IsExpCmpFilter l\nhb : b\u2081 < b\u2082\na\u2081 a\u2082 : \u2102\n\u22a2 (fun z => z ^ a\u2082 * cexp (\u2191b\u2082 * z)) =\u1da0[l] fun z => z ^ a\u2082 * cexp (\u2191b\u2082 * z)",
+                "tactic": "norm_cast"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Complex.IsExpCmpFilter.isLittleO_cpow_mul_exp"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+        "proof_id": "mathlib:Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              116,
+              52
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean",
+            "full_name": "Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re",
+            "start": [
+              107,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>pow_mul'<\\/a>]\"",
+                  "[{\"full_name\":\"pow_mul'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[722,7],\"def_end_pos\":[722,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 (fun z => (z.im ^ n) ^ 2) = fun z => z.im ^ (2 * n)",
+                "tactic": "simp only [pow_mul']"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>pow_one<\\/a>]\"",
+                  "[{\"full_name\":\"pow_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[663,7],\"def_end_pos\":[663,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\nn : \u2115\n\u22a2 (fun z => Real.exp z.re) = fun z => Real.exp z.re ^ 1",
+                "tactic": "simp only [pow_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Complex.IsExpCmpFilter.isLittleO_im_pow_exp_re"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000560-0000-39ddd7d0fa7455a7.parquet",
+      "sha256": "39ddd7d0fa7455a7314c104e359301e5ef5c9248a9b27e585598f934d5c6280c",
+      "bytes": 66799,
+      "rows": 16,
+      "revision": "25e226246b70de1e78b00c9d8e3db5bb5812aa98"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Complex.IsExpCmpFilter.abs_im_pow_eventuallyLE_exp_re",
+    "mathlib:Complex.IsExpCmpFilter.isLittleO_cpow_exp",
+    "mathlib:Complex.IsExpCmpFilter.isLittleO_exp_cpow",
+    "mathlib:Complex.IsExpCmpFilter.isLittleO_pow_mul_exp",
+    "mathlib:Complex.IsExpCmpFilter.isLittleO_zpow_mul_exp",
+    "mathlib:Complex.IsExpCmpFilter.isTheta_cpow_exp_re_mul_log",
+    "mathlib:Complex.IsExpCmpFilter.of_boundedUnder_abs_im",
+    "mathlib:Complex.IsExpCmpFilter.of_isBigO_im_re_rpow",
+    "mathlib:analyticAt_clog",
+    "mathlib:analyticOn_cexp"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.abs_im_pow_eventuallyLE_exp_re\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.isLittleO_cpow_exp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.isLittleO_exp_cpow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.isLittleO_pow_mul_exp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.isLittleO_zpow_mul_exp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.isTheta_cpow_exp_re_mul_log\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.of_boundedUnder_abs_im\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/CompareExp.lean\",null,null,\"Complex.IsExpCmpFilter.of_isBigO_im_re_rpow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean\",null,null,\"analyticAt_clog\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean\",null,null,\"analyticOn_cexp\"]"
+  ]
+}

manifests/batches/mathlib/000562.json

@@ -0,0 +1,1285 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000562",
+  "batch_fingerprint": "3a458e631f8178ddac1dae2e2654ee7eacaa44e1189af614f8a0dae982cde409",
+  "evidence": {
+    "batch_fingerprint": "3a458e631f8178ddac1dae2e2654ee7eacaa44e1189af614f8a0dae982cde409",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AddCircle.continuous_toCircle",
+        "id": "AddCircle.continuous_toCircle",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            187,
+            101
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "AddCircle.continuous_toCircle",
+          "start": [
+            186,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AddCircle.homeomorphCircle'_apply_mk",
+        "id": "AddCircle.homeomorphCircle'_apply_mk",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "AddCircle.homeomorphCircle'_apply_mk",
+          "start": [
+            216,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "AddCircle.scaled_exp_map_periodic",
+        "id": "AddCircle.scaled_exp_map_periodic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            81
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "AddCircle.scaled_exp_map_periodic",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>eq_or_ne<\\/a> T 0 with (rfl | hT)\"",
+                "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
+              ],
+              "state_after": "case inl\n\n\u22a2 Periodic (fun x => expMapCircle (2 * \u03c0 / 0 * x)) 0\n\ncase inr\nT : \u211d\nhT : T \u2260 0\n\u22a2 Periodic (fun x => expMapCircle (2 * \u03c0 / T * x)) T",
+              "state_before": "T : \u211d\n\u22a2 Periodic (fun x => expMapCircle (2 * \u03c0 / T * x)) T",
+              "tactic": "rcases eq_or_ne T 0 with (rfl | hT)"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x\"",
+                "[]"
+              ],
+              "state_after": "case inl\nx : \u211d\n\u22a2 (fun x => expMapCircle (2 * \u03c0 / 0 * x)) (x + 0) = (fun x => expMapCircle (2 * \u03c0 / 0 * x)) x",
+              "state_before": "case inl\n\n\u22a2 Periodic (fun x => expMapCircle (2 * \u03c0 / 0 * x)) 0",
+              "tactic": "intro x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nx : \u211d\n\u22a2 (fun x => expMapCircle (2 * \u03c0 / 0 * x)) (x + 0) = (fun x => expMapCircle (2 * \u03c0 / 0 * x)) x",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x\"",
+                "[]"
+              ],
+              "state_after": "case inr\nT : \u211d\nhT : T \u2260 0\nx : \u211d\n\u22a2 (fun x => expMapCircle (2 * \u03c0 / T * x)) (x + T) = (fun x => expMapCircle (2 * \u03c0 / T * x)) x",
+              "state_before": "case inr\nT : \u211d\nhT : T \u2260 0\n\u22a2 Periodic (fun x => expMapCircle (2 * \u03c0 / T * x)) T",
+              "tactic": "intro x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>mul_add<\\/a>]\"",
+                "[{\"full_name\":\"mul_add\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,14]}]"
+              ],
+              "state_after": "case inr\nT : \u211d\nhT : T \u2260 0\nx : \u211d\n\u22a2 expMapCircle (2 * \u03c0 / T * x + 2 * \u03c0 / T * T) = expMapCircle (2 * \u03c0 / T * x)",
+              "state_before": "case inr\nT : \u211d\nhT : T \u2260 0\nx : \u211d\n\u22a2 (fun x => expMapCircle (2 * \u03c0 / T * x)) (x + T) = (fun x => expMapCircle (2 * \u03c0 / T * x)) x",
+              "tactic": "simp_rw [mul_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>div_mul_cancel\\u2080<\\/a> _ hT, <a>periodic_expMapCircle<\\/a>]\"",
+                "[{\"full_name\":\"div_mul_cancel\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[342,15],\"def_end_pos\":[342,30]},{\"full_name\":\"periodic_expMapCircle\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Circle.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\nT : \u211d\nhT : T \u2260 0\nx : \u211d\n\u22a2 expMapCircle (2 * \u03c0 / T * x + 2 * \u03c0 / T * T) = expMapCircle (2 * \u03c0 / T * x)",
+              "tactic": "rw [div_mul_cancel\u2080 _ hT, periodic_expMapCircle]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.Angle.expMapCircle_coe",
+        "id": "Real.Angle.expMapCircle_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "Real.Angle.expMapCircle_coe",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "circle.arg_eq_arg",
+        "id": "circle.arg_eq_arg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            23
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "circle.arg_eq_arg",
+          "start": [
+            31,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "circle.injective_arg",
+        "id": "circle.injective_arg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            27,
+            82
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "circle.injective_arg",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expMapCircle_add_two_pi",
+        "id": "expMapCircle_add_two_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            110,
+            26
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "expMapCircle_add_two_pi",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expMapCircle_arg",
+        "id": "expMapCircle_arg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            75
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "expMapCircle_arg",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expMapCircle_sub_two_pi",
+        "id": "expMapCircle_sub_two_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            33
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "expMapCircle_sub_two_pi",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expMapCircle_two_pi",
+        "id": "expMapCircle_two_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            102,
+            51
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "expMapCircle_two_pi",
+          "start": [
+            101,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "invOn_arg_expMapCircle",
+        "id": "invOn_arg_expMapCircle",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            36
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "invOn_arg_expMapCircle",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'positivity'",
+        "full_name": "isLocalHomeomorph_expMapCircle",
+        "id": "isLocalHomeomorph_expMapCircle",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            81
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "isLocalHomeomorph_expMapCircle",
+          "start": [
+            235,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : <a>Fact<\\/a> (0 < 2 * \\u03c0) := \\u27e8by positivity\\u27e9\"",
+                "[{\"full_name\":\"Fact\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[114,7],\"def_end_pos\":[114,11]}]"
+              ],
+              "state_after": "this : Fact (0 < 2 * \u03c0)\n\u22a2 IsLocalHomeomorph \u21d1expMapCircle",
+              "state_before": "\u22a2 IsLocalHomeomorph \u21d1expMapCircle",
+              "tactic": "have : Fact (0 < 2 * \u03c0) := \u27e8by positivity\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact homeomorphCircle'.isLocalHomeomorph.comp (<a>isLocalHomeomorph_coe<\\/a> (2 * \\u03c0))\"",
+                "[{\"full_name\":\"AddCircle.isLocalHomeomorph_coe\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/AddCircle.lean\",\"def_pos\":[304,7],\"def_end_pos\":[304,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "this : Fact (0 < 2 * \u03c0)\n\u22a2 IsLocalHomeomorph \u21d1expMapCircle",
+              "tactic": "exact homeomorphCircle'.isLocalHomeomorph.comp (isLocalHomeomorph_coe (2 * \u03c0))"
+            },
+            {
+              "annotated_tactic": [
+                "\"positivity\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u22a2 0 < 2 * \u03c0",
+              "tactic": "positivity"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "leftInverse_expMapCircle_arg",
+        "id": "leftInverse_expMapCircle_arg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "leftInverse_expMapCircle_arg",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "surjOn_expMapCircle_neg_pi_pi",
+        "id": "surjOn_expMapCircle_neg_pi_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            37
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean",
+          "full_name": "surjOn_expMapCircle_neg_pi_pi",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa [two_mul, mul_add] using hn'",
+        "full_name": "Complex.exp_eq_one_iff",
+        "id": "Complex.exp_eq_one_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            165,
+            53
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Complex.exp_eq_one_iff",
+          "start": [
+            156,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nx : \u2102\n\u22a2 cexp x = 1 \u2192 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)\n\ncase mpr\nx : \u2102\n\u22a2 (\u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)) \u2192 cexp x = 1",
+              "state_before": "x : \u2102\n\u22a2 cexp x = 1 \u2194 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case mp\nx : \u2102\nh : cexp x = 1\n\u22a2 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "state_before": "case mp\nx : \u2102\n\u22a2 cexp x = 1 \u2192 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>existsUnique_add_zsmul_mem_Ioc<\\/a> <a>Real.two_pi_pos<\\/a> x.im (-\\u03c0) with \\u27e8n, hn, -\\u27e9\"",
+                "[{\"full_name\":\"existsUnique_add_zsmul_mem_Ioc\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Archimedean.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,39]},{\"full_name\":\"Real.two_pi_pos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Trigonometric\\/Basic.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,19]}]"
+              ],
+              "state_after": "case mp.intro.intro\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "state_before": "case mp\nx : \u2102\nh : cexp x = 1\n\u22a2 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "tactic": "rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos x.im (-\u03c0) with \u27e8n, hn, -\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"use -n\"",
+                "[]"
+              ],
+              "state_after": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 x = \u2191(-n) * (2 * \u2191\u03c0 * I)",
+              "state_before": "case mp.intro.intro\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 \u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)",
+              "tactic": "use -n"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Int.cast_neg<\\/a>, <a>neg_mul<\\/a>, <a>eq_neg_iff_add_eq_zero<\\/a>]\"",
+                "[{\"full_name\":\"Int.cast_neg\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,17]},{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]},{\"full_name\":\"eq_neg_iff_add_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[931,3],\"def_end_pos\":[931,14]}]"
+              ],
+              "state_after": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 x + \u2191n * (2 * \u2191\u03c0 * I) = 0",
+              "state_before": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 x = \u2191(-n) * (2 * \u2191\u03c0 * I)",
+              "tactic": "rw [Int.cast_neg, neg_mul, eq_neg_iff_add_eq_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : (x + n * (2 * \\u03c0 * <a>I<\\/a>)).<a>im<\\/a> \\u2208 <a>Set.Ioc<\\/a> (-\\u03c0) \\u03c0 := by simpa [<a>two_mul<\\/a>, <a>mul_add<\\/a>] using hn\"",
+                "[{\"full_name\":\"Complex.I\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[278,5],\"def_end_pos\":[278,6]},{\"full_name\":\"Complex.im\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[33,3],\"def_end_pos\":[33,5]},{\"full_name\":\"Set.Ioc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[69,5],\"def_end_pos\":[69,8]},{\"full_name\":\"two_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,16]},{\"full_name\":\"mul_add\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,14]}]"
+              ],
+              "state_after": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\nthis : (x + \u2191n * (2 * \u2191\u03c0 * I)).im \u2208 Set.Ioc (-\u03c0) \u03c0\n\u22a2 x + \u2191n * (2 * \u2191\u03c0 * I) = 0",
+              "state_before": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 x + \u2191n * (2 * \u2191\u03c0 * I) = 0",
+              "tactic": "have : (x + n * (2 * \u03c0 * I)).im \u2208 Set.Ioc (-\u03c0) \u03c0 := by simpa [two_mul, mul_add] using hn"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>log_exp<\\/a> this.1 this.2, exp_periodic.int_mul n, h, <a>log_one<\\/a>]\"",
+                "[{\"full_name\":\"Complex.log_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[60,9],\"def_end_pos\":[60,16]},{\"full_name\":\"Complex.log_one\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\nx : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\nthis : (x + \u2191n * (2 * \u2191\u03c0 * I)).im \u2208 Set.Ioc (-\u03c0) \u03c0\n\u22a2 x + \u2191n * (2 * \u2191\u03c0 * I) = 0",
+              "tactic": "rw [\u2190 log_exp this.1 this.2, exp_periodic.int_mul n, h, log_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>two_mul<\\/a>, <a>mul_add<\\/a>] using hn\"",
+                "[{\"full_name\":\"two_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,16]},{\"full_name\":\"mul_add\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x : \u2102\nh : cexp x = 1\nn : \u2124\nhn : x.im + n \u2022 (2 * \u03c0) \u2208 Set.Ioc (-\u03c0) (-\u03c0 + 2 * \u03c0)\n\u22a2 (x + \u2191n * (2 * \u2191\u03c0 * I)).im \u2208 Set.Ioc (-\u03c0) \u03c0",
+              "tactic": "simpa [two_mul, mul_add] using hn"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8n, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mpr.intro\nn : \u2124\n\u22a2 cexp (\u2191n * (2 * \u2191\u03c0 * I)) = 1",
+              "state_before": "case mpr\nx : \u2102\n\u22a2 (\u2203 n, x = \u2191n * (2 * \u2191\u03c0 * I)) \u2192 cexp x = 1",
+              "tactic": "rintro \u27e8n, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (exp_periodic.int_mul n).eq.trans <a>exp_zero<\\/a>\"",
+                "[{\"full_name\":\"Complex.exp_zero\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr.intro\nn : \u2124\n\u22a2 cexp (\u2191n * (2 * \u2191\u03c0 * I)) = 1",
+              "tactic": "exact (exp_periodic.int_mul n).eq.trans exp_zero"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [hx]'",
+        "full_name": "Complex.log_conj_eq_ite",
+        "id": "Complex.log_conj_eq_ite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            49
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Complex.log_conj_eq_ite",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>log<\\/a>, <a>abs_conj<\\/a>, <a>arg_conj<\\/a>, <a>map_add<\\/a>, <a>map_mul<\\/a>, <a>conj_ofReal<\\/a>]\"",
+                "[{\"full_name\":\"Complex.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[29,19],\"def_end_pos\":[29,22]},{\"full_name\":\"Complex.abs_conj\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[135,9],\"def_end_pos\":[135,17]},{\"full_name\":\"Complex.arg_conj\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Arg.lean\",\"def_pos\":[333,9],\"def_end_pos\":[333,17]},{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"Complex.conj_ofReal\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[572,9],\"def_end_pos\":[572,20]}]"
+              ],
+              "state_after": "x : \u2102\n\u22a2 \u2191(Real.log (abs x)) + \u2191(if x.arg = \u03c0 then \u03c0 else -x.arg) * I =\n    if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + \u2191x.arg * (starRingEnd \u2102) I",
+              "state_before": "x : \u2102\n\u22a2 log ((starRingEnd \u2102) x) = if x.arg = \u03c0 then log x else (starRingEnd \u2102) (log x)",
+              "tactic": "simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_ofReal]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with hx\"",
+                "[]"
+              ],
+              "state_after": "case pos\nx : \u2102\nhx : x.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191\u03c0 * I = \u2191(Real.log (abs x)) + \u2191x.arg * I\n\ncase neg\nx : \u2102\nhx : \u00acx.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191(-x.arg) * I = \u2191(Real.log (abs x)) + \u2191x.arg * (starRingEnd \u2102) I",
+              "state_before": "x : \u2102\n\u22a2 \u2191(Real.log (abs x)) + \u2191(if x.arg = \u03c0 then \u03c0 else -x.arg) * I =\n    if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + \u2191x.arg * (starRingEnd \u2102) I",
+              "tactic": "split_ifs with hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>ofReal_neg<\\/a>, <a>conj_I<\\/a>, <a>mul_neg<\\/a>, <a>neg_mul<\\/a>]\"",
+                "[{\"full_name\":\"Complex.ofReal_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,19]},{\"full_name\":\"Complex.conj_I\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[577,9],\"def_end_pos\":[577,15]},{\"full_name\":\"mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191(-x.arg) * I = \u2191(Real.log (abs x)) + \u2191x.arg * (starRingEnd \u2102) I",
+              "tactic": "simp_rw [ofReal_neg, conj_I, mul_neg, neg_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hx]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nx : \u2102\nhx : x.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191\u03c0 * I = \u2191(Real.log (abs x)) + \u2191x.arg * I",
+              "tactic": "rw [hx]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [hx]'",
+        "full_name": "Complex.log_inv_eq_ite",
+        "id": "Complex.log_inv_eq_ite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            22
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Complex.log_inv_eq_ite",
+          "start": [
+            135,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases hx : x = 0\"",
+                "[]"
+              ],
+              "state_after": "case pos\nx : \u2102\nhx : x = 0\n\u22a2 log x\u207b\u00b9 = if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x\n\ncase neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 log x\u207b\u00b9 = if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x",
+              "state_before": "x : \u2102\n\u22a2 log x\u207b\u00b9 = if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x",
+              "tactic": "by_cases hx : x = 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>inv_def<\\/a>, <a>log_mul_ofReal<\\/a>, <a>Real.log_inv<\\/a>, <a>ofReal_neg<\\/a>, \\u2190 <a>sub_eq_neg_add<\\/a>, <a>log_conj_eq_ite<\\/a>]\"",
+                "[{\"full_name\":\"Complex.inv_def\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[814,9],\"def_end_pos\":[814,16]},{\"full_name\":\"Complex.log_mul_ofReal\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,23]},{\"full_name\":\"Real.log_inv\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,16]},{\"full_name\":\"Complex.ofReal_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,19]},{\"full_name\":\"sub_eq_neg_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[735,3],\"def_end_pos\":[735,14]},{\"full_name\":\"Complex.log_conj_eq_ite\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,24]}]"
+              ],
+              "state_after": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then log x else (starRingEnd \u2102) (log x)) - \u2191(Real.log (normSq x)) =\n    if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x\n\ncase neg.hr\nx : \u2102\nhx : \u00acx = 0\n\u22a2 0 < (normSq x)\u207b\u00b9\n\ncase neg.hx\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (starRingEnd \u2102) x \u2260 0",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 log x\u207b\u00b9 = if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x",
+              "tactic": "rw [inv_def, log_mul_ofReal, Real.log_inv, ofReal_neg, \u2190 sub_eq_neg_add, log_conj_eq_ite]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hx]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nx : \u2102\nhx : x = 0\n\u22a2 log x\u207b\u00b9 = if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x",
+              "tactic": "simp [hx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>log<\\/a>, <a>map_add<\\/a>, <a>map_mul<\\/a>, <a>conj_ofReal<\\/a>, <a>conj_I<\\/a>, <a>normSq_eq_abs<\\/a>, <a>Real.log_pow<\\/a>,\\n      <a>Nat.cast_two<\\/a>, <a>ofReal_mul<\\/a>, <a>neg_add<\\/a>, <a>mul_neg<\\/a>, <a>neg_neg<\\/a>]\"",
+                "[{\"full_name\":\"Complex.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[29,19],\"def_end_pos\":[29,22]},{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"Complex.conj_ofReal\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[572,9],\"def_end_pos\":[572,20]},{\"full_name\":\"Complex.conj_I\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[577,9],\"def_end_pos\":[577,15]},{\"full_name\":\"Complex.normSq_eq_abs\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[249,9],\"def_end_pos\":[249,22]},{\"full_name\":\"Real.log_pow\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[300,9],\"def_end_pos\":[300,16]},{\"full_name\":\"Nat.cast_two\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[208,9],\"def_end_pos\":[208,17]},{\"full_name\":\"Complex.ofReal_mul\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[257,9],\"def_end_pos\":[257,19]},{\"full_name\":\"neg_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[725,15],\"def_end_pos\":[725,22]},{\"full_name\":\"mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"neg_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[869,3],\"def_end_pos\":[869,14]}]"
+              ],
+              "state_after": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      \u21912 * \u2191(Real.log (abs x)) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then log x else (starRingEnd \u2102) (log x)) - \u2191(Real.log (normSq x)) =\n    if x.arg = \u03c0 then -(starRingEnd \u2102) (log x) else -log x",
+              "tactic": "simp_rw [log, map_add, map_mul, conj_ofReal, conj_I, normSq_eq_abs, Real.log_pow,\n  Nat.cast_two, ofReal_mul, neg_add, mul_neg, neg_neg]"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      2 * \u2191(Real.log (abs x)) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      \u21912 * \u2191(Real.log (abs x)) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>two_mul<\\/a>]\"",
+                "[{\"full_name\":\"two_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,16]}]"
+              ],
+              "state_after": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      2 * \u2191(Real.log (abs x)) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "tactic": "rw [two_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs\"",
+                "[]"
+              ],
+              "state_after": "case pos\nx : \u2102\nhx : \u00acx = 0\nh\u271d : x.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191x.arg * I - (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) = -\u2191(Real.log (abs x)) + \u2191x.arg * I\n\ncase neg\nx : \u2102\nhx : \u00acx = 0\nh\u271d : \u00acx.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + -(\u2191x.arg * I) - (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) =\n    -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (if x.arg = \u03c0 then \u2191(Real.log (abs x)) + \u2191x.arg * I else \u2191(Real.log (abs x)) + -(\u2191x.arg * I)) -\n      (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) =\n    if x.arg = \u03c0 then -\u2191(Real.log (abs x)) + \u2191x.arg * I else -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "tactic": "split_ifs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>add_sub_right_comm<\\/a>, <a>sub_add_cancel_left<\\/a>]\"",
+                "[{\"full_name\":\"add_sub_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[785,3],\"def_end_pos\":[785,14]},{\"full_name\":\"sub_add_cancel_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1336,3],\"def_end_pos\":[1336,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nx : \u2102\nhx : \u00acx = 0\nh\u271d : x.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + \u2191x.arg * I - (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) = -\u2191(Real.log (abs x)) + \u2191x.arg * I",
+              "tactic": "rw [add_sub_right_comm, sub_add_cancel_left]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>add_sub_right_comm<\\/a>, <a>sub_add_cancel_left<\\/a>]\"",
+                "[{\"full_name\":\"add_sub_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[785,3],\"def_end_pos\":[785,14]},{\"full_name\":\"sub_add_cancel_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1336,3],\"def_end_pos\":[1336,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\nx : \u2102\nhx : \u00acx = 0\nh\u271d : \u00acx.arg = \u03c0\n\u22a2 \u2191(Real.log (abs x)) + -(\u2191x.arg * I) - (\u2191(Real.log (abs x)) + \u2191(Real.log (abs x))) =\n    -\u2191(Real.log (abs x)) + -(\u2191x.arg * I)",
+              "tactic": "rw [add_sub_right_comm, sub_add_cancel_left]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>inv_pos<\\/a>, <a>Complex.normSq_pos<\\/a>]\"",
+                "[{\"full_name\":\"inv_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[47,15],\"def_end_pos\":[47,22]},{\"full_name\":\"Complex.normSq_pos\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[717,9],\"def_end_pos\":[717,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.hr\nx : \u2102\nhx : \u00acx = 0\n\u22a2 0 < (normSq x)\u207b\u00b9",
+              "tactic": "rwa [inv_pos, Complex.normSq_pos]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>map_ne_zero<\\/a>]\"",
+                "[{\"full_name\":\"map_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Lemmas.lean\",\"def_pos\":[39,9],\"def_end_pos\":[39,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.hx\nx : \u2102\nhx : \u00acx = 0\n\u22a2 (starRingEnd \u2102) x \u2260 0",
+              "tactic": "rwa [map_ne_zero]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.natCast_log",
+        "id": "Complex.natCast_log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            94
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Complex.natCast_log",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.ofNat_log",
+        "id": "Complex.ofNat_log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            14
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Complex.ofNat_log",
+          "start": [
+            78,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.clog",
+        "id": "Continuous.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            289,
+            69
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Continuous.clog",
+          "start": [
+            287,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.clog",
+        "id": "ContinuousAt.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            270,
+            13
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "ContinuousAt.clog",
+          "start": [
+            268,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.clog",
+        "id": "ContinuousOn.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            283,
+            27
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "ContinuousOn.clog",
+          "start": [
+            281,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.clog",
+        "id": "ContinuousWithinAt.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            277,
+            13
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "ContinuousWithinAt.clog",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.clog",
+        "id": "Filter.Tendsto.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            262,
+            40
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+          "full_name": "Filter.Tendsto.clog",
+          "start": [
+            260,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean"
+      }
+    ],
+    "input_proofs": 64,
+    "input_units": 2,
+    "proofs": 40,
+    "published_proofs": 36,
+    "published_transitions": 62,
+    "replay_failures": [],
+    "transitions": 86,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+        "proof_id": "mathlib:Complex.countable_preimage_exp",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              189,
+              31
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+            "full_name": "Complex.countable_preimage_exp",
+            "start": [
+              177,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun hs => ?_, fun hs => ?_\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s.Countable\n\ncase refine_2\ns : Set \u2102\nhs : s.Countable\n\u22a2 (cexp \u207b\u00b9' s).Countable",
+                "state_before": "s : Set \u2102\n\u22a2 (cexp \u207b\u00b9' s).Countable \u2194 s.Countable",
+                "tactic": "refine \u27e8fun hs => ?_, fun hs => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine ((hs.image <a>exp<\\/a>).<a>insert<\\/a> 0).<a>mono<\\/a> ?_\"",
+                  "[{\"full_name\":\"Complex.exp\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[55,5],\"def_end_pos\":[55,8]},{\"full_name\":\"Set.Countable.insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[268,19],\"def_end_pos\":[268,35]},{\"full_name\":\"Set.Countable.mono\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[121,9],\"def_end_pos\":[121,23]}]"
+                ],
+                "state_after": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s \u2286 insert 0 (cexp '' (cexp \u207b\u00b9' s))",
+                "state_before": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s.Countable",
+                "tactic": "refine ((hs.image exp).insert 0).mono ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Set.image_preimage_eq_inter_range<\\/a>, <a>range_exp<\\/a>, \\u2190 <a>Set.diff_eq<\\/a>, \\u2190 <a>Set.union_singleton<\\/a>,\\n        <a>Set.diff_union_self<\\/a>]\"",
+                  "[{\"full_name\":\"Set.image_preimage_eq_inter_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[786,9],\"def_end_pos\":[786,38]},{\"full_name\":\"Complex.range_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[53,9],\"def_end_pos\":[53,18]},{\"full_name\":\"Set.diff_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,16]},{\"full_name\":\"Set.union_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1304,9],\"def_end_pos\":[1304,24]},{\"full_name\":\"Set.diff_union_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1990,9],\"def_end_pos\":[1990,24]}]"
+                ],
+                "state_after": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s \u2286 s \u222a {0}",
+                "state_before": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s \u2286 insert 0 (cexp '' (cexp \u207b\u00b9' s))",
+                "tactic": "rw [Set.image_preimage_eq_inter_range, range_exp, \u2190 Set.diff_eq, \u2190 Set.union_singleton,\n    Set.diff_union_self]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Set.subset_union_left<\\/a>\"",
+                  "[{\"full_name\":\"Set.subset_union_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[796,9],\"def_end_pos\":[796,26]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\ns : Set \u2102\nhs : (cexp \u207b\u00b9' s).Countable\n\u22a2 s \u2286 s \u222a {0}",
+                "tactic": "exact Set.subset_union_left"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>Set.biUnion_preimage_singleton<\\/a>]\"",
+                  "[{\"full_name\":\"Set.biUnion_preimage_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1753,9],\"def_end_pos\":[1753,35]}]"
+                ],
+                "state_after": "case refine_2\ns : Set \u2102\nhs : s.Countable\n\u22a2 (\u22c3 y \u2208 s, cexp \u207b\u00b9' {y}).Countable",
+                "state_before": "case refine_2\ns : Set \u2102\nhs : s.Countable\n\u22a2 (cexp \u207b\u00b9' s).Countable",
+                "tactic": "rw [\u2190 Set.biUnion_preimage_singleton]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine hs.biUnion fun z hz => ?_\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "state_before": "case refine_2\ns : Set \u2102\nhs : s.Countable\n\u22a2 (\u22c3 y \u2208 s, cexp \u207b\u00b9' {y}).Countable",
+                "tactic": "refine hs.biUnion fun z hz => ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>em<\\/a> (\\u2203 w, <a>exp<\\/a> w = z) with (\\u27e8w, rfl\\u27e9 | hne)\"",
+                  "[{\"full_name\":\"em\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[195,7],\"def_end_pos\":[195,9]},{\"full_name\":\"Complex.exp\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[55,5],\"def_end_pos\":[55,8]}]"
+                ],
+                "state_after": "case refine_2.inl.intro\ns : Set \u2102\nhs : s.Countable\nw : \u2102\nhz : cexp w \u2208 s\n\u22a2 (cexp \u207b\u00b9' {cexp w}).Countable\n\ncase refine_2.inr\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\nhne : \u00ac\u2203 w, cexp w = z\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "state_before": "case refine_2\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "tactic": "rcases em (\u2203 w, exp w = z) with (\u27e8w, rfl\u27e9 | hne)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>Set.preimage<\\/a>, <a>Set.mem_singleton_iff<\\/a>, <a>exp_eq_exp_iff_exists_int<\\/a>, <a>Set.setOf_exists<\\/a>]\"",
+                  "[{\"full_name\":\"Set.preimage\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[113,5],\"def_end_pos\":[113,13]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]},{\"full_name\":\"Complex.exp_eq_exp_iff_exists_int\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[172,9],\"def_end_pos\":[172,34]},{\"full_name\":\"Set.setOf_exists\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,21]}]"
+                ],
+                "state_after": "case refine_2.inl.intro\ns : Set \u2102\nhs : s.Countable\nw : \u2102\nhz : cexp w \u2208 s\n\u22a2 (\u22c3 i, {x | x = w + \u2191i * (2 * \u2191\u03c0 * I)}).Countable",
+                "state_before": "case refine_2.inl.intro\ns : Set \u2102\nhs : s.Countable\nw : \u2102\nhz : cexp w \u2208 s\n\u22a2 (cexp \u207b\u00b9' {cexp w}).Countable",
+                "tactic": "simp only [Set.preimage, Set.mem_singleton_iff, exp_eq_exp_iff_exists_int, Set.setOf_exists]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Set.countable_iUnion<\\/a> fun m => <a>Set.countable_singleton<\\/a> _\"",
+                  "[{\"full_name\":\"Set.countable_iUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[222,9],\"def_end_pos\":[222,25]},{\"full_name\":\"Set.countable_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[167,17],\"def_end_pos\":[167,36]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2.inl.intro\ns : Set \u2102\nhs : s.Countable\nw : \u2102\nhz : cexp w \u2208 s\n\u22a2 (\u22c3 i, {x | x = w + \u2191i * (2 * \u2191\u03c0 * I)}).Countable",
+                "tactic": "exact Set.countable_iUnion fun m => Set.countable_singleton _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"push_neg at hne\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2.inr\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\nhne : \u2200 (w : \u2102), cexp w \u2260 z\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "state_before": "case refine_2.inr\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\nhne : \u00ac\u2203 w, cexp w = z\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "tactic": "push_neg at hne"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Set.preimage<\\/a>, hne]\"",
+                  "[{\"full_name\":\"Set.preimage\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[113,5],\"def_end_pos\":[113,13]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2.inr\ns : Set \u2102\nhs : s.Countable\nz : \u2102\nhz : z \u2208 s\nhne : \u2200 (w : \u2102), cexp w \u2260 z\n\u22a2 (cexp \u207b\u00b9' {z}).Countable",
+                "tactic": "simp [Set.preimage, hne]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Complex.countable_preimage_exp"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+        "proof_id": "mathlib:Complex.ofReal_log",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              72,
+              57
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+            "full_name": "Complex.ofReal_log",
+            "start": [
+              70,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>log_re<\\/a>, <a>ofReal_re<\\/a>, <a>abs_of_nonneg<\\/a> hx]\"",
+                  "[{\"full_name\":\"Complex.log_re\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[33,9],\"def_end_pos\":[33,15]},{\"full_name\":\"Complex.ofReal_re\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[96,9],\"def_end_pos\":[96,18]},{\"full_name\":\"Complex.abs_of_nonneg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Abs.lean\",\"def_pos\":[79,16],\"def_end_pos\":[79,29]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "x : \u211d\nhx : 0 \u2264 x\n\u22a2 (\u2191(Real.log x)).re = (log \u2191x).re",
+                "tactic": "rw [log_re, ofReal_re, abs_of_nonneg hx]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>ofReal_im<\\/a>, <a>log_im<\\/a>, <a>arg_ofReal_of_nonneg<\\/a> hx]\"",
+                  "[{\"full_name\":\"Complex.ofReal_im\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,18]},{\"full_name\":\"Complex.log_im\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Log.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,15]},{\"full_name\":\"Complex.arg_ofReal_of_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Arg.lean\",\"def_pos\":[233,9],\"def_end_pos\":[233,29]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "x : \u211d\nhx : 0 \u2264 x\n\u22a2 (\u2191(Real.log x)).im = (log \u2191x).im",
+                "tactic": "rw [ofReal_im, log_im, arg_ofReal_of_nonneg hx]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Complex.ofReal_log"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+        "proof_id": "mathlib:Complex.tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              205,
+              23
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+            "full_name": "Complex.tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero",
+            "start": [
+              195,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"convert\\n    (continuous_ofReal.continuousAt.comp_continuousWithinAt\\n            (continuous_abs.continuousWithinAt.log _)).tendsto.add\\n      (((continuous_ofReal.tendsto _).<a>comp<\\/a> <|\\n            <a>tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero<\\/a> hre him).<a>mul<\\/a>\\n        <a>tendsto_const_nhds<\\/a>) using 1\"",
+                  "[{\"full_name\":\"Filter.Tendsto.comp\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[3098,9],\"def_end_pos\":[3098,21]},{\"full_name\":\"Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Complex\\/Arg.lean\",\"def_pos\":[642,9],\"def_end_pos\":[642,59]},{\"full_name\":\"Filter.Tendsto.mul\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Monoid.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,27]},{\"full_name\":\"tendsto_const_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[991,9],\"def_end_pos\":[991,27]}]"
+                ],
+                "state_after": "case h.e'_5\nz : \u2102\nhre : z.re < 0\nhim : z.im = 0\n\u22a2 \ud835\udcdd (\u2191(Real.log (abs z)) - \u2191\u03c0 * I) = \ud835\udcdd ((ofReal' \u2218 fun x => Real.log (abs x)) z + \u2191(-\u03c0) * I)\n\ncase convert_1\nz : \u2102\nhre : z.re < 0\nhim : z.im = 0\n\u22a2 abs z \u2260 0",
+                "state_before": "z : \u2102\nhre : z.re < 0\nhim : z.im = 0\n\u22a2 Tendsto log (\ud835\udcdd[{z | z.im < 0}] z) (\ud835\udcdd (\u2191(Real.log (abs z)) - \u2191\u03c0 * I))",
+                "tactic": "convert\n  (continuous_ofReal.continuousAt.comp_continuousWithinAt\n          (continuous_abs.continuousWithinAt.log _)).tendsto.add\n    (((continuous_ofReal.tendsto _).comp <|\n          tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero hre him).mul\n      tendsto_const_nhds) using 1"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>sub_eq_add_neg<\\/a>]\"",
+                  "[{\"full_name\":\"sub_eq_add_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1057,3],\"def_end_pos\":[1057,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.e'_5\nz : \u2102\nhre : z.re < 0\nhim : z.im = 0\n\u22a2 \ud835\udcdd (\u2191(Real.log (abs z)) - \u2191\u03c0 * I) = \ud835\udcdd ((ofReal' \u2218 fun x => Real.log (abs x)) z + \u2191(-\u03c0) * I)",
+                "tactic": "simp [sub_eq_add_neg]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"lift z to \\u211d using him\"",
+                  "[]"
+                ],
+                "state_after": "case convert_1.intro\nz : \u211d\nhre : (\u2191z).re < 0\n\u22a2 abs \u2191z \u2260 0",
+                "state_before": "case convert_1\nz : \u2102\nhre : z.re < 0\nhim : z.im = 0\n\u22a2 abs z \u2260 0",
+                "tactic": "lift z to \u211d using him"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa using hre.ne\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case convert_1.intro\nz : \u211d\nhre : (\u2191z).re < 0\n\u22a2 abs \u2191z \u2260 0",
+                "tactic": "simpa using hre.ne"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Complex.tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+        "proof_id": "mathlib:continuousAt_clog",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              257,
+              29
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean",
+            "full_name": "continuousAt_clog",
+            "start": [
+              250,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine <a>ContinuousAt.add<\\/a> ?_ ?_\"",
+                  "[{\"full_name\":\"ContinuousAt.add\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Monoid.lean\",\"def_pos\":[244,3],\"def_end_pos\":[244,14]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => \u2191(Real.log (Complex.abs x))) x\n\ncase refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => \u2191x.arg * I) x",
+                "state_before": "\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt log x",
+                "tactic": "refine ContinuousAt.add ?_ ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine continuous_ofReal.continuousAt.comp ?_\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => Real.log (Complex.abs x)) x",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => \u2191(Real.log (Complex.abs x))) x",
+                "tactic": "refine continuous_ofReal.continuousAt.comp ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>Real.continuousAt_log<\\/a> ?_).<a>comp<\\/a> Complex.continuous_abs.continuousAt\"",
+                  "[{\"full_name\":\"Real.continuousAt_log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[373,9],\"def_end_pos\":[373,25]},{\"full_name\":\"ContinuousAt.comp\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1633,16],\"def_end_pos\":[1633,33]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 Complex.abs x \u2260 0",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => Real.log (Complex.abs x)) x",
+                "tactic": "refine (Real.continuousAt_log ?_).comp Complex.continuous_abs.continuousAt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact Complex.abs.ne_zero_iff.mpr <| <a>slitPlane_ne_zero<\\/a> h\"",
+                  "[{\"full_name\":\"Complex.slitPlane_ne_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Complex\\/Basic.lean\",\"def_pos\":[748,7],\"def_end_pos\":[748,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 Complex.abs x \u2260 0",
+                "tactic": "exact Complex.abs.ne_zero_iff.mpr <| slitPlane_ne_zero h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have h_cont_mul : <a>Continuous<\\/a> fun x : \\u2102 => x * <a>I<\\/a> := continuous_id'.mul <a>continuous_const<\\/a>\"",
+                  "[{\"full_name\":\"Continuous\",\"def_path\":\"Mathlib\\/Topology\\/Defs\\/Basic.lean\",\"def_pos\":[137,11],\"def_end_pos\":[137,21]},{\"full_name\":\"Complex.I\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[278,5],\"def_end_pos\":[278,6]},{\"full_name\":\"continuous_const\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1675,9],\"def_end_pos\":[1675,25]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => \u2191x.arg * I) x",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\n\u22a2 ContinuousAt (fun x => \u2191x.arg * I) x",
+                "tactic": "have h_cont_mul : Continuous fun x : \u2102 => x * I := continuous_id'.mul continuous_const"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine h_cont_mul.continuousAt.comp (continuous_ofReal.continuousAt.comp ?_)\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt arg x",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => \u2191x.arg * I) x",
+                "tactic": "refine h_cont_mul.continuousAt.comp (continuous_ofReal.continuousAt.comp ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>continuousAt_arg<\\/a> h\"",
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+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nx : \u2102\nh : x \u2208 slitPlane\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt arg x",
+                "tactic": "exact continuousAt_arg h"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "continuousAt_clog"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
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+  "proof_ids": [
+    "mathlib:AddCircle.homeomorphCircle_apply",
+    "mathlib:AddCircle.injective_toCircle",
+    "mathlib:AddCircle.toCircle_add",
+    "mathlib:Complex.continuousWithinAt_log_of_re_neg_of_im_zero",
+    "mathlib:Complex.exp_eq_exp_iff_exists_int",
+    "mathlib:Complex.exp_eq_exp_iff_exp_sub_eq_one",
+    "mathlib:Complex.exp_inj_of_neg_pi_lt_of_le_pi",
+    "mathlib:Complex.exp_log",
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+    "mathlib:Complex.log_conj",
+    "mathlib:Complex.log_exp",
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+    "mathlib:Complex.log_zero",
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+    "mathlib:Complex.map_exp_comap_re_atTop",
+    "mathlib:Complex.neg_pi_lt_log_im",
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+    "mathlib:Complex.tendsto_log_nhdsWithin_im_nonneg_of_re_neg_of_im_zero",
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+    "mathlib:Real.Angle.coe_expMapCircle",
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+    "mathlib:Real.Angle.expMapCircle_zero",
+    "mathlib:arg_expMapCircle",
+    "mathlib:expMapCircle_eq_expMapCircle",
+    "mathlib:periodic_expMapCircle"
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+  ]
+}

manifests/batches/numina/001320.json

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manifests/batches/numina/001321.json

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manifests/batches/numina/001322.json

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manifests/batches/numina/001323.json

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manifests/batches/numina/001324.json

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manifests/batches/numina/001325.json

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manifests/batches/numina/001326.json

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manifests/batches/numina/001327.json

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manifests/batches/numina/001328.json

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manifests/batches/numina/001329.json

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+      "bytes": 78897,
+      "rows": 5,
+      "revision": "25e226246b70de1e78b00c9d8e3db5bb5812aa98"
+    },
+    {
+      "path": "data/internal_test/numina-001329-0000-75ea463462fd9184.parquet",
+      "sha256": "75ea463462fd9184000982856726ef7ef0fbed3b55da3addb56554543fbffa46",
+      "bytes": 32252,
+      "rows": 3,
+      "revision": "25e226246b70de1e78b00c9d8e3db5bb5812aa98"
+    }
+  ],
+  "proof_ids": [
+    "numina:09c6790c-3df6-5951-963e-79424577e83d",
+    "numina:4a8cb5e0-dfc1-5d5e-81af-8f50a6091e79",
+    "numina:6c494065-1e2e-5397-b640-dbd0b6b2d432",
+    "numina:a3240c77-3300-578f-aca3-0333050ff0b9",
+    "numina:fd918a2a-a646-5ba8-8839-e4f97d484735"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006645.lean\",\"93323690472137fe733ce4d008ee9efaf581b2c65979dd907cd7c4a73ff2b18c\",0,\"number_theory_611156\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006646.lean\",\"8c51438a368b8c7b17e9d9c72aee67b7648cbb998cd4b4d4588a34848b468300\",1,\"number_theory_4966_2\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006647.lean\",\"20f6f45e1ccc827c2f9dfcd3fc4a67cd4c3f394eb4e94164b1a203c95f7a65f2\",0,\"algebra_170741\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006648.lean\",\"30a86c13c2519888f8b7cda19e39a120b9fd1d02941cb73a65d115a74ba7d02b\",0,\"algebra_606074\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006649.lean\",\"72161985bd128419ca5983461134947ba6590ade2b69f87cc54cc2e6f39612bf\",0,\"number_theory_96872\"]"
+  ]
+}