Record 32 verified HPC batch manifests

manifests/batches/mathlib/001925.json

@@ -0,0 +1,1360 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001925",
+  "batch_fingerprint": "a3d0ea8645ac63677f7e8c559015a3e58e38a96d87961a1e5055ad38a702813f",
+  "evidence": {
+    "batch_fingerprint": "a3d0ea8645ac63677f7e8c559015a3e58e38a96d87961a1e5055ad38a702813f",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProperSpace.of_isCompact_closedBall_of_le",
+        "id": "ProperSpace.of_isCompact_closedBall_of_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            55
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+          "full_name": "ProperSpace.of_isCompact_closedBall_of_le",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ProperSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProperSpace.of_seq_closedBall",
+        "id": "ProperSpace.of_seq_closedBall",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            78
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+          "full_name": "ProperSpace.of_seq_closedBall",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ProperSpace.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact \u27e8r / 2, \u27e8half_pos hr, half_lt_self hr\u27e9, empty_subset _\u27e9'",
+        "full_name": "exists_pos_lt_subset_ball",
+        "id": "exists_pos_lt_subset_ball",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            144,
+            89
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+          "full_name": "exists_pos_lt_subset_ball",
+          "start": [
+            134,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>eq_empty_or_nonempty<\\/a> s with (rfl | hne)\"",
+                "[{\"full_name\":\"Set.eq_empty_or_nonempty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[605,9],\"def_end_pos\":[605,29]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : 0 < r\nhs : IsClosed \u2205\nh : \u2205 \u2286 ball x r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, \u2205 \u2286 ball x r'\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "rcases eq_empty_or_nonempty s with (rfl | hne)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>IsCompact<\\/a> s :=\\n    (<a>isCompact_closedBall<\\/a> x r).<a>of_isClosed_subset<\\/a> hs (h.trans <a>ball_subset_closedBall<\\/a>)\"",
+                "[{\"full_name\":\"IsCompact\",\"def_path\":\"Mathlib\\/Topology\\/Defs\\/Filter.lean\",\"def_pos\":[273,5],\"def_end_pos\":[273,14]},{\"full_name\":\"ProperSpace.isCompact_closedBall\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/ProperSpace.lean\",\"def_pos\":[34,3],\"def_end_pos\":[34,23]},{\"full_name\":\"IsCompact.of_isClosed_subset\",\"def_path\":\"Mathlib\\/Topology\\/Compactness\\/Compact.lean\",\"def_pos\":[99,9],\"def_end_pos\":[99,37]},{\"full_name\":\"Metric.ball_subset_closedBall\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[524,9],\"def_end_pos\":[524,31]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "have : IsCompact s :=\n  (isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8y, hys, hy\\u27e9 : \\u2203 y \\u2208 s, s \\u2286 <a>closedBall<\\/a> x (<a>dist<\\/a> y x) :=\\n    this.exists_isMaxOn hne (continuous_id.dist <a>continuous_const<\\/a>).<a>continuousOn<\\/a>\"",
+                "[{\"full_name\":\"Metric.closedBall\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[464,5],\"def_end_pos\":[464,15]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"continuous_const\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1675,9],\"def_end_pos\":[1675,25]},{\"full_name\":\"Continuous.continuousOn\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[968,9],\"def_end_pos\":[968,32]}]"
+              ],
+              "state_after": "case inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "obtain \u27e8y, hys, hy\u27e9 : \u2203 y \u2208 s, s \u2286 closedBall x (dist y x) :=\n  this.exists_isMaxOn hne (continuous_id.dist continuous_const).continuousOn"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hyr : <a>dist<\\/a> y x < r := h hys\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]}]"
+              ],
+              "state_after": "case inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "state_before": "case inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "have hyr : dist y x < r := h hys"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>exists_between<\\/a> hyr with \\u27e8r', hyr', hrr'\\u27e9\"",
+                "[{\"full_name\":\"exists_between\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[1369,9],\"def_end_pos\":[1369,23]}]"
+              ],
+              "state_after": "case inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\nr' : \u211d\nhyr' : dist y x < r'\nhrr' : r' < r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "state_before": "case inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "rcases exists_between hyr with \u27e8r', hyr', hrr'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8r', \\u27e8dist_nonneg.trans_lt hyr', hrr'\\u27e9, hy.trans <| <a>closedBall_subset_ball<\\/a> hyr'\\u27e9\"",
+                "[{\"full_name\":\"Metric.closedBall_subset_ball\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[616,9],\"def_end_pos\":[616,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhr : 0 < r\nhs : IsClosed s\nh : s \u2286 ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : \u03b1\nhys : y \u2208 s\nhy : s \u2286 closedBall x (dist y x)\nhyr : dist y x < r\nr' : \u211d\nhyr' : dist y x < r'\nhrr' : r' < r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, s \u2286 ball x r'",
+              "tactic": "exact \u27e8r', \u27e8dist_nonneg.trans_lt hyr', hrr'\u27e9, hy.trans <| closedBall_subset_ball hyr'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8r \\/ 2, \\u27e8<a>half_pos<\\/a> hr, <a>half_lt_self<\\/a> hr\\u27e9, <a>empty_subset<\\/a> _\\u27e9\"",
+                "[{\"full_name\":\"half_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[445,9],\"def_end_pos\":[445,17]},{\"full_name\":\"half_lt_self\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[466,11],\"def_end_pos\":[466,23]},{\"full_name\":\"Set.empty_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[544,9],\"def_end_pos\":[544,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : 0 < r\nhs : IsClosed \u2205\nh : \u2205 \u2286 ball x r\n\u22a2 \u2203 r' \u2208 Ioo 0 r, \u2205 \u2286 ball x r'",
+              "tactic": "exact \u27e8r / 2, \u27e8half_pos hr, half_lt_self hr\u27e9, empty_subset _\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ProperSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isCompact_sphere",
+        "id": "isCompact_sphere",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            42,
+            89
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+          "full_name": "isCompact_sphere",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ProperSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.dist",
+        "id": "Continuous.dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            236,
+            43
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Continuous.dist",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.nndist",
+        "id": "Continuous.nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            268,
+            45
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Continuous.nndist",
+          "start": [
+            265,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.dist",
+        "id": "Filter.Tendsto.dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            242,
+            61
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Filter.Tendsto.dist",
+          "start": [
+            239,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.nndist",
+        "id": "Filter.Tendsto.nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            274,
+            63
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Filter.Tendsto.nndist",
+          "start": [
+            271,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulOpposite.dist_op",
+        "id": "MulOpposite.dist_op",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            75,
+            63
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "MulOpposite.dist_op",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulOpposite.dist_unop",
+        "id": "MulOpposite.dist_unop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            72
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "MulOpposite.dist_unop",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulOpposite.nndist_op",
+        "id": "MulOpposite.nndist_op",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            69
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "MulOpposite.nndist_op",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulOpposite.nndist_unop",
+        "id": "MulOpposite.nndist_unop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            78
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "MulOpposite.nndist_unop",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"convert NNReal.ball_zero_eq_Ico' \u27e8c, c_pos.le\u27e9\"",
+        "full_name": "NNReal.ball_zero_eq_Ico",
+        "id": "NNReal.ball_zero_eq_Ico",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            131,
+            25
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "NNReal.ball_zero_eq_Ico",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases c_pos : 0 < c\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : 0 < c\n\u22a2 ball 0 c = Ico 0 c.toNNReal\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : \u00ac0 < c\n\u22a2 ball 0 c = Ico 0 c.toNNReal",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\n\u22a2 ball 0 c = Ico 0 c.toNNReal",
+              "tactic": "by_cases c_pos : 0 < c"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [not_lt.mp c_pos]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : \u00ac0 < c\n\u22a2 ball 0 c = Ico 0 c.toNNReal",
+              "tactic": "simp [not_lt.mp c_pos]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>NNReal.ball_zero_eq_Ico'<\\/a> \\u27e8c, c_pos.le\\u27e9\"",
+                "[{\"full_name\":\"NNReal.ball_zero_eq_Ico'\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[123,7],\"def_end_pos\":[123,31]}]"
+              ],
+              "state_after": "case h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : 0 < c\n\u22a2 c.toNNReal = \u27e8c, \u22ef\u27e9",
+              "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : 0 < c\n\u22a2 ball 0 c = Ico 0 c.toNNReal",
+              "tactic": "convert NNReal.ball_zero_eq_Ico' \u27e8c, c_pos.le\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>Real.toNNReal<\\/a>, c_pos.le]\"",
+                "[{\"full_name\":\"Real.toNNReal\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[120,19],\"def_end_pos\":[120,39]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nc : \u211d\nc_pos : 0 < c\n\u22a2 c.toNNReal = \u27e8c, \u22ef\u27e9",
+              "tactic": "simp [Real.toNNReal, c_pos.le]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.dist_eq",
+        "id": "NNReal.dist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            67
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "NNReal.dist_eq",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]'",
+        "full_name": "NNReal.le_add_nndist",
+        "id": "NNReal.le_add_nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            38
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "NNReal.le_add_nndist",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices (a : \\u211d) \\u2264 (b : \\u211d) + <a>dist<\\/a> a b by\\n    rwa [\\u2190 <a>NNReal.coe_le_coe<\\/a>, <a>NNReal.coe_add<\\/a>, <a>coe_nndist<\\/a>]\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"NNReal.coe_le_coe\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[372,26],\"def_end_pos\":[372,36]},{\"full_name\":\"NNReal.coe_add\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[180,19],\"def_end_pos\":[180,26]},{\"full_name\":\"coe_nndist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[302,9],\"def_end_pos\":[302,19]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a \u2264 \u2191b + dist a b",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 a \u2264 b + nndist a b",
+              "tactic": "suffices (a : \u211d) \u2264 (b : \u211d) + dist a b by\n  rwa [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>sub_le_iff_le_add'<\\/a>]\"",
+                "[{\"full_name\":\"sub_le_iff_le_add'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[815,3],\"def_end_pos\":[815,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a - \u2191b \u2264 dist a b",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a \u2264 \u2191b + dist a b",
+              "tactic": "rw [\u2190 sub_le_iff_le_add']"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_of_abs_le<\\/a> (<a>dist_eq<\\/a> a b).<a>ge<\\/a>\"",
+                "[{\"full_name\":\"le_of_abs_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[392,9],\"def_end_pos\":[392,21]},{\"full_name\":\"NNReal.dist_eq\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[95,7],\"def_end_pos\":[95,21]},{\"full_name\":\"Eq.ge\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[218,19],\"def_end_pos\":[218,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\n\u22a2 \u2191a - \u2191b \u2264 dist a b",
+              "tactic": "exact le_of_abs_le (dist_eq a b).ge"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>NNReal.coe_le_coe<\\/a>, <a>NNReal.coe_add<\\/a>, <a>coe_nndist<\\/a>]\"",
+                "[{\"full_name\":\"NNReal.coe_le_coe\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[372,26],\"def_end_pos\":[372,36]},{\"full_name\":\"NNReal.coe_add\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[180,19],\"def_end_pos\":[180,26]},{\"full_name\":\"coe_nndist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[302,9],\"def_end_pos\":[302,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\na b : \u211d\u22650\nthis : \u2191a \u2264 \u2191b + dist a b\n\u22a2 a \u2264 b + nndist a b",
+              "tactic": "rwa [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_nndist]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Prod.dist_eq",
+        "id": "Prod.dist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            87
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Prod.dist_eq",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Subtype.dist_eq",
+        "id": "Subtype.dist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            90
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Subtype.dist_eq",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Subtype.nndist_eq",
+        "id": "Subtype.nndist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            96
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "Subtype.nndist_eq",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ULift.dist_eq",
+        "id": "ULift.dist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            148,
+            69
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "ULift.dist_eq",
+          "start": [
+            148,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ULift.dist_up_up",
+        "id": "ULift.dist_up_up",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            154,
+            86
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "ULift.dist_up_up",
+          "start": [
+            154,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ULift.nndist_eq",
+        "id": "ULift.nndist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            75
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "ULift.nndist_eq",
+          "start": [
+            151,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ULift.nndist_up_up",
+        "id": "ULift.nndist_up_up",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            157,
+            92
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "ULift.nndist_up_up",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UniformContinuous.dist",
+        "id": "UniformContinuous.dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            226,
+            46
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "UniformContinuous.dist",
+          "start": [
+            224,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UniformContinuous.nndist",
+        "id": "UniformContinuous.nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            258,
+            48
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "UniformContinuous.nndist",
+          "start": [
+            256,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_dist",
+        "id": "continuous_dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            230,
+            101
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "continuous_dist",
+          "start": [
+            229,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_iff_continuous_dist",
+        "id": "continuous_iff_continuous_dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            249,
+            85
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "continuous_iff_continuous_dist",
+          "start": [
+            245,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_nndist",
+        "id": "continuous_nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            262,
+            38
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "continuous_nndist",
+          "start": [
+            261,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_pi_const_le",
+        "id": "dist_pi_const_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            355,
+            49
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "dist_pi_const_le",
+          "start": [
+            354,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_pi_def",
+        "id": "dist_pi_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            306,
+            100
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "dist_pi_def",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_pi_const",
+        "id": "nndist_pi_const",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            369,
+            33
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "nndist_pi_const",
+          "start": [
+            367,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_pi_const_le",
+        "id": "nndist_pi_const_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            359,
+            37
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "nndist_pi_const_le",
+          "start": [
+            358,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_pi_def",
+        "id": "nndist_pi_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            303,
+            96
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "nndist_pi_def",
+          "start": [
+            303,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rintro rfl'",
+        "full_name": "sphere_prod",
+        "id": "sphere_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            209,
+            30
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "sphere_prod",
+          "start": [
+            199,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain hr | rfl | hr := <a>lt_trichotomy<\\/a> r 0\"",
+                "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : r < 0\n\u22a2 sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r\n\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 sphere x 0 = sphere x.1 0 \u00d7\u02e2 closedBall x.2 0 \u222a closedBall x.1 0 \u00d7\u02e2 sphere x.2 0\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\n\u22a2 sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\n\u22a2 sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "tactic": "obtain hr | rfl | hr := lt_trichotomy r 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hr]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : r < 0\n\u22a2 sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "tactic": "simp [hr]"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases x\"",
+                "[]"
+              ],
+              "state_after": "case inr.inl.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 sphere (fst\u271d, snd\u271d) 0 =\n    sphere (fst\u271d, snd\u271d).1 0 \u00d7\u02e2 closedBall (fst\u271d, snd\u271d).2 0 \u222a closedBall (fst\u271d, snd\u271d).1 0 \u00d7\u02e2 sphere (fst\u271d, snd\u271d).2 0",
+              "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 sphere x 0 = sphere x.1 0 \u00d7\u02e2 closedBall x.2 0 \u222a closedBall x.1 0 \u00d7\u02e2 sphere x.2 0",
+              "tactic": "cases x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [\\u2190 <a>closedBall_eq_sphere_of_nonpos<\\/a> <a>le_rfl<\\/a>, <a>union_self<\\/a>, <a>closedBall_prod_same<\\/a>]\"",
+                "[{\"full_name\":\"Metric.closedBall_eq_sphere_of_nonpos\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[520,9],\"def_end_pos\":[520,39]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"Set.union_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,19]},{\"full_name\":\"closedBall_prod_same\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[195,7],\"def_end_pos\":[195,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 sphere (fst\u271d, snd\u271d) 0 =\n    sphere (fst\u271d, snd\u271d).1 0 \u00d7\u02e2 closedBall (fst\u271d, snd\u271d).2 0 \u222a closedBall (fst\u271d, snd\u271d).1 0 \u00d7\u02e2 sphere (fst\u271d, snd\u271d).2 0",
+              "tactic": "simp_rw [\u2190 closedBall_eq_sphere_of_nonpos le_rfl, union_self, closedBall_prod_same]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext \\u27e8x', y'\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case inr.inr.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 (x', y') \u2208 sphere x r \u2194 (x', y') \u2208 sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "state_before": "case inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\n\u22a2 sphere x r = sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "tactic": "ext \u27e8x', y'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>Set.mem_union<\\/a>, <a>Set.mem_prod<\\/a>, <a>Metric.mem_closedBall<\\/a>, <a>Metric.mem_sphere<\\/a>, <a>Prod.dist_eq<\\/a>,\\n      <a>max_eq_iff<\\/a>]\"",
+                "[{\"full_name\":\"Set.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[734,9],\"def_end_pos\":[734,18]},{\"full_name\":\"Set.mem_prod\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,17]},{\"full_name\":\"Metric.mem_closedBall\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[468,17],\"def_end_pos\":[468,31]},{\"full_name\":\"Metric.mem_sphere\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[478,17],\"def_end_pos\":[478,27]},{\"full_name\":\"Prod.dist_eq\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[178,7],\"def_end_pos\":[178,19]},{\"full_name\":\"max_eq_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,19]}]"
+              ],
+              "state_after": "case inr.inr.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.1 = r \u2227 dist y' x.2 \u2264 dist x' x.1 \u2228 dist y' x.2 = r \u2227 dist x' x.1 \u2264 dist y' x.2 \u2194\n    dist x' x.1 = r \u2227 dist y' x.2 \u2264 r \u2228 dist x' x.1 \u2264 r \u2227 dist y' x.2 = r",
+              "state_before": "case inr.inr.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 (x', y') \u2208 sphere x r \u2194 (x', y') \u2208 sphere x.1 r \u00d7\u02e2 closedBall x.2 r \u222a closedBall x.1 r \u00d7\u02e2 sphere x.2 r",
+              "tactic": "simp_rw [Set.mem_union, Set.mem_prod, Metric.mem_closedBall, Metric.mem_sphere, Prod.dist_eq,\n  max_eq_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>or_congr<\\/a> (<a>and_congr_right<\\/a> ?_) (and_comm.trans (<a>and_congr_left<\\/a> ?_))\"",
+                "[{\"full_name\":\"or_congr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[77,9],\"def_end_pos\":[77,17]},{\"full_name\":\"and_congr_right\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,24]},{\"full_name\":\"and_congr_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[145,9],\"def_end_pos\":[145,23]}]"
+              ],
+              "state_after": "case inr.inr.h.mk.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.1 = r \u2192 (dist y' x.2 \u2264 dist x' x.1 \u2194 dist y' x.2 \u2264 r)\n\ncase inr.inr.h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist y' x.2 = r \u2192 (dist x' x.1 \u2264 dist y' x.2 \u2194 dist x' x.1 \u2264 r)",
+              "state_before": "case inr.inr.h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.1 = r \u2227 dist y' x.2 \u2264 dist x' x.1 \u2228 dist y' x.2 = r \u2227 dist x' x.1 \u2264 dist y' x.2 \u2194\n    dist x' x.1 = r \u2227 dist y' x.2 \u2264 r \u2228 dist x' x.1 \u2264 r \u2227 dist y' x.2 = r",
+              "tactic": "refine or_congr (and_congr_right ?_) (and_comm.trans (and_congr_left ?_))"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals rintro rfl; rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr.h.mk.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist x' x.1 = r \u2192 (dist y' x.2 \u2264 dist x' x.1 \u2194 dist y' x.2 \u2264 r)\n\ncase inr.inr.h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist y' x.2 = r \u2192 (dist x' x.1 \u2264 dist y' x.2 \u2194 dist x' x.1 \u2264 r)",
+              "tactic": "all_goals rintro rfl; rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro rfl\"",
+                "[]"
+              ],
+              "state_after": "case inr.inr.h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nhr : 0 < dist y' x.2\n\u22a2 dist x' x.1 \u2264 dist y' x.2 \u2194 dist x' x.1 \u2264 dist y' x.2",
+              "state_before": "case inr.inr.h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nr : \u211d\nhr : 0 < r\nx' : \u03b1\ny' : \u03b2\n\u22a2 dist y' x.2 = r \u2192 (dist x' x.1 \u2264 dist y' x.2 \u2194 dist x' x.1 \u2264 r)",
+              "tactic": "rintro rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr.h.mk.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nx : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nhr : 0 < dist y' x.2\n\u22a2 dist x' x.1 \u2264 dist y' x.2 \u2194 dist x' x.1 \u2264 dist y' x.2",
+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "uniformContinuous_dist",
+        "id": "uniformContinuous_dist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            221,
+            29
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "uniformContinuous_dist",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "uniformContinuous_nndist",
+        "id": "uniformContinuous_nndist",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            253,
+            38
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+          "full_name": "uniformContinuous_nndist",
+          "start": [
+            252,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean"
+      }
+    ],
+    "input_proofs": 63,
+    "input_units": 2,
+    "proofs": 28,
+    "published_proofs": 25,
+    "published_transitions": 41,
+    "replay_failures": [],
+    "transitions": 61,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+        "proof_id": "mathlib:exists_lt_subset_ball",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              154,
+              81
+            ],
+            "file_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean",
+            "full_name": "exists_lt_subset_ball",
+            "start": [
+              149,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>le_or_lt<\\/a> r 0 with hr | hr\"",
+                  "[{\"full_name\":\"le_or_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[342,9],\"def_end_pos\":[342,17]}]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : r \u2264 0\n\u22a2 \u2203 r' < r, s \u2286 ball x r'\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : 0 < r\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "tactic": "rcases le_or_lt r 0 with hr | hr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>ball_eq_empty<\\/a>.2 hr, <a>subset_empty_iff<\\/a>] at h\"",
+                  "[{\"full_name\":\"Metric.ball_eq_empty\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[425,9],\"def_end_pos\":[425,22]},{\"full_name\":\"Set.subset_empty_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[548,9],\"def_end_pos\":[548,25]}]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s = \u2205\nhr : r \u2264 0\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : r \u2264 0\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "tactic": "rw [ball_eq_empty.2 hr, subset_empty_iff] at h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"subst s\"",
+                  "[]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : r \u2264 0\nhs : IsClosed \u2205\n\u22a2 \u2203 r' < r, \u2205 \u2286 ball x r'",
+                "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s = \u2205\nhr : r \u2264 0\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "tactic": "subst s"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>exists_lt<\\/a> r).<a>imp<\\/a> fun r' hr' => \\u27e8hr', <a>empty_subset<\\/a> _\\u27e9\"",
+                  "[{\"full_name\":\"NoMinOrder.exists_lt\",\"def_path\":\"Mathlib\\/Order\\/Max.lean\",\"def_pos\":[55,3],\"def_end_pos\":[55,12]},{\"full_name\":\"Exists.imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[193,9],\"def_end_pos\":[193,19]},{\"full_name\":\"Set.empty_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[544,9],\"def_end_pos\":[544,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\nhr : r \u2264 0\nhs : IsClosed \u2205\n\u22a2 \u2203 r' < r, \u2205 \u2286 ball x r'",
+                "tactic": "exact (exists_lt r).imp fun r' hr' => \u27e8hr', empty_subset _\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>exists_pos_lt_subset_ball<\\/a> hr hs h).<a>imp<\\/a> fun r' hr' => \\u27e8hr'.1.2, hr'.2\\u27e9\"",
+                  "[{\"full_name\":\"exists_pos_lt_subset_ball\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/ProperSpace.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,34]},{\"full_name\":\"Exists.imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[193,9],\"def_end_pos\":[193,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nh : s \u2286 ball x r\nhr : 0 < r\n\u22a2 \u2203 r' < r, s \u2286 ball x r'",
+                "tactic": "exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => \u27e8hr'.1.2, hr'.2\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "exists_lt_subset_ball"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+        "proof_id": "mathlib:dist_pi_le_iff'",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              351,
+              60
+            ],
+            "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+            "full_name": "dist_pi_le_iff'",
+            "start": [
+              346,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases hr : 0 \\u2264 r\"",
+                  "[]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r",
+                "tactic": "by_cases hr : 0 \u2264 r"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>dist_pi_le_iff<\\/a> hr\"",
+                  "[{\"full_name\":\"dist_pi_le_iff\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[334,7],\"def_end_pos\":[334,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r",
+                "tactic": "exact dist_pi_le_iff hr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>iff_of_false<\\/a> (fun h => hr <| dist_nonneg.trans h) fun h =>\\n      hr <| dist_nonneg.trans <| h <| <a>Classical.arbitrary<\\/a> _\"",
+                  "[{\"full_name\":\"iff_of_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1354,9],\"def_end_pos\":[1354,21]},{\"full_name\":\"Classical.arbitrary\",\"def_path\":\"Mathlib\\/Logic\\/Nonempty.lean\",\"def_pos\":[99,32],\"def_end_pos\":[99,51]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\ninst\u271d : Nonempty \u03b2\nf g : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nhr : \u00ac0 \u2264 r\n\u22a2 dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r",
+                "tactic": "exact iff_of_false (fun h => hr <| dist_nonneg.trans h) fun h =>\n  hr <| dist_nonneg.trans <| h <| Classical.arbitrary _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "dist_pi_le_iff'"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+        "proof_id": "mathlib:sphere_pi",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              426,
+              51
+            ],
+            "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean",
+            "full_name": "sphere_pi",
+            "start": [
+              411,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hr | rfl | hr := <a>lt_trichotomy<\\/a> r 0\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : r < 0\n\u22a2 sphere x r = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r\n\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0) \u2229 closedBall x 0\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\n\u22a2 sphere x r = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\n\u22a2 sphere x r = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "tactic": "obtain hr | rfl | hr := lt_trichotomy r 0"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [hr]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : r < 0\n\u22a2 sphere x r = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "tactic": "simp [hr]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>closedBall_eq_sphere_of_nonpos<\\/a> <a>le_rfl<\\/a>, <a>eq_comm<\\/a>, <a>Set.inter_eq_right<\\/a>]\"",
+                  "[{\"full_name\":\"Metric.closedBall_eq_sphere_of_nonpos\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[520,9],\"def_end_pos\":[520,39]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"eq_comm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,16]},{\"full_name\":\"Set.inter_eq_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[949,15],\"def_end_pos\":[949,29]}]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0) \u2229 closedBall x 0",
+                "tactic": "rw [closedBall_eq_sphere_of_nonpos le_rfl, eq_comm, Set.inter_eq_right]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"letI := h.resolve_left (<a>lt_irrefl<\\/a> _)\"",
+                  "[{\"full_name\":\"lt_irrefl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,18]}]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "tactic": "letI := h.resolve_left (lt_irrefl _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"inhabit \\u03b2\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "tactic": "inhabit \u03b2"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>subset_iUnion_of_subset<\\/a> <a>default<\\/a> ?_\"",
+                  "[{\"full_name\":\"Set.subset_iUnion_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,32]},{\"full_name\":\"Inhabited.default\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[697,3],\"def_end_pos\":[697,10]}]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 Function.eval default \u207b\u00b9' sphere (x default) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 \u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) 0",
+                "tactic": "refine subset_iUnion_of_subset default ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro x hx\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : x \u2208 sphere x\u271d 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\n\u22a2 sphere x 0 \u2286 Function.eval default \u207b\u00b9' sphere (x default) 0",
+                "tactic": "intro x hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"replace hx := hx.le\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : dist x x\u271d \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : x \u2208 sphere x\u271d 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "tactic": "replace hx := hx.le"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>dist_pi_le_iff<\\/a> <a>le_rfl<\\/a>] at hx\"",
+                  "[{\"full_name\":\"dist_pi_le_iff\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[334,7],\"def_end_pos\":[334,21]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+                ],
+                "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : \u2200 (b : \u03b2), dist (x b) (x\u271d b) \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : dist x x\u271d \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "tactic": "rw [dist_pi_le_iff le_rfl] at hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>le_antisymm<\\/a> (hx <a>default<\\/a>) <a>dist_nonneg<\\/a>\"",
+                  "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Inhabited.default\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[697,3],\"def_end_pos\":[697,10]},{\"full_name\":\"dist_nonneg\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\nh : 0 < 0 \u2228 Nonempty \u03b2\nthis : Nonempty \u03b2 := Or.resolve_left h (lt_irrefl 0)\ninhabited_h : Inhabited \u03b2\nx : (x : \u03b2) \u2192 \u03c0 x\nhx : \u2200 (b : \u03b2), dist (x b) (x\u271d b) \u2264 0\n\u22a2 x \u2208 Function.eval default \u207b\u00b9' sphere (x\u271d default) 0",
+                "tactic": "exact le_antisymm (hx default) dist_nonneg"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 x\u271d \u2208 sphere x r \u2194 x\u271d \u2208 (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "state_before": "case inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\n\u22a2 sphere x r = (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "tactic": "ext"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>dist_pi_eq_iff<\\/a> hr, <a>dist_pi_le_iff<\\/a> hr.le]\"",
+                  "[{\"full_name\":\"dist_pi_eq_iff\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[340,7],\"def_end_pos\":[340,21]},{\"full_name\":\"dist_pi_le_iff\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Constructions.lean\",\"def_pos\":[334,7],\"def_end_pos\":[334,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nx : (b : \u03b2) \u2192 \u03c0 b\nr : \u211d\nh : 0 < r \u2228 Nonempty \u03b2\nhr : 0 < r\nx\u271d : (b : \u03b2) \u2192 \u03c0 b\n\u22a2 x\u271d \u2208 sphere x r \u2194 x\u271d \u2208 (\u22c3 i, Function.eval i \u207b\u00b9' sphere (x i) r) \u2229 closedBall x r",
+                "tactic": "simp [dist_pi_eq_iff hr, dist_pi_le_iff hr.le]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "sphere_pi"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001925-0000-9fa36bac28e04902.parquet",
+      "sha256": "9fa36bac28e049029e9af0e77a7c48772c72e9d12556074b7a8678f9edf313ed",
+      "bytes": 140884,
+      "rows": 34,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/dev/mathlib-001925-0000-4a6ec6ec52c55571.parquet",
+      "sha256": "4a6ec6ec52c55571e4baf1c49d2eb245a23f7e0fb6225c603c766330f2d8c1d2",
+      "bytes": 48792,
+      "rows": 2,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/internal_test/mathlib-001925-0000-9da0632328c9fcd3.parquet",
+      "sha256": "9da0632328c9fcd31abae801b78f898f1840887c7e3bd2d8635ba6c3aae44c29",
+      "bytes": 65237,
+      "rows": 5,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Fin.dist_insertNth_insertNth",
+    "mathlib:Fin.nndist_insertNth_insertNth",
+    "mathlib:Metric.exists_isLocalMin_mem_ball",
+    "mathlib:NNReal.closedBall_zero_eq_Icc",
+    "mathlib:NNReal.closedBall_zero_eq_Icc'",
+    "mathlib:NNReal.nndist_eq",
+    "mathlib:NNReal.nndist_zero_eq_val",
+    "mathlib:NNReal.nndist_zero_eq_val'",
+    "mathlib:ball_pi",
+    "mathlib:ball_pi'",
+    "mathlib:ball_prod_same",
+    "mathlib:closedBall_pi",
+    "mathlib:closedBall_pi'",
+    "mathlib:closedBall_prod_same",
+    "mathlib:dist_le_pi_dist",
+    "mathlib:dist_pi_const",
+    "mathlib:dist_pi_eq_iff",
+    "mathlib:dist_pi_le_iff",
+    "mathlib:dist_pi_lt_iff",
+    "mathlib:dist_prod_same_left",
+    "mathlib:dist_prod_same_right",
+    "mathlib:nndist_le_pi_nndist",
+    "mathlib:nndist_pi_eq_iff",
+    "mathlib:nndist_pi_le_iff",
+    "mathlib:nndist_pi_lt_iff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/ProperSpace.lean\",null,null,\"Metric.exists_isLocalMin_mem_ball\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"Fin.dist_insertNth_insertNth\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"Fin.nndist_insertNth_insertNth\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"NNReal.closedBall_zero_eq_Icc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"NNReal.closedBall_zero_eq_Icc'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"NNReal.nndist_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"NNReal.nndist_zero_eq_val\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"NNReal.nndist_zero_eq_val'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"ball_pi\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"ball_pi'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"ball_prod_same\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"closedBall_pi\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"closedBall_pi'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"closedBall_prod_same\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_le_pi_dist\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_pi_const\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_pi_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_pi_le_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_pi_lt_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_prod_same_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"dist_prod_same_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"nndist_le_pi_nndist\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"nndist_pi_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"nndist_pi_le_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean\",null,null,\"nndist_pi_lt_iff\"]"
+  ]
+}

manifests/batches/mathlib/001927.json

@@ -0,0 +1,172 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001927",
+  "batch_fingerprint": "fd42fef7eb0a6f1d81606938f19c0eeefbdc9d5a85d307e48eeb2710f994fbfa",
+  "evidence": {
+    "batch_fingerprint": "fd42fef7eb0a6f1d81606938f19c0eeefbdc9d5a85d307e48eeb2710f994fbfa",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "SeqCompact.lebesgue_number_lemma_of_metric",
+        "id": "SeqCompact.lebesgue_number_lemma_of_metric",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            24,
+            55
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Sequences.lean",
+          "full_name": "SeqCompact.lebesgue_number_lemma_of_metric",
+          "start": [
+            21,
+            8
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Sequences.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_subseq_of_bounded",
+        "id": "tendsto_subseq_of_bounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            69
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Sequences.lean",
+          "full_name": "tendsto_subseq_of_bounded",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Sequences.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_subseq_of_frequently_bounded",
+        "id": "tendsto_subseq_of_frequently_bounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            34
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/Sequences.lean",
+          "full_name": "tendsto_subseq_of_frequently_bounded",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/Sequences.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "exists_iUnion_ball_eq_radius_lt",
+        "id": "exists_iUnion_ball_eq_radius_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            33
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean",
+          "full_name": "exists_iUnion_ball_eq_radius_lt",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "exists_iUnion_ball_eq_radius_pos_lt",
+        "id": "exists_iUnion_ball_eq_radius_pos_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            33
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean",
+          "full_name": "exists_iUnion_ball_eq_radius_pos_lt",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "exists_locallyFinite_iUnion_eq_ball_radius_lt",
+        "id": "exists_locallyFinite_iUnion_eq_ball_radius_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            66
+          ],
+          "file_path": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean",
+          "full_name": "exists_locallyFinite_iUnion_eq_ball_radius_lt",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean"
+      }
+    ],
+    "input_proofs": 9,
+    "input_units": 2,
+    "proofs": 3,
+    "published_proofs": 3,
+    "published_transitions": 12,
+    "replay_failures": [],
+    "transitions": 12,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001927-0000-6c28d5f0a3848367.parquet",
+      "sha256": "6c28d5f0a38483679489cf5085e3d16b7c49d62c832cdaa3e9a922fd3505de51",
+      "bytes": 81483,
+      "rows": 12,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:exists_locallyFinite_subset_iUnion_ball_radius_lt",
+    "mathlib:exists_subset_iUnion_ball_radius_lt",
+    "mathlib:exists_subset_iUnion_ball_radius_pos_lt"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/ShrinkingLemma.lean\",null,null,\"exists_locallyFinite_subset_iUnion_ball_radius_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/ShrinkingLemma.lean\",null,null,\"exists_subset_iUnion_ball_radius_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/MetricSpace/ShrinkingLemma.lean\",null,null,\"exists_subset_iUnion_ball_radius_pos_lt\"]"
+  ]
+}

manifests/batches/mathlib/001931.json

@@ -0,0 +1,799 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001931",
+  "batch_fingerprint": "126dfaa81561dab1b673357c45cbc26449d315549bef790bc9963d3a51b938d0",
+  "evidence": {
+    "batch_fingerprint": "126dfaa81561dab1b673357c45cbc26449d315549bef790bc9963d3a51b938d0",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Inducing.noetherianSpace",
+        "id": "Inducing.noetherianSpace",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            92
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "Inducing.noetherianSpace",
+          "start": [
+            74,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.NoetherianSpace.discrete",
+        "id": "TopologicalSpace.NoetherianSpace.discrete",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            91
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.NoetherianSpace.discrete",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.NoetherianSpace.finite",
+        "id": "TopologicalSpace.NoetherianSpace.finite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            80
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.NoetherianSpace.finite",
+          "start": [
+            166,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.NoetherianSpace.range",
+        "id": "TopologicalSpace.NoetherianSpace.range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            136,
+            30
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.NoetherianSpace.range",
+          "start": [
+            133,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.NoetherianSpace.wellFounded_closeds",
+        "id": "TopologicalSpace.NoetherianSpace.wellFounded_closeds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            48
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.NoetherianSpace.wellFounded_closeds",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'refine (noetherianSpace_iff \u03b1).trans (Opens.compl_bijective.2.wellFounded_iff ?_)'",
+        "full_name": "TopologicalSpace.noetherianSpace_TFAE",
+        "id": "TopologicalSpace.noetherianSpace_TFAE",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            14
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.noetherianSpace_TFAE",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"tfae_have 1 \\u2194 2\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_iff_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "tactic": "tfae_have 1 \u2194 2"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_have 1 \\u2194 4\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_iff_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\n\u22a2 NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "tactic": "tfae_have 1 \u2194 4"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_have 1 \\u2192 3\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_to_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "tactic": "tfae_have 1 \u2192 3"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_have 3 \\u2192 4\"",
+                "[]"
+              ],
+              "state_after": "case tfae_3_to_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\n\u22a2 (\u2200 (s : Set \u03b1), IsCompact s) \u2192 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\ntfae_3_to_4 : (\u2200 (s : Set \u03b1), IsCompact s) \u2192 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "tactic": "tfae_have 3 \u2192 4"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_finish\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\ntfae_3_to_4 : (\u2200 (s : Set \u03b1), IsCompact s) \u2192 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 [NoetherianSpace \u03b1, WellFounded fun s t => s < t, \u2200 (s : Set \u03b1), IsCompact s, \u2200 (s : Opens \u03b1), IsCompact \u2191s].TFAE",
+              "tactic": "tfae_finish"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (<a>noetherianSpace_iff<\\/a> \\u03b1).<a>trans<\\/a> (<a>Opens.compl_bijective<\\/a>.2.<a>wellFounded_iff<\\/a> ?_)\"",
+                "[{\"full_name\":\"TopologicalSpace.noetherianSpace_iff\",\"def_path\":\"Mathlib\\/Topology\\/NoetherianSpace.lean\",\"def_pos\":[48,3],\"def_end_pos\":[48,9]},{\"full_name\":\"Iff.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[803,9],\"def_end_pos\":[803,18]},{\"full_name\":\"TopologicalSpace.Opens.compl_bijective\",\"def_path\":\"Mathlib\\/Topology\\/Sets\\/Closeds.lean\",\"def_pos\":[226,9],\"def_end_pos\":[226,30]},{\"full_name\":\"Function.Surjective.wellFounded_iff\",\"def_path\":\"Mathlib\\/Order\\/RelIso\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,44]}]"
+              ],
+              "state_after": "case tfae_1_iff_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 {a b : Opens \u03b1}, a > b \u2194 a.compl < b.compl",
+              "state_before": "case tfae_1_iff_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t",
+              "tactic": "refine (noetherianSpace_iff \u03b1).trans (Opens.compl_bijective.2.wellFounded_iff ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (@<a>OrderIso.compl<\\/a> (<a>Set<\\/a> \\u03b1)).lt_iff_lt.symm\"",
+                "[{\"full_name\":\"OrderIso.compl\",\"def_path\":\"Mathlib\\/Order\\/Hom\\/Set.lean\",\"def_pos\":[147,5],\"def_end_pos\":[147,19]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_1_iff_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u22a2 \u2200 {a b : Opens \u03b1}, a > b \u2194 a.compl < b.compl",
+              "tactic": "exact (@OrderIso.compl (Set \u03b1)).lt_iff_lt.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>noetherianSpace_iff_opens<\\/a> \\u03b1\"",
+                "[{\"full_name\":\"TopologicalSpace.noetherianSpace_iff_opens\",\"def_path\":\"Mathlib\\/Topology\\/NoetherianSpace.lean\",\"def_pos\":[53,9],\"def_end_pos\":[53,34]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_1_iff_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\n\u22a2 NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s",
+              "tactic": "exact noetherianSpace_iff_opens \u03b1"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact @<a>NoetherianSpace.isCompact<\\/a> \\u03b1 _\"",
+                "[{\"full_name\":\"TopologicalSpace.NoetherianSpace.isCompact\",\"def_path\":\"Mathlib\\/Topology\\/NoetherianSpace.lean\",\"def_pos\":[66,19],\"def_end_pos\":[66,44]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_1_to_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\n\u22a2 NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s",
+              "tactic": "exact @NoetherianSpace.isCompact \u03b1 _"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun h s => h s\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_3_to_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ntfae_1_iff_2 : NoetherianSpace \u03b1 \u2194 WellFounded fun s t => s < t\ntfae_1_iff_4 : NoetherianSpace \u03b1 \u2194 \u2200 (s : Opens \u03b1), IsCompact \u2191s\ntfae_1_to_3 : NoetherianSpace \u03b1 \u2192 \u2200 (s : Set \u03b1), IsCompact s\n\u22a2 (\u2200 (s : Set \u03b1), IsCompact s) \u2192 \u2200 (s : Opens \u03b1), IsCompact \u2191s",
+              "tactic": "exact fun h s => h s"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.noetherianSpace_iff_isCompact",
+        "id": "TopologicalSpace.noetherianSpace_iff_isCompact",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            35
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.noetherianSpace_iff_isCompact",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.noetherianSpace_iff_of_homeomorph",
+        "id": "TopologicalSpace.noetherianSpace_iff_of_homeomorph",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            130,
+            87
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.noetherianSpace_iff_of_homeomorph",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.noetherianSpace_of_surjective",
+        "id": "TopologicalSpace.noetherianSpace_of_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            43
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.noetherianSpace_of_surjective",
+          "start": [
+            122,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "TopologicalSpace.noetherian_univ_iff",
+        "id": "TopologicalSpace.noetherian_univ_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            147,
+            60
+          ],
+          "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+          "full_name": "TopologicalSpace.noetherian_univ_iff",
+          "start": [
+            146,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/NoetherianSpace.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Scott.IsOpen.inter",
+        "id": "Scott.IsOpen.inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            34
+          ],
+          "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+          "full_name": "Scott.IsOpen.inter",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/OmegaCompletePartialOrder.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Scott.IsOpen.isUpperSet",
+        "id": "Scott.IsOpen.isUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            81
+          ],
+          "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+          "full_name": "Scott.IsOpen.isUpperSet",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/OmegaCompletePartialOrder.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Scott.isOpen_univ",
+        "id": "Scott.isOpen_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            73
+          ],
+          "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+          "full_name": "Scott.isOpen_univ",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/OmegaCompletePartialOrder.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have hf : IsUpperSet {x | \u00acf x \u2264 f y} := ((notBelow_isOpen (f y)).preimage hf).isUpperSet'",
+        "full_name": "scottContinuous_of_continuous",
+        "id": "scottContinuous_of_continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            129,
+            8
+          ],
+          "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+          "full_name": "scottContinuous_of_continuous",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have h : <a>Monotone<\\/a> f := fun x y h \\u21a6 by\\n    have hf : <a>IsUpperSet<\\/a> {x | \\u00acf x \\u2264 f y} := ((<a>notBelow_isOpen<\\/a> (f y)).<a>preimage<\\/a> hf).<a>isUpperSet<\\/a>\\n    simpa only [<a>mem_setOf_eq<\\/a>, <a>le_refl<\\/a>, <a>not_true<\\/a>, <a>imp_false<\\/a>, <a>not_not<\\/a>] using hf h\"",
+                "[{\"full_name\":\"Monotone\",\"def_path\":\"Mathlib\\/Order\\/Monotone\\/Basic.lean\",\"def_pos\":[77,5],\"def_end_pos\":[77,13]},{\"full_name\":\"IsUpperSet\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[61,5],\"def_end_pos\":[61,15]},{\"full_name\":\"notBelow_isOpen\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[97,9],\"def_end_pos\":[97,24]},{\"full_name\":\"IsOpen.preimage\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1546,9],\"def_end_pos\":[1546,24]},{\"full_name\":\"Scott.IsOpen.isUpperSet\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[69,9],\"def_end_pos\":[69,26]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"le_refl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,16]},{\"full_name\":\"not_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1369,9],\"def_end_pos\":[1369,17]},{\"full_name\":\"imp_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1420,17],\"def_end_pos\":[1420,26]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\nh : Monotone f\n\u22a2 Continuous' f",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\n\u22a2 Continuous' f",
+              "tactic": "have h : Monotone f := fun x y h \u21a6 by\n  have hf : IsUpperSet {x | \u00acf x \u2264 f y} := ((notBelow_isOpen (f y)).preimage hf).isUpperSet\n  simpa only [mem_setOf_eq, le_refl, not_true, imp_false, not_not] using hf h"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8h, fun c \\u21a6 <a>eq_of_forall_ge_iff<\\/a> fun z \\u21a6 ?_\\u27e9\"",
+                "[{\"full_name\":\"eq_of_forall_ge_iff\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[543,9],\"def_end_pos\":[543,28]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\nh : Monotone f\n\u22a2 Continuous' f",
+              "tactic": "refine \u27e8h, fun c \u21a6 eq_of_forall_ge_iff fun z \u21a6 ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases (<a>notBelow_isOpen<\\/a> z).<a>preimage<\\/a> hf with \\u27e8hf, hf'\\u27e9\"",
+                "[{\"full_name\":\"notBelow_isOpen\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[97,9],\"def_end_pos\":[97,24]},{\"full_name\":\"IsOpen.preimage\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1546,9],\"def_end_pos\":[1546,24]}]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : OmegaCompletePartialOrder.Continuous { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf }\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "tactic": "rcases (notBelow_isOpen z).preimage hf with \u27e8hf, hf'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"specialize hf' c\"",
+                "[]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' :\n  { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf } (\u03c9Sup c) =\n    \u03c9Sup (c.map { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf })\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : OmegaCompletePartialOrder.Continuous { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf }\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "tactic": "specialize hf' c"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>OrderHom.coe_mk<\\/a>, <a>mem_preimage<\\/a>, <a>notBelow<\\/a>, <a>mem_setOf_eq<\\/a>] at hf'\"",
+                "[{\"full_name\":\"OrderHom.coe_mk\",\"def_path\":\"Mathlib\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[225,17],\"def_end_pos\":[225,23]},{\"full_name\":\"Set.mem_preimage\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,21]},{\"full_name\":\"notBelow\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[93,5],\"def_end_pos\":[93,13]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' :\n  { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf } (\u03c9Sup c) =\n    \u03c9Sup (c.map { toFun := fun x => x \u2208 f \u207b\u00b9' notBelow z, monotone' := hf })\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "tactic": "simp only [OrderHom.coe_mk, mem_preimage, notBelow, mem_setOf_eq] at hf'"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>not_iff_not<\\/a>]\"",
+                "[{\"full_name\":\"not_iff_not\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[447,9],\"def_end_pos\":[447,20]}]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 \u00ac{ toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u00ac\u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 { toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "tactic": "rw [\u2190 not_iff_not]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>\\u03c9Sup_le_iff<\\/a>, hf', <a>\\u03c9Sup<\\/a>, <a>iSup<\\/a>, <a>sSup<\\/a>, <a>mem_range<\\/a>, <a>Chain.map_coe<\\/a>, <a>Function.comp_apply<\\/a>,\\n    <a>eq_iff_iff<\\/a>, <a>not_forall<\\/a>, <a>OrderHom.coe_mk<\\/a>]\"",
+                "[{\"full_name\":\"OmegaCompletePartialOrder.\\u03c9Sup_le_iff\",\"def_path\":\"Mathlib\\/Order\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,20]},{\"full_name\":\"OmegaCompletePartialOrder.\\u03c9Sup\",\"def_path\":\"Mathlib\\/Order\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[186,3],\"def_end_pos\":[186,7]},{\"full_name\":\"iSup\",\"def_path\":\"Mathlib\\/Order\\/SetNotation.lean\",\"def_pos\":[60,5],\"def_end_pos\":[60,9]},{\"full_name\":\"SupSet.sSup\",\"def_path\":\"Mathlib\\/Order\\/SetNotation.lean\",\"def_pos\":[42,3],\"def_end_pos\":[42,7]},{\"full_name\":\"Set.mem_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[160,17],\"def_end_pos\":[160,26]},{\"full_name\":\"OmegaCompletePartialOrder.Chain.map_coe\",\"def_path\":\"Mathlib\\/Order\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[124,17],\"def_end_pos\":[124,24]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]},{\"full_name\":\"eq_iff_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1378,17],\"def_end_pos\":[1378,27]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]},{\"full_name\":\"OrderHom.coe_mk\",\"def_path\":\"Mathlib\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[225,17],\"def_end_pos\":[225,23]}]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 (\u2203 a, (\u2203 y, \u00acf (c y) \u2264 z \u2194 a) \u2227 a) \u2194 \u2203 x, \u00acf (c x) \u2264 z",
+              "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 \u00ac{ toFun := f, monotone' := h } (\u03c9Sup c) \u2264 z \u2194 \u00ac\u03c9Sup (c.map { toFun := f, monotone' := h }) \u2264 z",
+              "tactic": "simp only [\u03c9Sup_le_iff, hf', \u03c9Sup, iSup, sSup, mem_range, Chain.map_coe, Function.comp_apply,\n  eq_iff_iff, not_forall, OrderHom.coe_mk]"
+            },
+            {
+              "annotated_tactic": [
+                "\"tauto\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nh : Monotone f\nc : Chain (Scott \u03b1)\nz : Scott \u03b2\nhf : Monotone fun x => x \u2208 f \u207b\u00b9' notBelow z\nhf' : (\u00acf (\u03c9Sup c) \u2264 z) = \u03c9Sup (c.map { toFun := fun x => \u00acf x \u2264 z, monotone' := \u22ef })\n\u22a2 (\u2203 a, (\u2203 y, \u00acf (c y) \u2264 z \u2194 a) \u2227 a) \u2194 \u2203 x, \u00acf (c x) \u2264 z",
+              "tactic": "tauto"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hf : <a>IsUpperSet<\\/a> {x | \\u00acf x \\u2264 f y} := ((<a>notBelow_isOpen<\\/a> (f y)).<a>preimage<\\/a> hf).<a>isUpperSet<\\/a>\"",
+                "[{\"full_name\":\"IsUpperSet\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[61,5],\"def_end_pos\":[61,15]},{\"full_name\":\"notBelow_isOpen\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[97,9],\"def_end_pos\":[97,24]},{\"full_name\":\"IsOpen.preimage\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1546,9],\"def_end_pos\":[1546,24]},{\"full_name\":\"Scott.IsOpen.isUpperSet\",\"def_path\":\"Mathlib\\/Topology\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[69,9],\"def_end_pos\":[69,26]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nx y : Scott \u03b1\nh : x \u2264 y\nhf : IsUpperSet {x | \u00acf x \u2264 f y}\n\u22a2 f x \u2264 f y",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf : _root_.Continuous f\nx y : Scott \u03b1\nh : x \u2264 y\n\u22a2 f x \u2264 f y",
+              "tactic": "have hf : IsUpperSet {x | \u00acf x \u2264 f y} := ((notBelow_isOpen (f y)).preimage hf).isUpperSet"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>mem_setOf_eq<\\/a>, <a>le_refl<\\/a>, <a>not_true<\\/a>, <a>imp_false<\\/a>, <a>not_not<\\/a>] using hf h\"",
+                "[{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"le_refl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,16]},{\"full_name\":\"not_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1369,9],\"def_end_pos\":[1369,17]},{\"full_name\":\"imp_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1420,17],\"def_end_pos\":[1420,26]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : OmegaCompletePartialOrder \u03b2\nf : Scott \u03b1 \u2192 Scott \u03b2\nhf\u271d : _root_.Continuous f\nx y : Scott \u03b1\nh : x \u2264 y\nhf : IsUpperSet {x | \u00acf x \u2264 f y}\n\u22a2 f x \u2264 f y",
+              "tactic": "simpa only [mem_setOf_eq, le_refl, not_true, imp_false, not_not] using hf h"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/OmegaCompletePartialOrder.lean"
+      }
+    ],
+    "input_proofs": 27,
+    "input_units": 2,
+    "proofs": 13,
+    "published_proofs": 11,
+    "published_transitions": 35,
+    "replay_failures": [],
+    "transitions": 56,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+        "proof_id": "mathlib:TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              191,
+              64
+            ],
+            "file_path": "Mathlib/Topology/NoetherianSpace.lean",
+            "full_name": "TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible",
+            "start": [
+              175,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"apply wellFounded_closeds.induction s\"",
+                  "[]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\n\u22a2 \u2200 (x : Closeds \u03b1),\n    (\u2200 y < x, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S) \u2192\n      \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 x = sSup S",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "apply wellFounded_closeds.induction s"
+              },
+              {
+                "annotated_tactic": [
+                  "\"clear s\"",
+                  "[]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\n\u22a2 \u2200 (x : Closeds \u03b1),\n    (\u2200 y < x, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S) \u2192\n      \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 x = sSup S",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\n\u22a2 \u2200 (x : Closeds \u03b1),\n    (\u2200 y < x, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S) \u2192\n      \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 x = sSup S",
+                "tactic": "clear s"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro s H\"",
+                  "[]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\n\u22a2 \u2200 (x : Closeds \u03b1),\n    (\u2200 y < x, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S) \u2192\n      \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 x = sSup S",
+                "tactic": "intro s H"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>eq_or_ne<\\/a> s \\u22a5 with rfl | h\\u2080\"",
+                  "[{\"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\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nH : \u2200 y < \u22a5, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 \u22a5 = sSup S\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "rcases eq_or_ne s \u22a5 with rfl | h\u2080"
+              },
+              {
+                "annotated_tactic": [
+                  "\"use \\u2205\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nH : \u2200 y < \u22a5, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2205.Finite \u2227 (\u2200 t \u2208 \u2205, IsIrreducible \u2191t) \u2227 \u22a5 = sSup \u2205",
+                "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nH : \u2200 y < \u22a5, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 \u22a5 = sSup S",
+                "tactic": "use \u2205"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\nH : \u2200 y < \u22a5, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\n\u22a2 \u2205.Finite \u2227 (\u2200 t \u2208 \u2205, IsIrreducible \u2191t) \u2227 \u22a5 = sSup \u2205",
+                "tactic": "simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"by_cases h\\u2081 : <a>IsPreirreducible<\\/a> (s : <a>Set<\\/a> \\u03b1)\"",
+                  "[{\"full_name\":\"IsPreirreducible\",\"def_path\":\"Mathlib\\/Topology\\/Irreducible.lean\",\"def_pos\":[37,5],\"def_end_pos\":[37,21]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsPreirreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : \u00acIsPreirreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "by_cases h\u2081 : IsPreirreducible (s : Set \u03b1)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"replace h\\u2081 : <a>IsIrreducible<\\/a> (s : <a>Set<\\/a> \\u03b1) := \\u27e8<a>Closeds.coe_nonempty<\\/a>.2 h\\u2080, h\\u2081\\u27e9\"",
+                  "[{\"full_name\":\"IsIrreducible\",\"def_path\":\"Mathlib\\/Topology\\/Irreducible.lean\",\"def_pos\":[43,5],\"def_end_pos\":[43,18]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"TopologicalSpace.Closeds.coe_nonempty\",\"def_path\":\"Mathlib\\/Topology\\/Sets\\/Closeds.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,21]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsIrreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsPreirreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "replace h\u2081 : IsIrreducible (s : Set \u03b1) := \u27e8Closeds.coe_nonempty.2 h\u2080, h\u2081\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"use {s}\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsIrreducible \u2191s\n\u22a2 {s}.Finite \u2227 (\u2200 t \u2208 {s}, IsIrreducible \u2191t) \u2227 s = sSup {s}",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsIrreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "use {s}"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [h\\u2081]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : IsIrreducible \u2191s\n\u22a2 {s}.Finite \u2227 (\u2200 t \u2208 {s}, IsIrreducible \u2191t) \u2227 s = sSup {s}",
+                "tactic": "simp [h\u2081]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>isPreirreducible_iff_closed_union_closed<\\/a>, <a>not_forall<\\/a>, <a>not_or<\\/a>] at h\\u2081\"",
+                  "[{\"full_name\":\"isPreirreducible_iff_closed_union_closed\",\"def_path\":\"Mathlib\\/Topology\\/Irreducible.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,49]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]},{\"full_name\":\"not_or\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[134,17],\"def_end_pos\":[134,23]}]"
+                ],
+                "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : \u2203 x x_1, \u2203 (_ : IsClosed x) (_ : IsClosed x_1) (_ : \u2191s \u2286 x \u222a x_1), \u00ac\u2191s \u2286 x \u2227 \u00ac\u2191s \u2286 x_1\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : \u00acIsPreirreducible \u2191s\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "simp only [isPreirreducible_iff_closed_union_closed, not_forall, not_or] at h\u2081"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8z\\u2081, z\\u2082, hz\\u2081, hz\\u2082, h, hz\\u2081', hz\\u2082'\\u27e9 := h\\u2081\"",
+                  "[]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 z\u2082 : Set \u03b1\nhz\u2081 : IsClosed z\u2081\nhz\u2082 : IsClosed z\u2082\nh : \u2191s \u2286 z\u2081 \u222a z\u2082\nhz\u2081' : \u00ac\u2191s \u2286 z\u2081\nhz\u2082' : \u00ac\u2191s \u2286 z\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nh\u2081 : \u2203 x x_1, \u2203 (_ : IsClosed x) (_ : IsClosed x_1) (_ : \u2191s \u2286 x \u222a x_1), \u00ac\u2191s \u2286 x \u2227 \u00ac\u2191s \u2286 x_1\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "obtain \u27e8z\u2081, z\u2082, hz\u2081, hz\u2082, h, hz\u2081', hz\u2082'\u27e9 := h\u2081"
+              },
+              {
+                "annotated_tactic": [
+                  "\"lift z\\u2081 to <a>Closeds<\\/a> \\u03b1 using hz\\u2081\"",
+                  "[{\"full_name\":\"TopologicalSpace.Closeds\",\"def_path\":\"Mathlib\\/Topology\\/Sets\\/Closeds.lean\",\"def_pos\":[33,11],\"def_end_pos\":[33,18]}]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2082 : Set \u03b1\nhz\u2082 : IsClosed z\u2082\nhz\u2082' : \u00ac\u2191s \u2286 z\u2082\nz\u2081 : Closeds \u03b1\nh : \u2191s \u2286 \u2191z\u2081 \u222a z\u2082\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 z\u2082 : Set \u03b1\nhz\u2081 : IsClosed z\u2081\nhz\u2082 : IsClosed z\u2082\nh : \u2191s \u2286 z\u2081 \u222a z\u2082\nhz\u2081' : \u00ac\u2191s \u2286 z\u2081\nhz\u2082' : \u00ac\u2191s \u2286 z\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "lift z\u2081 to Closeds \u03b1 using hz\u2081"
+              },
+              {
+                "annotated_tactic": [
+                  "\"lift z\\u2082 to <a>Closeds<\\/a> \\u03b1 using hz\\u2082\"",
+                  "[{\"full_name\":\"TopologicalSpace.Closeds\",\"def_path\":\"Mathlib\\/Topology\\/Sets\\/Closeds.lean\",\"def_pos\":[33,11],\"def_end_pos\":[33,18]}]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2082 : Set \u03b1\nhz\u2082 : IsClosed z\u2082\nhz\u2082' : \u00ac\u2191s \u2286 z\u2082\nz\u2081 : Closeds \u03b1\nh : \u2191s \u2286 \u2191z\u2081 \u222a z\u2082\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "lift z\u2082 to Closeds \u03b1 using hz\u2082"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases H (s \\u2293 z\\u2081) (<a>inf_lt_left<\\/a>.2 hz\\u2081') with \\u27e8S\\u2081, hSf\\u2081, hS\\u2081, h\\u2081\\u27e9\"",
+                  "[{\"full_name\":\"inf_lt_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[428,9],\"def_end_pos\":[428,20]}]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "rcases H (s \u2293 z\u2081) (inf_lt_left.2 hz\u2081') with \u27e8S\u2081, hSf\u2081, hS\u2081, h\u2081\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases H (s \\u2293 z\\u2082) (<a>inf_lt_left<\\/a>.2 hz\\u2082') with \\u27e8S\\u2082, hSf\\u2082, hS\\u2082, h\\u2082\\u27e9\"",
+                  "[{\"full_name\":\"inf_lt_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[428,9],\"def_end_pos\":[428,20]}]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\nS\u2082 : Set (Closeds \u03b1)\nhSf\u2082 : S\u2082.Finite\nhS\u2082 : \u2200 t \u2208 S\u2082, IsIrreducible \u2191t\nh\u2082 : s \u2293 z\u2082 = sSup S\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "rcases H (s \u2293 z\u2082) (inf_lt_left.2 hz\u2082') with \u27e8S\u2082, hSf\u2082, hS\u2082, h\u2082\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8S\\u2081 \\u222a S\\u2082, hSf\\u2081.union hSf\\u2082, <a>Set.union_subset_iff<\\/a>.2 \\u27e8hS\\u2081, hS\\u2082\\u27e9, ?_\\u27e9\"",
+                  "[{\"full_name\":\"Set.union_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[808,9],\"def_end_pos\":[808,25]}]"
+                ],
+                "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\nS\u2082 : Set (Closeds \u03b1)\nhSf\u2082 : S\u2082.Finite\nhS\u2082 : \u2200 t \u2208 S\u2082, IsIrreducible \u2191t\nh\u2082 : s \u2293 z\u2082 = sSup S\u2082\n\u22a2 s = sSup (S\u2081 \u222a S\u2082)",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\nS\u2082 : Set (Closeds \u03b1)\nhSf\u2082 : S\u2082.Finite\nhS\u2082 : \u2200 t \u2208 S\u2082, IsIrreducible \u2191t\nh\u2082 : s \u2293 z\u2082 = sSup S\u2082\n\u22a2 \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 s = sSup S",
+                "tactic": "refine \u27e8S\u2081 \u222a S\u2082, hSf\u2081.union hSf\u2082, Set.union_subset_iff.2 \u27e8hS\u2081, hS\u2082\u27e9, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>sSup_union<\\/a>, \\u2190 h\\u2081, \\u2190 h\\u2082, \\u2190 <a>inf_sup_left<\\/a>, <a>left_eq_inf<\\/a>]\"",
+                  "[{\"full_name\":\"sSup_union\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[402,9],\"def_end_pos\":[402,19]},{\"full_name\":\"inf_sup_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[726,9],\"def_end_pos\":[726,21]},{\"full_name\":\"left_eq_inf\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[409,9],\"def_end_pos\":[409,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : NoetherianSpace \u03b1\ns : Closeds \u03b1\nH : \u2200 y < s, \u2203 S, S.Finite \u2227 (\u2200 t \u2208 S, IsIrreducible \u2191t) \u2227 y = sSup S\nh\u2080 : s \u2260 \u22a5\nz\u2081 : Closeds \u03b1\nhz\u2081' : \u00ac\u2191s \u2286 \u2191z\u2081\nz\u2082 : Closeds \u03b1\nhz\u2082' : \u00ac\u2191s \u2286 \u2191z\u2082\nh : \u2191s \u2286 \u2191z\u2081 \u222a \u2191z\u2082\nS\u2081 : Set (Closeds \u03b1)\nhSf\u2081 : S\u2081.Finite\nhS\u2081 : \u2200 t \u2208 S\u2081, IsIrreducible \u2191t\nh\u2081 : s \u2293 z\u2081 = sSup S\u2081\nS\u2082 : Set (Closeds \u03b1)\nhSf\u2082 : S\u2082.Finite\nhS\u2082 : \u2200 t \u2208 S\u2082, IsIrreducible \u2191t\nh\u2082 : s \u2293 z\u2082 = sSup S\u2082\n\u22a2 s = sSup (S\u2081 \u222a S\u2082)",
+                "tactic": "rwa [sSup_union, \u2190 h\u2081, \u2190 h\u2082, \u2190 inf_sup_left, left_eq_inf]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "TopologicalSpace.NoetherianSpace.exists_finite_set_closeds_irreducible"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+        "proof_id": "mathlib:is\u03c9Sup_\u03c9Sup",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              113,
+              18
+            ],
+            "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean",
+            "full_name": "is\u03c9Sup_\u03c9Sup",
+            "start": [
+              110,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case left\n\u03b1 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\n\u22a2 \u2200 (i : \u2115), c i \u2264 \u03c9Sup c\n\ncase right\n\u03b1 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\n\u22a2 \u2200 (y : \u03b1), (\u2200 (i : \u2115), c i \u2264 y) \u2192 \u03c9Sup c \u2264 y",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\n\u22a2 Is\u03c9Sup c (\u03c9Sup c)",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>le_\\u03c9Sup<\\/a>\"",
+                  "[{\"full_name\":\"OmegaCompletePartialOrder.le_\\u03c9Sup\",\"def_path\":\"Mathlib\\/Order\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[188,3],\"def_end_pos\":[188,10]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case left\n\u03b1 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\n\u22a2 \u2200 (i : \u2115), c i \u2264 \u03c9Sup c",
+                "tactic": "apply le_\u03c9Sup"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>\\u03c9Sup_le<\\/a>\"",
+                  "[{\"full_name\":\"OmegaCompletePartialOrder.\\u03c9Sup_le\",\"def_path\":\"Mathlib\\/Order\\/OmegaCompletePartialOrder.lean\",\"def_pos\":[190,3],\"def_end_pos\":[190,10]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case right\n\u03b1 : Type u_1\ninst\u271d : OmegaCompletePartialOrder \u03b1\nc : Chain \u03b1\n\u22a2 \u2200 (y : \u03b1), (\u2200 (i : \u2115), c i \u2264 y) \u2192 \u03c9Sup c \u2264 y",
+                "tactic": "apply \u03c9Sup_le"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "is\u03c9Sup_\u03c9Sup"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001931-0000-a1afc16f3d2a7418.parquet",
+      "sha256": "a1afc16f3d2a7418cb94b92e9d29a15a252cd2238641c0d5903d6ba0d1573548",
+      "bytes": 108654,
+      "rows": 32,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/internal_test/mathlib-001931-0000-e66148aff12a13ef.parquet",
+      "sha256": "e66148aff12a13ef85417760b6eec364d978f68000e4b014bd507b4105734010",
+      "bytes": 62239,
+      "rows": 3,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Scott.isOpen_sUnion",
+    "mathlib:Scott.is\u03c9Sup_iff_isLUB",
+    "mathlib:TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible",
+    "mathlib:TopologicalSpace.NoetherianSpace.exists_finset_irreducible",
+    "mathlib:TopologicalSpace.NoetherianSpace.finite_irreducibleComponents",
+    "mathlib:TopologicalSpace.NoetherianSpace.iUnion",
+    "mathlib:TopologicalSpace.NoetherianSpace.isCompact",
+    "mathlib:TopologicalSpace.noetherianSpace_iff_opens",
+    "mathlib:TopologicalSpace.noetherianSpace_set_iff",
+    "mathlib:continuous_of_scottContinuous",
+    "mathlib:notBelow_isOpen"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.NoetherianSpace.exists_finite_set_isClosed_irreducible\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.NoetherianSpace.exists_finset_irreducible\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.NoetherianSpace.finite_irreducibleComponents\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.NoetherianSpace.iUnion\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.NoetherianSpace.isCompact\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.noetherianSpace_iff_opens\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/NoetherianSpace.lean\",null,null,\"TopologicalSpace.noetherianSpace_set_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/OmegaCompletePartialOrder.lean\",null,null,\"Scott.isOpen_sUnion\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/OmegaCompletePartialOrder.lean\",null,null,\"Scott.is\u03c9Sup_iff_isLUB\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/OmegaCompletePartialOrder.lean\",null,null,\"continuous_of_scottContinuous\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/OmegaCompletePartialOrder.lean\",null,null,\"notBelow_isOpen\"]"
+  ]
+}

manifests/batches/mathlib/001933.json

@@ -0,0 +1,360 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001933",
+  "batch_fingerprint": "c944ddea3eb9976a4a8c8059f8d7b14c8f094a99c75b17765086dcb6e1577ab9",
+  "evidence": {
+    "batch_fingerprint": "c944ddea3eb9976a4a8c8059f8d7b14c8f094a99c75b17765086dcb6e1577ab9",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "BddAbove.isBounded",
+        "id": "BddAbove.isBounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            47
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "BddAbove.isBounded",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "BddAbove.isBounded_inter",
+        "id": "BddAbove.isBounded_inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            69
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "BddAbove.isBounded_inter",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "BddBelow.isBounded",
+        "id": "BddBelow.isBounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            47
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "BddBelow.isBounded",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "BddBelow.isBounded_inter",
+        "id": "BddBelow.isBounded_inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            69
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "BddBelow.isBounded_inter",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Bornology.IsBounded.bddAbove",
+        "id": "Bornology.IsBounded.bddAbove",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            43
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "Bornology.IsBounded.bddAbove",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Bornology.IsBounded.bddBelow",
+        "id": "Bornology.IsBounded.bddBelow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            43
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "Bornology.IsBounded.bddBelow",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Bornology.IsBounded.subset_Icc_sInf_sSup",
+        "id": "Bornology.IsBounded.subset_Icc_sInf_sSup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            108,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "Bornology.IsBounded.subset_Icc_sInf_sSup",
+          "start": [
+            107,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isBounded_iff_bddBelow_bddAbove",
+        "id": "isBounded_iff_bddBelow_bddAbove",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            53
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "isBounded_iff_bddBelow_bddAbove",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h, orderBornology_isBounded]'",
+        "full_name": "isOrderBornology_iff_eq_orderBornology",
+        "id": "isOrderBornology_iff_eq_orderBornology",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            51,
+            47
+          ],
+          "file_path": "Mathlib/Topology/Order/Bornology.lean",
+          "full_name": "isOrderBornology_iff_eq_orderBornology",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun h \\u21a6 ?_, fun h \\u21a6 \\u27e8fun s \\u21a6 by rw [h, <a>orderBornology_isBounded<\\/a>]\\u27e9\\u27e9\"",
+                "[{\"full_name\":\"orderBornology_isBounded\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Bornology.lean\",\"def_pos\":[36,15],\"def_end_pos\":[36,39]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\nh : IsOrderBornology \u03b1\n\u22a2 inst\u271d\u00b2 = orderBornology",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 IsOrderBornology \u03b1 \u2194 inst\u271d\u00b2 = orderBornology",
+              "tactic": "refine \u27e8fun h \u21a6 ?_, fun h \u21a6 \u27e8fun s \u21a6 by rw [h, orderBornology_isBounded]\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext s\"",
+                "[]"
+              ],
+              "state_after": "case h_cobounded.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\nh : IsOrderBornology \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 cobounded \u03b1 \u2194 s \u2208 cobounded \u03b1",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\nh : IsOrderBornology \u03b1\n\u22a2 inst\u271d\u00b2 = orderBornology",
+              "tactic": "ext s"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact isBounded_compl_iff.symm.trans (h.1 _)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_cobounded.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\nh : IsOrderBornology \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 cobounded \u03b1 \u2194 s \u2208 cobounded \u03b1",
+              "tactic": "exact isBounded_compl_iff.symm.trans (h.1 _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h, <a>orderBornology_isBounded<\\/a>]\"",
+                "[{\"full_name\":\"orderBornology_isBounded\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Bornology.lean\",\"def_pos\":[36,15],\"def_end_pos\":[36,39]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\nh : inst\u271d\u00b2 = orderBornology\ns : Set \u03b1\n\u22a2 IsBounded s \u2194 BddBelow s \u2227 BddAbove s",
+              "tactic": "rw [h, orderBornology_isBounded]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bornology.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa [mem_lowerBounds] using hc'",
+        "full_name": "Filter.isBounded_ge_map_of_bounded_range",
+        "id": "Filter.isBounded_ge_map_of_bounded_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            38,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/Bounded.lean",
+          "full_name": "Filter.isBounded_ge_map_of_bounded_range",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8c, hc\\u27e9 := h.bddBelow\"",
+                "[]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 lowerBounds (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2265 x_1) (map f F)",
+              "state_before": "\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2265 x_1) (map f F)",
+              "tactic": "obtain \u27e8c, hc\u27e9 := h.bddBelow"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>isBoundedUnder_of<\\/a> \\u27e8c, by simpa [<a>mem_lowerBounds<\\/a>] using hc\\u27e9\"",
+                "[{\"full_name\":\"Filter.isBoundedUnder_of\",\"def_path\":\"Mathlib\\/Order\\/LiminfLimsup.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,26]},{\"full_name\":\"mem_lowerBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 lowerBounds (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2265 x_1) (map f F)",
+              "tactic": "apply isBoundedUnder_of \u27e8c, by simpa [mem_lowerBounds] using hc\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>mem_lowerBounds<\\/a>] using hc\"",
+                "[{\"full_name\":\"mem_lowerBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 lowerBounds (range f)\n\u22a2 \u2200 (x : \u03b9), f x \u2265 c",
+              "tactic": "simpa [mem_lowerBounds] using hc"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bounded.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa [mem_upperBounds] using hc'",
+        "full_name": "Filter.isBounded_le_map_of_bounded_range",
+        "id": "Filter.isBounded_le_map_of_bounded_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            31,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/Bounded.lean",
+          "full_name": "Filter.isBounded_le_map_of_bounded_range",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8c, hc\\u27e9 := h.bddAbove\"",
+                "[]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 upperBounds (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f F)",
+              "state_before": "\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f F)",
+              "tactic": "obtain \u27e8c, hc\u27e9 := h.bddAbove"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>isBoundedUnder_of<\\/a> \\u27e8c, by simpa [<a>mem_upperBounds<\\/a>] using hc\\u27e9\"",
+                "[{\"full_name\":\"Filter.isBoundedUnder_of\",\"def_path\":\"Mathlib\\/Order\\/LiminfLimsup.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,26]},{\"full_name\":\"mem_upperBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 upperBounds (range f)\n\u22a2 IsBounded (fun x x_1 => x \u2264 x_1) (map f F)",
+              "tactic": "exact isBoundedUnder_of \u27e8c, by simpa [mem_upperBounds] using hc\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>mem_upperBounds<\\/a>] using hc\"",
+                "[{\"full_name\":\"mem_upperBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b9 : Type u_1\nF : Filter \u03b9\nf : \u03b9 \u2192 \u211d\nh : Bornology.IsBounded (range f)\nc : \u211d\nhc : c \u2208 upperBounds (range f)\n\u22a2 \u2200 (x : \u03b9), f x \u2264 c",
+              "tactic": "simpa [mem_upperBounds] using hc"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Bounded.lean"
+      }
+    ],
+    "input_proofs": 11,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001934.json

@@ -0,0 +1,113 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001934",
+  "batch_fingerprint": "9d6082bae3b939fe9100e9f94f660ecb01012dada8db7aec5cdddde46f344d4d",
+  "evidence": {
+    "batch_fingerprint": "9d6082bae3b939fe9100e9f94f660ecb01012dada8db7aec5cdddde46f344d4d",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AlexDisc.coe_forgetToTop",
+        "id": "AlexDisc.coe_forgetToTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            39,
+            86
+          ],
+          "file_path": "Mathlib/Topology/Order/Category/AlexDisc.lean",
+          "full_name": "AlexDisc.coe_forgetToTop",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Category/AlexDisc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AlexDisc.coe_of",
+        "id": "AlexDisc.coe_of",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            98
+          ],
+          "file_path": "Mathlib/Topology/Order/Category/AlexDisc.lean",
+          "full_name": "AlexDisc.coe_of",
+          "start": [
+            44,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Category/AlexDisc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AlexDisc.forgetToTop_of",
+        "id": "AlexDisc.forgetToTop_of",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            60
+          ],
+          "file_path": "Mathlib/Topology/Order/Category/AlexDisc.lean",
+          "full_name": "AlexDisc.forgetToTop_of",
+          "start": [
+            45,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Category/AlexDisc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Locale.PT.isOpen_iff",
+        "id": "Locale.PT.isOpen_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            75,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/Category/FrameAdjunction.lean",
+          "full_name": "Locale.PT.isOpen_iff",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Category/FrameAdjunction.lean"
+      }
+    ],
+    "input_proofs": 4,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001936.json

@@ -0,0 +1,313 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001936",
+  "batch_fingerprint": "fa5fa70c43e5a9c5abb0f614608b3db99d513714e63484c11a1178383a69bf5b",
+  "evidence": {
+    "batch_fingerprint": "fa5fa70c43e5a9c5abb0f614608b3db99d513714e63484c11a1178383a69bf5b",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsExtrOn.closure",
+        "id": "IsExtrOn.closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            39,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+          "full_name": "IsExtrOn.closure",
+          "start": [
+            37,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ExtrClosure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalExtrOn.closure",
+        "id": "IsLocalExtrOn.closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+          "full_name": "IsLocalExtrOn.closure",
+          "start": [
+            59,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ExtrClosure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalMinOn.closure",
+        "id": "IsLocalMinOn.closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            33
+          ],
+          "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+          "full_name": "IsLocalMinOn.closure",
+          "start": [
+            54,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ExtrClosure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsMaxOn.closure",
+        "id": "IsMaxOn.closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            29,
+            94
+          ],
+          "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+          "full_name": "IsMaxOn.closure",
+          "start": [
+            27,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ExtrClosure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsMinOn.closure",
+        "id": "IsMinOn.closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            34,
+            20
+          ],
+          "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+          "full_name": "IsMinOn.closure",
+          "start": [
+            32,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ExtrClosure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.nhds_atBot",
+        "id": "Filter.Tendsto.nhds_atBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            40,
+            42
+          ],
+          "file_path": "Mathlib/Topology/Order/Filter.lean",
+          "full_name": "Filter.Tendsto.nhds_atBot",
+          "start": [
+            38,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Filter.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.nhds_atTop",
+        "id": "Filter.Tendsto.nhds_atTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            35,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Filter.lean",
+          "full_name": "Filter.Tendsto.nhds_atTop",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Filter.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.tendsto_nhds_atBot",
+        "id": "Filter.tendsto_nhds_atBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            30,
+            41
+          ],
+          "file_path": "Mathlib/Topology/Order/Filter.lean",
+          "full_name": "Filter.tendsto_nhds_atBot",
+          "start": [
+            29,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Filter.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.tendsto_nhds_atTop",
+        "id": "Filter.tendsto_nhds_atTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            93
+          ],
+          "file_path": "Mathlib/Topology/Order/Filter.lean",
+          "full_name": "Filter.tendsto_nhds_atTop",
+          "start": [
+            25,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Filter.lean"
+      }
+    ],
+    "input_proofs": 10,
+    "input_units": 2,
+    "proofs": 1,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 7,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+        "proof_id": "mathlib:IsLocalMaxOn.closure",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              51,
+              68
+            ],
+            "file_path": "Mathlib/Topology/Order/ExtrClosure.lean",
+            "full_name": "IsLocalMaxOn.closure",
+            "start": [
+              42,
+              11
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>mem_nhdsWithin<\\/a>.1 h with \\u27e8U, Uo, aU, hU\\u27e9\"",
+                  "[{\"full_name\":\"mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,23]}]"
+                ],
+                "state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\n\u22a2 IsLocalMaxOn f (closure s) a",
+                "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\n\u22a2 IsLocalMaxOn f (closure s) a",
+                "tactic": "rcases mem_nhdsWithin.1 h with \u27e8U, Uo, aU, hU\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>mem_nhdsWithin<\\/a>.2 \\u27e8U, Uo, aU, ?_\\u27e9\"",
+                  "[{\"full_name\":\"mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,23]}]"
+                ],
+                "state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\n\u22a2 U \u2229 closure s \u2286 {x | (fun x => f x \u2264 f a) x}",
+                "state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\n\u22a2 IsLocalMaxOn f (closure s) a",
+                "tactic": "refine mem_nhdsWithin.2 \u27e8U, Uo, aU, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro x \\u27e8hxU, hxs\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 x \u2208 {x | (fun x => f x \u2264 f a) x}",
+                "state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\n\u22a2 U \u2229 closure s \u2286 {x | (fun x => f x \u2264 f a) x}",
+                "tactic": "rintro x \u27e8hxU, hxs\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>ContinuousWithinAt.closure_le<\\/a> ?_ ?_ <a>continuousWithinAt_const<\\/a> hU\"",
+                  "[{\"full_name\":\"ContinuousWithinAt.closure_le\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[685,9],\"def_end_pos\":[685,38]},{\"full_name\":\"continuousWithinAt_const\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[1064,9],\"def_end_pos\":[1064,33]}]"
+                ],
+                "state_after": "case intro.intro.intro.intro.refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 x \u2208 closure (U \u2229 s)\n\ncase intro.intro.intro.intro.refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 ContinuousWithinAt f (U \u2229 s) x",
+                "state_before": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 x \u2208 {x | (fun x => f x \u2264 f a) x}",
+                "tactic": "refine ContinuousWithinAt.closure_le ?_ ?_ continuousWithinAt_const hU"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>mem_closure_iff_nhdsWithin_neBot<\\/a>, <a>nhdsWithin_inter_of_mem<\\/a>, \\u2190\\n      <a>mem_closure_iff_nhdsWithin_neBot<\\/a>]\"",
+                  "[{\"full_name\":\"mem_closure_iff_nhdsWithin_neBot\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1266,9],\"def_end_pos\":[1266,41]},{\"full_name\":\"nhdsWithin_inter_of_mem\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[262,9],\"def_end_pos\":[262,32]},{\"full_name\":\"mem_closure_iff_nhdsWithin_neBot\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1266,9],\"def_end_pos\":[1266,41]}]"
+                ],
+                "state_after": "case intro.intro.intro.intro.refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 U \u2208 \ud835\udcdd[s] x",
+                "state_before": "case intro.intro.intro.intro.refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 x \u2208 closure (U \u2229 s)",
+                "tactic": "rwa [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_inter_of_mem, \u2190\n  mem_closure_iff_nhdsWithin_neBot]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>nhdsWithin_le_nhds<\\/a> (Uo.mem_nhds hxU)\"",
+                  "[{\"full_name\":\"nhdsWithin_le_nhds\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[196,9],\"def_end_pos\":[196,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.intro.intro.intro.refine_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 U \u2208 \ud835\udcdd[s] x",
+                "tactic": "exact nhdsWithin_le_nhds (Uo.mem_nhds hxU)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (hc _ hxs).<a>mono<\\/a> (inter_subset_right.trans <a>subset_closure<\\/a>)\"",
+                  "[{\"full_name\":\"ContinuousWithinAt.mono\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[747,9],\"def_end_pos\":[747,32]},{\"full_name\":\"subset_closure\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[403,9],\"def_end_pos\":[403,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.intro.intro.intro.refine_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Preorder Y\ninst\u271d : OrderClosedTopology Y\nf g : X \u2192 Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a \u2208 U\nhU : U \u2229 s \u2286 {x | (fun x => f x \u2264 f a) x}\nx : X\nhxU : x \u2208 U\nhxs : x \u2208 closure s\n\u22a2 ContinuousWithinAt f (U \u2229 s) x",
+                "tactic": "exact (hc _ hxs).mono (inter_subset_right.trans subset_closure)"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "IsLocalMaxOn.closure"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001937.json

@@ -0,0 +1,1006 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001937",
+  "batch_fingerprint": "9f50e267bb2e30235b5596bcd6539c68ae78d14679063e3449b90b4218a487f5",
+  "evidence": {
+    "batch_fingerprint": "9f50e267bb2e30235b5596bcd6539c68ae78d14679063e3449b90b4218a487f5",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.cancel_right",
+        "id": "ContinuousOrderHom.cancel_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            192,
+            85
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.cancel_right",
+          "start": [
+            190,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.coe_comp",
+        "id": "ContinuousOrderHom.coe_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            165,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.coe_comp",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.coe_id",
+        "id": "ContinuousOrderHom.coe_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            148,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.coe_id",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.coe_toOrderHom",
+        "id": "ContinuousOrderHom.coe_toOrderHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.coe_toOrderHom",
+          "start": [
+            111,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.comp_apply",
+        "id": "ContinuousOrderHom.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.comp_apply",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.comp_assoc",
+        "id": "ContinuousOrderHom.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            176,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.comp_assoc",
+          "start": [
+            174,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.comp_id",
+        "id": "ContinuousOrderHom.comp_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            181,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.comp_id",
+          "start": [
+            180,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.copy_eq",
+        "id": "ContinuousOrderHom.copy_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.copy_eq",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.ext",
+        "id": "ContinuousOrderHom.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            21
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.ext",
+          "start": [
+            117,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.id_apply",
+        "id": "ContinuousOrderHom.id_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.id_apply",
+          "start": [
+            154,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.id_comp",
+        "id": "ContinuousOrderHom.id_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            186,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.id_comp",
+          "start": [
+            185,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOrderHom.toFun_eq_coe",
+        "id": "ContinuousOrderHom.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            113,
+            66
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Basic.lean",
+          "full_name": "ContinuousOrderHom.toFun_eq_coe",
+          "start": [
+            113,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.cancel_right",
+        "id": "EsakiaHom.cancel_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            356,
+            78
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.cancel_right",
+          "start": [
+            354,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_comp",
+        "id": "EsakiaHom.coe_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            331,
+            93
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_comp",
+          "start": [
+            331,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_comp_continuousOrderHom",
+        "id": "EsakiaHom.coe_comp_continuousOrderHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            322,
+            54
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_comp_continuousOrderHom",
+          "start": [
+            321,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_comp_pseudoEpimorphism",
+        "id": "EsakiaHom.coe_comp_pseudoEpimorphism",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            327,
+            83
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_comp_pseudoEpimorphism",
+          "start": [
+            326,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_copy",
+        "id": "EsakiaHom.coe_copy",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            276,
+            92
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_copy",
+          "start": [
+            276,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_id",
+        "id": "EsakiaHom.coe_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            294,
+            47
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_id",
+          "start": [
+            294,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_id_continuousOrderHom",
+        "id": "EsakiaHom.coe_id_continuousOrderHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            309,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_id_continuousOrderHom",
+          "start": [
+            309,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.coe_id_pseudoEpimorphism",
+        "id": "EsakiaHom.coe_id_pseudoEpimorphism",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            299,
+            77
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.coe_id_pseudoEpimorphism",
+          "start": [
+            298,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.comp_apply",
+        "id": "EsakiaHom.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            335,
+            99
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.comp_apply",
+          "start": [
+            335,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.comp_assoc",
+        "id": "EsakiaHom.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            340,
+            49
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.comp_assoc",
+          "start": [
+            339,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.comp_id",
+        "id": "EsakiaHom.comp_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            345,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.comp_id",
+          "start": [
+            344,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.copy_eq",
+        "id": "EsakiaHom.copy_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            280,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.copy_eq",
+          "start": [
+            279,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.ext",
+        "id": "EsakiaHom.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            265,
+            21
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.ext",
+          "start": [
+            264,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.id_apply",
+        "id": "EsakiaHom.id_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            305,
+            55
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.id_apply",
+          "start": [
+            305,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.id_comp",
+        "id": "EsakiaHom.id_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            350,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.id_comp",
+          "start": [
+            349,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.toContinuousOrderHom_coe",
+        "id": "EsakiaHom.toContinuousOrderHom_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            257,
+            48
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.toContinuousOrderHom_coe",
+          "start": [
+            256,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "EsakiaHom.toFun_eq_coe",
+        "id": "EsakiaHom.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            260,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "EsakiaHom.toFun_eq_coe",
+          "start": [
+            260,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.cancel_right",
+        "id": "PseudoEpimorphism.cancel_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            219,
+            78
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.cancel_right",
+          "start": [
+            217,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.coe_comp",
+        "id": "PseudoEpimorphism.coe_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            188,
+            38
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.coe_comp",
+          "start": [
+            187,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.coe_comp_orderHom",
+        "id": "PseudoEpimorphism.coe_comp_orderHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            193,
+            53
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.coe_comp_orderHom",
+          "start": [
+            192,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.coe_copy",
+        "id": "PseudoEpimorphism.coe_copy",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            147,
+            100
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.coe_copy",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.coe_id",
+        "id": "PseudoEpimorphism.coe_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            165,
+            55
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.coe_id",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.coe_id_orderHom",
+        "id": "PseudoEpimorphism.coe_id_orderHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.coe_id_orderHom",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.comp_apply",
+        "id": "PseudoEpimorphism.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            198,
+            34
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.comp_apply",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.comp_assoc",
+        "id": "PseudoEpimorphism.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            79
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.comp_assoc",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.comp_id",
+        "id": "PseudoEpimorphism.comp_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            208,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.comp_id",
+          "start": [
+            207,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.copy_eq",
+        "id": "PseudoEpimorphism.copy_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.copy_eq",
+          "start": [
+            150,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.ext",
+        "id": "PseudoEpimorphism.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            137,
+            21
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.ext",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.id_apply",
+        "id": "PseudoEpimorphism.id_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.id_apply",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.id_comp",
+        "id": "PseudoEpimorphism.id_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            19
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.id_comp",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.toFun_eq_coe",
+        "id": "PseudoEpimorphism.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.toFun_eq_coe",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PseudoEpimorphism.toOrderHom_eq_coe",
+        "id": "PseudoEpimorphism.toOrderHom_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            130,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean",
+          "full_name": "PseudoEpimorphism.toOrderHom_eq_coe",
+          "start": [
+            130,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Hom/Esakia.lean"
+      }
+    ],
+    "input_proofs": 46,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 2,
+    "replay_failures": [],
+    "transitions": 2,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001937-0000-bc879af216b56b80.parquet",
+      "sha256": "bc879af216b56b80da53497b6725f04d46c695f7f145df2aff32ed391c1f7556",
+      "bytes": 48317,
+      "rows": 2,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ContinuousOrderHom.cancel_left",
+    "mathlib:PseudoEpimorphism.cancel_left"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Hom/Basic.lean\",null,null,\"ContinuousOrderHom.cancel_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Hom/Esakia.lean\",null,null,\"PseudoEpimorphism.cancel_left\"]"
+  ]
+}

manifests/batches/mathlib/001939.json

@@ -0,0 +1,1753 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001939",
+  "batch_fingerprint": "23acc18a354f0b8a9610738ac919837fda180c938bfecc3277438472a86b9fa2",
+  "evidence": {
+    "batch_fingerprint": "23acc18a354f0b8a9610738ac919837fda180c938bfecc3277438472a86b9fa2",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_inf",
+        "id": "Continuous.finset_inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            410,
+            98
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_inf",
+          "start": [
+            409,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_inf'",
+        "id": "Continuous.finset_inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            370,
+            101
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_inf'",
+          "start": [
+            368,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_inf'_apply",
+        "id": "Continuous.finset_inf'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            366,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_inf'_apply",
+          "start": [
+            363,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_inf_apply",
+        "id": "Continuous.finset_inf_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            407,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_inf_apply",
+          "start": [
+            404,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_sup",
+        "id": "Continuous.finset_sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            329,
+            98
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_sup",
+          "start": [
+            328,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_sup'",
+        "id": "Continuous.finset_sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            289,
+            101
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_sup'",
+          "start": [
+            287,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_sup'_apply",
+        "id": "Continuous.finset_sup'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            285,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_sup'_apply",
+          "start": [
+            282,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.finset_sup_apply",
+        "id": "Continuous.finset_sup_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            326,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.finset_sup_apply",
+          "start": [
+            323,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.inf",
+        "id": "Continuous.inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            42
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.inf",
+          "start": [
+            88,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.inf'",
+        "id": "Continuous.inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            248,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.inf'",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.sup",
+        "id": "Continuous.sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            42
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.sup",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.sup'",
+        "id": "Continuous.sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Continuous.sup'",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.finset_inf'_apply",
+        "id": "ContinuousAt.finset_inf'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            340,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.finset_inf'_apply",
+          "start": [
+            338,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.finset_inf_apply",
+        "id": "ContinuousAt.finset_inf_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            381,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.finset_inf_apply",
+          "start": [
+            379,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.finset_sup'_apply",
+        "id": "ContinuousAt.finset_sup'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            259,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.finset_sup'_apply",
+          "start": [
+            257,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.finset_sup_apply",
+        "id": "ContinuousAt.finset_sup_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            300,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.finset_sup_apply",
+          "start": [
+            298,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.inf",
+        "id": "ContinuousAt.inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            229,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.inf",
+          "start": [
+            226,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.inf'",
+        "id": "ContinuousAt.inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            224,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.inf'",
+          "start": [
+            222,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.sup",
+        "id": "ContinuousAt.sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.sup",
+          "start": [
+            192,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.sup'",
+        "id": "ContinuousAt.sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            190,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousAt.sup'",
+          "start": [
+            188,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_inf",
+        "id": "ContinuousOn.finset_inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            402,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_inf",
+          "start": [
+            400,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_inf'",
+        "id": "ContinuousOn.finset_inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            361,
+            61
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_inf'",
+          "start": [
+            359,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_inf'_apply",
+        "id": "ContinuousOn.finset_inf'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            357,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_inf'_apply",
+          "start": [
+            355,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_inf_apply",
+        "id": "ContinuousOn.finset_inf_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            398,
+            62
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_inf_apply",
+          "start": [
+            396,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_sup",
+        "id": "ContinuousOn.finset_sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            321,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_sup",
+          "start": [
+            319,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_sup'",
+        "id": "ContinuousOn.finset_sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            280,
+            61
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_sup'",
+          "start": [
+            278,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_sup'_apply",
+        "id": "ContinuousOn.finset_sup'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            276,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_sup'_apply",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.finset_sup_apply",
+        "id": "ContinuousOn.finset_sup_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            317,
+            62
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.finset_sup_apply",
+          "start": [
+            315,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.inf",
+        "id": "ContinuousOn.inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            246,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.inf",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.inf'",
+        "id": "ContinuousOn.inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            241,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.inf'",
+          "start": [
+            239,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.sup",
+        "id": "ContinuousOn.sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            212,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.sup",
+          "start": [
+            209,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.sup'",
+        "id": "ContinuousOn.sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            207,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousOn.sup'",
+          "start": [
+            205,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.finset_inf'_apply",
+        "id": "ContinuousWithinAt.finset_inf'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            349,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.finset_inf'_apply",
+          "start": [
+            346,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.finset_inf_apply",
+        "id": "ContinuousWithinAt.finset_inf_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            390,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.finset_inf_apply",
+          "start": [
+            387,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.finset_sup'_apply",
+        "id": "ContinuousWithinAt.finset_sup'_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            268,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.finset_sup'_apply",
+          "start": [
+            265,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.finset_sup_apply",
+        "id": "ContinuousWithinAt.finset_sup_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            309,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.finset_sup_apply",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.inf",
+        "id": "ContinuousWithinAt.inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.inf",
+          "start": [
+            235,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.inf'",
+        "id": "ContinuousWithinAt.inf'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            233,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.inf'",
+          "start": [
+            231,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.sup",
+        "id": "ContinuousWithinAt.sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            13
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.sup",
+          "start": [
+            201,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.sup'",
+        "id": "ContinuousWithinAt.sup'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            199,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "ContinuousWithinAt.sup'",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.finset_inf'_nhds",
+        "id": "Filter.Tendsto.finset_inf'_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            154,
+            37
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.finset_inf'_nhds",
+          "start": [
+            151,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.finset_inf'_nhds_apply",
+        "id": "Filter.Tendsto.finset_inf'_nhds_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            43
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.finset_inf'_nhds_apply",
+          "start": [
+            156,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.finset_inf_nhds",
+        "id": "Filter.Tendsto.finset_inf_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            32
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.finset_inf_nhds",
+          "start": [
+            173,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa using hs'",
+        "full_name": "Filter.Tendsto.finset_sup'_nhds",
+        "id": "Filter.Tendsto.finset_sup'_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            144,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.finset_sup'_nhds",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction hne using <a>Finset.Nonempty.cons_induction<\\/a> with\\n  | singleton => simpa using hs\\n  | cons a s ha hne ihs =>\\n    rw [<a>forall_mem_cons<\\/a>] at hs\\n    simp only [<a>sup'_cons<\\/a>, hne]\\n    exact hs.1.<a>sup_nhds<\\/a> (ihs hs.2)\"",
+                "[{\"full_name\":\"Finset.Nonempty.cons_induction\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1302,9],\"def_end_pos\":[1302,32]},{\"full_name\":\"Finset.forall_mem_cons\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[890,9],\"def_end_pos\":[890,24]},{\"full_name\":\"Finset.sup'_cons\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Lattice.lean\",\"def_pos\":[826,9],\"def_end_pos\":[826,18]},{\"full_name\":\"Filter.Tendsto.sup_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Lattice.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "L : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\nhne : s.Nonempty\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))",
+              "tactic": "induction hne using Finset.Nonempty.cons_induction with\n| singleton => simpa using hs\n| cons a s ha hne ihs =>\n  rw [forall_mem_cons] at hs\n  simp only [sup'_cons, hne]\n  exact hs.1.sup_nhds (ihs hs.2)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using hs\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case singleton\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na\u271d : \u03b9\nhs : \u2200 i \u2208 {a\u271d}, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto ({a\u271d}.sup' \u22ef f) l (\ud835\udcdd ({a\u271d}.sup' \u22ef g))",
+              "tactic": "simpa using hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>forall_mem_cons<\\/a>] at hs\"",
+                "[{\"full_name\":\"Finset.forall_mem_cons\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[890,9],\"def_end_pos\":[890,24]}]"
+              ],
+              "state_after": "case cons\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nhne : s.Nonempty\nihs : (\u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) \u2192 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))\nhs : Tendsto (f a) l (\ud835\udcdd (g a)) \u2227 \u2200 x \u2208 s, Tendsto (f x) l (\ud835\udcdd (g x))\n\u22a2 Tendsto ((cons a s ha).sup' \u22ef f) l (\ud835\udcdd ((cons a s ha).sup' \u22ef g))",
+              "state_before": "case cons\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nhne : s.Nonempty\nihs : (\u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) \u2192 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))\nhs : \u2200 i \u2208 cons a s ha, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto ((cons a s ha).sup' \u22ef f) l (\ud835\udcdd ((cons a s ha).sup' \u22ef g))",
+              "tactic": "rw [forall_mem_cons] at hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>sup'_cons<\\/a>, hne]\"",
+                "[{\"full_name\":\"Finset.sup'_cons\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Lattice.lean\",\"def_pos\":[826,9],\"def_end_pos\":[826,18]}]"
+              ],
+              "state_after": "case cons\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nhne : s.Nonempty\nihs : (\u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) \u2192 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))\nhs : Tendsto (f a) l (\ud835\udcdd (g a)) \u2227 \u2200 x \u2208 s, Tendsto (f x) l (\ud835\udcdd (g x))\n\u22a2 Tendsto (f a \u2294 s.sup' \u22ef f) l (\ud835\udcdd (g a \u2294 s.sup' \u22ef g))",
+              "state_before": "case cons\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nhne : s.Nonempty\nihs : (\u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) \u2192 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))\nhs : Tendsto (f a) l (\ud835\udcdd (g a)) \u2227 \u2200 x \u2208 s, Tendsto (f x) l (\ud835\udcdd (g x))\n\u22a2 Tendsto ((cons a s ha).sup' \u22ef f) l (\ud835\udcdd ((cons a s ha).sup' \u22ef g))",
+              "tactic": "simp only [sup'_cons, hne]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hs.1.<a>sup_nhds<\\/a> (ihs hs.2)\"",
+                "[{\"full_name\":\"Filter.Tendsto.sup_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Lattice.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case cons\nL : Type u_1\nX : Type u_2\ninst\u271d\u00b3 : TopologicalSpace L\ninst\u271d\u00b2 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b9 : SemilatticeSup L\ninst\u271d : ContinuousSup L\na : \u03b9\ns : Finset \u03b9\nha : a \u2209 s\nhne : s.Nonempty\nihs : (\u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))) \u2192 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))\nhs : Tendsto (f a) l (\ud835\udcdd (g a)) \u2227 \u2200 x \u2208 s, Tendsto (f x) l (\ud835\udcdd (g x))\n\u22a2 Tendsto (f a \u2294 s.sup' \u22ef f) l (\ud835\udcdd (g a \u2294 s.sup' \u22ef g))",
+              "tactic": "exact hs.1.sup_nhds (ihs hs.2)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.inf_nhds",
+        "id": "Filter.Tendsto.inf_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.inf_nhds",
+          "start": [
+            125,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.inf_nhds'",
+        "id": "Filter.Tendsto.inf_nhds'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.inf_nhds'",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.sup_nhds",
+        "id": "Filter.Tendsto.sup_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            18
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.sup_nhds",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.sup_nhds'",
+        "id": "Filter.Tendsto.sup_nhds'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "Filter.Tendsto.sup_nhds'",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_inf",
+        "id": "continuous_inf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            31
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "continuous_inf",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_sup",
+        "id": "continuous_sup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            31
+          ],
+          "file_path": "Mathlib/Topology/Order/Lattice.lean",
+          "full_name": "continuous_sup",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Lattice.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [lawsonBasis, image2, IsLower.lowerBasis]'",
+        "full_name": "Topology.IsLawson.isTopologicalBasis",
+        "id": "Topology.IsLawson.isTopologicalBasis",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.IsLawson.isTopologicalBasis",
+          "start": [
+            86,
+            11
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have lawsonBasis_image2 : <a>lawsonBasis<\\/a> \\u03b1 =\\n      (<a>image2<\\/a> (fun x x_1 \\u21a6 \\u21d1<a>WithLower.toLower<\\/a> \\u207b\\u00b9' x \\u2229 \\u21d1<a>WithScott.toScott<\\/a> \\u207b\\u00b9' x_1)\\n        (<a>IsLower.lowerBasis<\\/a> (<a>WithLower<\\/a> \\u03b1)) {U | IsOpen[<a>scott<\\/a> \\u03b1] U}) := by\\n    rw [<a>lawsonBasis<\\/a>, <a>image2<\\/a>, <a>IsLower.lowerBasis<\\/a>]\\n    simp_rw [<a>diff_eq_compl_inter<\\/a>]\\n    aesop\"",
+                "[{\"full_name\":\"Topology.IsLawson.lawsonBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LawsonTopology.lean\",\"def_pos\":[83,5],\"def_end_pos\":[83,16]},{\"full_name\":\"Set.image2\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[322,5],\"def_end_pos\":[322,11]},{\"full_name\":\"Topology.WithLower.toLower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[79,22],\"def_end_pos\":[79,29]},{\"full_name\":\"Topology.WithScott.toScott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[304,22],\"def_end_pos\":[304,29]},{\"full_name\":\"Topology.IsLower.lowerBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[201,5],\"def_end_pos\":[201,15]},{\"full_name\":\"Topology.WithLower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[71,5],\"def_end_pos\":[71,14]},{\"full_name\":\"Topology.scott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[193,5],\"def_end_pos\":[193,10]},{\"full_name\":\"Topology.IsLawson.lawsonBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LawsonTopology.lean\",\"def_pos\":[83,5],\"def_end_pos\":[83,16]},{\"full_name\":\"Set.image2\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[322,5],\"def_end_pos\":[322,11]},{\"full_name\":\"Topology.IsLower.lowerBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[201,5],\"def_end_pos\":[201,15]},{\"full_name\":\"Set.diff_eq_compl_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1775,9],\"def_end_pos\":[1775,28]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 IsTopologicalBasis (lawsonBasis \u03b1)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 IsTopologicalBasis (lawsonBasis \u03b1)",
+              "tactic": "have lawsonBasis_image2 : lawsonBasis \u03b1 =\n    (image2 (fun x x_1 \u21a6 \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1)\n      (IsLower.lowerBasis (WithLower \u03b1)) {U | IsOpen[scott \u03b1] U}) := by\n  rw [lawsonBasis, image2, IsLower.lowerBasis]\n  simp_rw [diff_eq_compl_inter]\n  aesop"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [lawsonBasis_image2]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 IsTopologicalBasis\n    (image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U})",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 IsTopologicalBasis (lawsonBasis \u03b1)",
+              "tactic": "rw [lawsonBasis_image2]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>IsTopologicalBasis.inf_induced<\\/a> <a>IsLower.isTopologicalBasis<\\/a>\\n    (<a>isTopologicalBasis_opens<\\/a> (\\u03b1 := <a>WithScott<\\/a> \\u03b1))\\n    <a>WithLower.toLower<\\/a> <a>WithScott.toScott<\\/a>\"",
+                "[{\"full_name\":\"TopologicalSpace.IsTopologicalBasis.inf_induced\",\"def_path\":\"Mathlib\\/Topology\\/Bases.lean\",\"def_pos\":[268,9],\"def_end_pos\":[268,39]},{\"full_name\":\"Topology.IsLower.isTopologicalBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[267,19],\"def_end_pos\":[267,37]},{\"full_name\":\"TopologicalSpace.isTopologicalBasis_opens\",\"def_path\":\"Mathlib\\/Topology\\/Bases.lean\",\"def_pos\":[244,9],\"def_end_pos\":[244,33]},{\"full_name\":\"Topology.WithScott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[299,5],\"def_end_pos\":[299,14]},{\"full_name\":\"Topology.WithLower.toLower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[79,22],\"def_end_pos\":[79,29]},{\"full_name\":\"Topology.WithScott.toScott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[304,22],\"def_end_pos\":[304,29]}]"
+              ],
+              "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 inst\u271d\u00b9 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 IsTopologicalBasis\n    (image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U})",
+              "tactic": "convert IsTopologicalBasis.inf_induced IsLower.isTopologicalBasis\n  (isTopologicalBasis_opens (\u03b1 := WithScott \u03b1))\n  WithLower.toLower WithScott.toScott"
+            },
+            {
+              "annotated_tactic": [
+                "\"erw [@<a>topology_eq_lawson<\\/a> \\u03b1 _ _ _]\"",
+                "[{\"full_name\":\"Topology.IsLawson.topology_eq_lawson\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LawsonTopology.lean\",\"def_pos\":[73,3],\"def_end_pos\":[73,21]}]"
+              ],
+              "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lawson \u03b1 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 inst\u271d\u00b9 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "tactic": "erw [@topology_eq_lawson \u03b1 _ _ _]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>lawson<\\/a>]\"",
+                "[{\"full_name\":\"Topology.lawson\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LawsonTopology.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,11]}]"
+              ],
+              "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lower \u03b1 \u2293 scott \u03b1 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lawson \u03b1 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "tactic": "rw [lawson]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply (<a>congrArg\\u2082<\\/a> <a>Inf.inf<\\/a> _) _\"",
+                "[{\"full_name\":\"congrArg\\u2082\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,18]},{\"full_name\":\"Inf.inf\",\"def_path\":\"Mathlib\\/Order\\/Notation.lean\",\"def_pos\":[56,3],\"def_end_pos\":[56,6]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lower \u03b1 = induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 scott \u03b1 = induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lower \u03b1 \u2293 scott \u03b1 =\n    induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace \u2293\n      induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "tactic": "apply (congrArg\u2082 Inf.inf _) _"
+            },
+            {
+              "annotated_tactic": [
+                "\"letI _ := <a>scott<\\/a> \\u03b1\"",
+                "[{\"full_name\":\"Topology.scott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[193,5],\"def_end_pos\":[193,10]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\nx\u271d : TopologicalSpace \u03b1 := scott \u03b1\n\u22a2 scott \u03b1 = induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 scott \u03b1 = induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "tactic": "letI _ := scott \u03b1"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact @<a>IsScott.withScottHomeomorph<\\/a> \\u03b1 _ (<a>scott<\\/a> \\u03b1) \\u27e8<a>rfl<\\/a>\\u27e9 |>.inducing.induced\"",
+                "[{\"full_name\":\"Topology.IsScott.withScottHomeomorph\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[350,5],\"def_end_pos\":[350,32]},{\"full_name\":\"Topology.scott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[193,5],\"def_end_pos\":[193,10]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\nx\u271d : TopologicalSpace \u03b1 := scott \u03b1\n\u22a2 scott \u03b1 = induced (\u21d1WithScott.toScott) WithScott.instTopologicalSpace",
+              "tactic": "exact @IsScott.withScottHomeomorph \u03b1 _ (scott \u03b1) \u27e8rfl\u27e9 |>.inducing.induced"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>lawsonBasis<\\/a>, <a>image2<\\/a>, <a>IsLower.lowerBasis<\\/a>]\"",
+                "[{\"full_name\":\"Topology.IsLawson.lawsonBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LawsonTopology.lean\",\"def_pos\":[83,5],\"def_end_pos\":[83,16]},{\"full_name\":\"Set.image2\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[322,5],\"def_end_pos\":[322,11]},{\"full_name\":\"Topology.IsLower.lowerBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[201,5],\"def_end_pos\":[201,15]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u2203 u, IsOpen u \u2227 u \\ \u2191(upperClosure t) = s} =\n    {c |\n      \u2203 a \u2208 {s | \u2203 t, t.Finite \u2227 (\u2191(upperClosure t))\u1d9c = s},\n        \u2203 b \u2208 {U | IsOpen U}, \u21d1WithLower.toLower \u207b\u00b9' a \u2229 \u21d1WithScott.toScott \u207b\u00b9' b = c}",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}",
+              "tactic": "rw [lawsonBasis, image2, IsLower.lowerBasis]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>diff_eq_compl_inter<\\/a>]\"",
+                "[{\"full_name\":\"Set.diff_eq_compl_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1775,9],\"def_end_pos\":[1775,28]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u2203 u, IsOpen u \u2227 (\u2191(upperClosure t))\u1d9c \u2229 u = s} =\n    {c |\n      \u2203 a \u2208 {s | \u2203 t, t.Finite \u2227 (\u2191(upperClosure t))\u1d9c = s},\n        \u2203 b \u2208 {U | IsOpen U}, \u21d1WithLower.toLower \u207b\u00b9' a \u2229 \u21d1WithScott.toScott \u207b\u00b9' b = c}",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u2203 u, IsOpen u \u2227 u \\ \u2191(upperClosure t) = s} =\n    {c |\n      \u2203 a \u2208 {s | \u2203 t, t.Finite \u2227 (\u2191(upperClosure t))\u1d9c = s},\n        \u2203 b \u2208 {U | IsOpen U}, \u21d1WithLower.toLower \u207b\u00b9' a \u2229 \u21d1WithScott.toScott \u207b\u00b9' b = c}",
+              "tactic": "simp_rw [diff_eq_compl_inter]"
+            },
+            {
+              "annotated_tactic": [
+                "\"aesop\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u2203 u, IsOpen u \u2227 (\u2191(upperClosure t))\u1d9c \u2229 u = s} =\n    {c |\n      \u2203 a \u2208 {s | \u2203 t, t.Finite \u2227 (\u2191(upperClosure t))\u1d9c = s},\n        \u2203 b \u2208 {U | IsOpen U}, \u21d1WithLower.toLower \u207b\u00b9' a \u2229 \u21d1WithScott.toScott \u207b\u00b9' b = c}",
+              "tactic": "aesop"
+            },
+            {
+              "annotated_tactic": [
+                "\"letI _ := <a>lower<\\/a> \\u03b1\"",
+                "[{\"full_name\":\"Topology.lower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[62,5],\"def_end_pos\":[62,10]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\nx\u271d : TopologicalSpace \u03b1 := lower \u03b1\n\u22a2 lower \u03b1 = induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\n\u22a2 lower \u03b1 = induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace",
+              "tactic": "letI _ := lower \u03b1"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact @<a>IsLower.withLowerHomeomorph<\\/a> \\u03b1 \\u2039_\\u203a (<a>lower<\\/a> \\u03b1) \\u27e8<a>rfl<\\/a>\\u27e9 |>.inducing.induced\"",
+                "[{\"full_name\":\"Topology.IsLower.withLowerHomeomorph\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[216,5],\"def_end_pos\":[216,24]},{\"full_name\":\"Topology.lower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[62,5],\"def_end_pos\":[62,10]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLawson \u03b1\nlawsonBasis_image2 :\n  lawsonBasis \u03b1 =\n    image2 (fun x x_1 => \u21d1WithLower.toLower \u207b\u00b9' x \u2229 \u21d1WithScott.toScott \u207b\u00b9' x_1) (IsLower.lowerBasis (WithLower \u03b1))\n      {U | IsOpen U}\nx\u271d : TopologicalSpace \u03b1 := lower \u03b1\n\u22a2 lower \u03b1 = induced (\u21d1WithLower.toLower) WithLower.instTopologicalSpace",
+              "tactic": "exact @IsLower.withLowerHomeomorph \u03b1 \u2039_\u203a (lower \u03b1) \u27e8rfl\u27e9 |>.inducing.induced"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.isClosed_preimage_ofLawson",
+        "id": "Topology.WithLawson.isClosed_preimage_ofLawson",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            153,
+            64
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.isClosed_preimage_ofLawson",
+          "start": [
+            152,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.isOpen_def",
+        "id": "Topology.WithLawson.isOpen_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            156,
+            61
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.isOpen_def",
+          "start": [
+            155,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.isOpen_preimage_ofLawson",
+        "id": "Topology.WithLawson.isOpen_preimage_ofLawson",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            150,
+            61
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.isOpen_preimage_ofLawson",
+          "start": [
+            149,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.ofLawson_inj",
+        "id": "Topology.WithLawson.ofLawson_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            85
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.ofLawson_inj",
+          "start": [
+            127,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.ofLawson_toLawson",
+        "id": "Topology.WithLawson.ofLawson_toLawson",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.ofLawson_toLawson",
+          "start": [
+            122,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.of_Lawson_symm_eq",
+        "id": "Topology.WithLawson.of_Lawson_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.of_Lawson_symm_eq",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.toLawson_inj",
+        "id": "Topology.WithLawson.toLawson_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            74
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.toLawson_inj",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.toLawson_ofLawson",
+        "id": "Topology.WithLawson.toLawson_ofLawson",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            86
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.toLawson_ofLawson",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLawson.to_Lawson_symm_eq",
+        "id": "Topology.WithLawson.to_Lawson_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            119,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.WithLawson.to_Lawson_symm_eq",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.lawsonClosed_of_lowerClosed",
+        "id": "Topology.lawsonClosed_of_lowerClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            176,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.lawsonClosed_of_lowerClosed",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.lawsonClosed_of_scottClosed",
+        "id": "Topology.lawsonClosed_of_scottClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.lawsonClosed_of_scottClosed",
+          "start": [
+            172,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.lawsonOpen_iff_scottOpen_of_isUpperSet",
+        "id": "Topology.lawsonOpen_iff_scottOpen_of_isUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            182,
+            61
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.lawsonOpen_iff_scottOpen_of_isUpperSet",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.lawson_le_lower",
+        "id": "Topology.lawson_le_lower",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            58
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.lawson_le_lower",
+          "start": [
+            167,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.lawson_le_scott",
+        "id": "Topology.lawson_le_scott",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            165,
+            59
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.lawson_le_scott",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.scottHausdorff_le_lawson",
+        "id": "Topology.scottHausdorff_le_lawson",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            57
+          ],
+          "file_path": "Mathlib/Topology/Order/LawsonTopology.lean",
+          "full_name": "Topology.scottHausdorff_le_lawson",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LawsonTopology.lean"
+      }
+    ],
+    "input_proofs": 83,
+    "input_units": 2,
+    "proofs": 17,
+    "published_proofs": 16,
+    "published_transitions": 23,
+    "replay_failures": [],
+    "transitions": 27,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/Lattice.lean",
+        "proof_id": "mathlib:Filter.Tendsto.finset_sup_nhds",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              166,
+              34
+            ],
+            "file_path": "Mathlib/Topology/Order/Lattice.lean",
+            "full_name": "Filter.Tendsto.finset_sup_nhds",
+            "start": [
+              161,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases s.eq_empty_or_nonempty with rfl | hne\"",
+                  "[]"
+                ],
+                "state_after": "case inl\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 \u2205, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto (\u2205.sup f) l (\ud835\udcdd (\u2205.sup g))\n\ncase inr\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\nhne : s.Nonempty\n\u22a2 Tendsto (s.sup f) l (\ud835\udcdd (s.sup g))",
+                "state_before": "L : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto (s.sup f) l (\ud835\udcdd (s.sup g))",
+                "tactic": "rcases s.eq_empty_or_nonempty with rfl | hne"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa using <a>tendsto_const_nhds<\\/a>\"",
+                  "[{\"full_name\":\"tendsto_const_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[991,9],\"def_end_pos\":[991,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 \u2205, Tendsto (f i) l (\ud835\udcdd (g i))\n\u22a2 Tendsto (\u2205.sup f) l (\ud835\udcdd (\u2205.sup g))",
+                "tactic": "simpa using tendsto_const_nhds"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [\\u2190 <a>sup'_eq_sup<\\/a> hne]\"",
+                  "[{\"full_name\":\"Finset.sup'_eq_sup\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Lattice.lean\",\"def_pos\":[1184,9],\"def_end_pos\":[1184,20]}]"
+                ],
+                "state_after": "case inr\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\nhne : s.Nonempty\n\u22a2 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))",
+                "state_before": "case inr\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\nhne : s.Nonempty\n\u22a2 Tendsto (s.sup f) l (\ud835\udcdd (s.sup g))",
+                "tactic": "simp only [\u2190 sup'_eq_sup hne]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>finset_sup'_nhds<\\/a> hne hs\"",
+                  "[{\"full_name\":\"Filter.Tendsto.finset_sup'_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Lattice.lean\",\"def_pos\":[136,7],\"def_end_pos\":[136,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\nL : Type u_1\nX : Type u_2\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : TopologicalSpace X\n\u03b9 : Type u_3\n\u03b1 : Type u_4\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 L\nl : Filter \u03b1\ng : \u03b9 \u2192 L\ninst\u271d\u00b2 : SemilatticeSup L\ninst\u271d\u00b9 : OrderBot L\ninst\u271d : ContinuousSup L\nhs : \u2200 i \u2208 s, Tendsto (f i) l (\ud835\udcdd (g i))\nhne : s.Nonempty\n\u22a2 Tendsto (s.sup' hne f) l (\ud835\udcdd (s.sup' hne g))",
+                "tactic": "exact finset_sup'_nhds hne hs"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Filter.Tendsto.finset_sup_nhds"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001939-0000-b703cc3b7215c9a9.parquet",
+      "sha256": "b703cc3b7215c9a9213b33bc00da6dccb5ef9c4656436c534c72ce66335c571f",
+      "bytes": 73308,
+      "rows": 22,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/internal_test/mathlib-001939-0000-d8a34aa9a86a6582.parquet",
+      "sha256": "d8a34aa9a86a6582d9c235a4ef3f4c60db5a5b88e3323a52d77206d81aa23fee",
+      "bytes": 49432,
+      "rows": 1,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ContinuousAt.finset_inf",
+    "mathlib:ContinuousAt.finset_inf'",
+    "mathlib:ContinuousAt.finset_sup",
+    "mathlib:ContinuousAt.finset_sup'",
+    "mathlib:ContinuousWithinAt.finset_inf",
+    "mathlib:ContinuousWithinAt.finset_inf'",
+    "mathlib:ContinuousWithinAt.finset_sup",
+    "mathlib:ContinuousWithinAt.finset_sup'",
+    "mathlib:Filter.Tendsto.finset_sup'_nhds_apply",
+    "mathlib:Filter.Tendsto.finset_sup_nhds_apply",
+    "mathlib:Topology.IsLawson.singleton_isClosed",
+    "mathlib:Topology.isLawson_le_isScott",
+    "mathlib:Topology.lawsonClosed_iff_dirSupClosed_of_isLowerSet",
+    "mathlib:Topology.lawsonClosed_iff_scottClosed_of_isLowerSet",
+    "mathlib:Topology.lawsonOpen_iff_scottOpen_of_isUpperSet'",
+    "mathlib:Topology.scottHausdorff_le_isLawson"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousAt.finset_inf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousAt.finset_inf'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousAt.finset_sup\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousAt.finset_sup'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousWithinAt.finset_inf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousWithinAt.finset_inf'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousWithinAt.finset_sup\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"ContinuousWithinAt.finset_sup'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"Filter.Tendsto.finset_sup'_nhds_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Lattice.lean\",null,null,\"Filter.Tendsto.finset_sup_nhds_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.IsLawson.singleton_isClosed\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.isLawson_le_isScott\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.lawsonClosed_iff_dirSupClosed_of_isLowerSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.lawsonClosed_iff_scottClosed_of_isLowerSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.lawsonOpen_iff_scottOpen_of_isUpperSet'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LawsonTopology.lean\",null,null,\"Topology.scottHausdorff_le_isLawson\"]"
+  ]
+}

manifests/batches/mathlib/001942.json

@@ -0,0 +1,1661 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001942",
+  "batch_fingerprint": "cba3eecca4ef1cebb7c89f98e4f78a45442bd3b95d0fb0c2fa044dcce93ee48e",
+  "evidence": {
+    "batch_fingerprint": "cba3eecca4ef1cebb7c89f98e4f78a45442bd3b95d0fb0c2fa044dcce93ee48e",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLower.closure_singleton",
+        "id": "Topology.IsLower.closure_singleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            264,
+            79
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsLower.closure_singleton",
+          "start": [
+            262,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"obtain \u27e8a, rfl\u27e9 := h'\"",
+        "full_name": "Topology.IsLower.isLowerSet_of_isOpen",
+        "id": "Topology.IsLower.isLowerSet_of_isOpen",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            250,
+            48
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsLower.isLowerSet_of_isOpen",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"replace h := <a>isOpen_iff_generate_Ici_compl<\\/a>.1 h\"",
+                "[{\"full_name\":\"Topology.IsLower.isOpen_iff_generate_Ici_compl\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,38]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nh : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\n\u22a2 IsLowerSet s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nh : IsOpen s\n\u22a2 IsLowerSet s",
+              "tactic": "replace h := isOpen_iff_generate_Ici_compl.1 h"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction h with\\n  | <a>basic<\\/a> u h' => obtain \\u27e8a, rfl\\u27e9 := h'; exact (<a>isUpperSet_Ici<\\/a> a).<a>compl<\\/a>\\n  | <a>univ<\\/a> => exact <a>isLowerSet_univ<\\/a>\\n  | <a>inter<\\/a> u v _ _ hu2 hv2 => exact hu2.inter hv2\\n  | <a>sUnion<\\/a> _ _ ih => exact <a>isLowerSet_sUnion<\\/a> ih\"",
+                "[{\"full_name\":\"TopologicalSpace.GenerateOpen.basic\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,10]},{\"full_name\":\"isUpperSet_Ici\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,23]},{\"full_name\":\"IsUpperSet.compl\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,25]},{\"full_name\":\"TopologicalSpace.GenerateOpen.univ\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[60,5],\"def_end_pos\":[60,9]},{\"full_name\":\"isLowerSet_univ\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,24]},{\"full_name\":\"TopologicalSpace.GenerateOpen.inter\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[61,5],\"def_end_pos\":[61,10]},{\"full_name\":\"TopologicalSpace.GenerateOpen.sUnion\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[62,5],\"def_end_pos\":[62,11]},{\"full_name\":\"isLowerSet_sUnion\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nh : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\n\u22a2 IsLowerSet s",
+              "tactic": "induction h with\n| basic u h' => obtain \u27e8a, rfl\u27e9 := h'; exact (isUpperSet_Ici a).compl\n| univ => exact isLowerSet_univ\n| inter u v _ _ hu2 hv2 => exact hu2.inter hv2\n| sUnion _ _ ih => exact isLowerSet_sUnion ih"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8a, rfl\\u27e9 := h'\"",
+                "[]"
+              ],
+              "state_after": "case basic.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 IsLowerSet (Ici a)\u1d9c",
+              "state_before": "case basic\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns u : Set \u03b1\nh' : u \u2208 {t | \u2203 a, (Ici a)\u1d9c = t}\n\u22a2 IsLowerSet u",
+              "tactic": "obtain \u27e8a, rfl\u27e9 := h'"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>isUpperSet_Ici<\\/a> a).<a>compl<\\/a>\"",
+                "[{\"full_name\":\"isUpperSet_Ici\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,23]},{\"full_name\":\"IsUpperSet.compl\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[84,9],\"def_end_pos\":[84,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case basic.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 IsLowerSet (Ici a)\u1d9c",
+              "tactic": "exact (isUpperSet_Ici a).compl"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>isLowerSet_univ<\\/a>\"",
+                "[{\"full_name\":\"isLowerSet_univ\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case univ\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet univ",
+              "tactic": "exact isLowerSet_univ"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hu2.inter hv2\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inter\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns u v : Set \u03b1\na\u271d\u00b9 : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} u\na\u271d : GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} v\nhu2 : IsLowerSet u\nhv2 : IsLowerSet v\n\u22a2 IsLowerSet (u \u2229 v)",
+              "tactic": "exact hu2.inter hv2"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>isLowerSet_sUnion<\\/a> ih\"",
+                "[{\"full_name\":\"isLowerSet_sUnion\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case sUnion\n\u03b1 : Type u_1\n\u03b2 : ?m.6004\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 s \u2208 S\u271d, GenerateOpen {t | \u2203 a, (Ici a)\u1d9c = t} s\nih : \u2200 s \u2208 S\u271d, IsLowerSet s\n\u22a2 IsLowerSet (\u22c3\u2080 S\u271d)",
+              "tactic": "exact isLowerSet_sUnion ih"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp_rw [biInter_range, hf, sInter_eq_iInter]'",
+        "full_name": "Topology.IsLower.isTopologicalBasis",
+        "id": "Topology.IsLower.isTopologicalBasis",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            279,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsLower.isTopologicalBasis",
+          "start": [
+            267,
+            11
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"convert <a>isTopologicalBasis_of_subbasis<\\/a> (<a>topology_eq<\\/a> \\u03b1)\"",
+                "[{\"full_name\":\"TopologicalSpace.isTopologicalBasis_of_subbasis\",\"def_path\":\"Mathlib\\/Topology\\/Bases.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,39]},{\"full_name\":\"Topology.IsLower.topology_eq\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[210,7],\"def_end_pos\":[210,18]}]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 lowerBasis \u03b1 = (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 IsTopologicalBasis (lowerBasis \u03b1)",
+              "tactic": "convert isTopologicalBasis_of_subbasis (topology_eq \u03b1)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>lowerBasis<\\/a>, <a>coe_upperClosure<\\/a>, <a>compl_iUnion<\\/a>]\"",
+                "[{\"full_name\":\"Topology.IsLower.lowerBasis\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[201,5],\"def_end_pos\":[201,15]},{\"full_name\":\"coe_upperClosure\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[1409,9],\"def_end_pos\":[1409,25]},{\"full_name\":\"Set.compl_iUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,21]}]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} = (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 lowerBasis \u03b1 = (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "tactic": "simp_rw [lowerBasis, coe_upperClosure, compl_iUnion]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext s\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} \u2194 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\n\u22a2 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} = (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "tactic": "ext s"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h.mp\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} \u2192 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}\n\ncase h.e'_3.h.mpr\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}} \u2192 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} \u2194 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8F, hF, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns F : Set \u03b1\nhF : F.Finite\n\u22a2 \u22c2 i \u2208 F, (Ici i)\u1d9c \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "state_before": "case h.e'_3.h.mp\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s} \u2192 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "tactic": "rintro \u27e8F, hF, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8(fun a => (<a>Ici<\\/a> a)\\u1d9c) '' F, \\u27e8hF.image _, <a>image_subset_iff<\\/a>.2 fun _ _ => \\u27e8_, <a>rfl<\\/a>\\u27e9\\u27e9, ?_\\u27e9\"",
+                "[{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]},{\"full_name\":\"Set.image_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,25]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "case h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns F : Set \u03b1\nhF : F.Finite\n\u22a2 (fun f => \u22c2\u2080 f) ((fun a => (Ici a)\u1d9c) '' F) = \u22c2 i \u2208 F, (Ici i)\u1d9c",
+              "state_before": "case h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns F : Set \u03b1\nhF : F.Finite\n\u22a2 \u22c2 i \u2208 F, (Ici i)\u1d9c \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}}",
+              "tactic": "refine \u27e8(fun a => (Ici a)\u1d9c) '' F, \u27e8hF.image _, image_subset_iff.2 fun _ _ => \u27e8_, rfl\u27e9\u27e9, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>sInter_image<\\/a>]\"",
+                "[{\"full_name\":\"Set.sInter_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1164,9],\"def_end_pos\":[1164,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns F : Set \u03b1\nhF : F.Finite\n\u22a2 (fun f => \u22c2\u2080 f) ((fun a => (Ici a)\u1d9c) '' F) = \u22c2 i \u2208 F, (Ici i)\u1d9c",
+              "tactic": "simp only [sInter_image]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8F, \\u27e8hF, hs\\u27e9, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "state_before": "case h.e'_3.h.mpr\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns\u271d s : Set \u03b1\n\u22a2 s \u2208 (fun f => \u22c2\u2080 f) '' {f | f.Finite \u2227 f \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}} \u2192 s \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "tactic": "rintro \u27e8F, \u27e8hF, hs\u27e9, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"haveI := hF.to_subtype\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "state_before": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "tactic": "haveI := hF.to_subtype"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>subset_def<\\/a>, <a>Subtype.forall'<\\/a>] at hs\"",
+                "[{\"full_name\":\"Set.subset_def\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[322,9],\"def_end_pos\":[322,19]},{\"full_name\":\"Subtype.forall'\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[58,19],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : \u2200 (x : { a // a \u2208 F }), \u2191x \u2208 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "state_before": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : F \u2286 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "tactic": "rw [subset_def, Subtype.forall'] at hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"choose f hf using hs\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nthis : Finite \u2191F\nf : { a // a \u2208 F } \u2192 \u03b1\nhf : \u2200 (x : { a // a \u2208 F }), (Ici (f x))\u1d9c = \u2191x\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "state_before": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nhs : \u2200 (x : { a // a \u2208 F }), \u2191x \u2208 {s | \u2203 a, (Ici a)\u1d9c = s}\nthis : Finite \u2191F\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "tactic": "choose f hf using hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, <a>finite_range<\\/a> f, by simp_rw [<a>biInter_range<\\/a>, hf, <a>sInter_eq_iInter<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"Set.finite_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[868,9],\"def_end_pos\":[868,21]},{\"full_name\":\"Set.biInter_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1685,9],\"def_end_pos\":[1685,22]},{\"full_name\":\"Set.sInter_eq_iInter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1316,9],\"def_end_pos\":[1316,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nthis : Finite \u2191F\nf : { a // a \u2208 F } \u2192 \u03b1\nhf : \u2200 (x : { a // a \u2208 F }), (Ici (f x))\u1d9c = \u2191x\n\u22a2 (fun f => \u22c2\u2080 f) F \u2208 {s | \u2203 t, t.Finite \u2227 \u22c2 i \u2208 t, (Ici i)\u1d9c = s}",
+              "tactic": "exact \u27e8_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>biInter_range<\\/a>, hf, <a>sInter_eq_iInter<\\/a>]\"",
+                "[{\"full_name\":\"Set.biInter_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1685,9],\"def_end_pos\":[1685,22]},{\"full_name\":\"Set.sInter_eq_iInter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1316,9],\"def_end_pos\":[1316,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : ?m.7266\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsLower \u03b1\ns : Set \u03b1\nF : Set (Set \u03b1)\nhF : F.Finite\nthis : Finite \u2191F\nf : { a // a \u2208 F } \u2192 \u03b1\nhf : \u2200 (x : { a // a \u2208 F }), (Ici (f x))\u1d9c = \u2191x\n\u22a2 \u22c2 i \u2208 range f, (Ici i)\u1d9c = (fun f => \u22c2\u2080 f) F",
+              "tactic": "simp_rw [biInter_range, hf, sInter_eq_iInter]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLower.isUpperSet_of_isClosed",
+        "id": "Topology.IsLower.isUpperSet_of_isClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            254,
+            60
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsLower.isUpperSet_of_isClosed",
+          "start": [
+            253,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLower.topology_eq",
+        "id": "Topology.IsLower.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            210,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsLower.topology_eq",
+          "start": [
+            210,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.closure_singleton",
+        "id": "Topology.IsUpper.closure_singleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            360,
+            41
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.closure_singleton",
+          "start": [
+            359,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.continuous_iff_Iic",
+        "id": "Topology.IsUpper.continuous_iff_Iic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            369,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.continuous_iff_Iic",
+          "start": [
+            365,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.continuous_of_Iic",
+        "id": "Topology.IsUpper.continuous_of_Iic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            376,
+            25
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.continuous_of_Iic",
+          "start": [
+            371,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.isClosed_lowerClosure",
+        "id": "Topology.IsUpper.isClosed_lowerClosure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            345,
+            45
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.isClosed_lowerClosure",
+          "start": [
+            344,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.isLowerSet_of_isClosed",
+        "id": "Topology.IsUpper.isLowerSet_of_isClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            352,
+            60
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.isLowerSet_of_isClosed",
+          "start": [
+            351,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.isTopologicalBasis",
+        "id": "Topology.IsUpper.isTopologicalBasis",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            363,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.isTopologicalBasis",
+          "start": [
+            362,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.isUpperSet_of_isOpen",
+        "id": "Topology.IsUpper.isUpperSet_of_isOpen",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            349,
+            44
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.isUpperSet_of_isOpen",
+          "start": [
+            348,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpper.topology_eq",
+        "id": "Topology.IsUpper.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            323,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.IsUpper.topology_eq",
+          "start": [
+            323,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.isOpen_def",
+        "id": "Topology.WithLower.isOpen_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            10
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.isOpen_def",
+          "start": [
+            123,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.isOpen_preimage_ofLower",
+        "id": "Topology.WithLower.isOpen_preimage_ofLower",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            87
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.isOpen_preimage_ofLower",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.ofLower_inj",
+        "id": "Topology.WithLower.ofLower_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            10
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.ofLower_inj",
+          "start": [
+            102,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.ofLower_toLower",
+        "id": "Topology.WithLower.ofLower_toLower",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.ofLower_toLower",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.of_WithLower_symm_eq",
+        "id": "Topology.WithLower.of_WithLower_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            89,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.of_WithLower_symm_eq",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.toLower_inj",
+        "id": "Topology.WithLower.toLower_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.toLower_inj",
+          "start": [
+            98,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.toLower_ofLower",
+        "id": "Topology.WithLower.toLower_ofLower",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            92,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.toLower_ofLower",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLower.to_WithLower_symm_eq",
+        "id": "Topology.WithLower.to_WithLower_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithLower.to_WithLower_symm_eq",
+          "start": [
+            86,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.isOpen_def",
+        "id": "Topology.WithUpper.isOpen_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            161,
+            98
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.isOpen_def",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.isOpen_preimage_ofUpper",
+        "id": "Topology.WithUpper.isOpen_preimage_ofUpper",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            87
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.isOpen_preimage_ofUpper",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.ofUpper_inj",
+        "id": "Topology.WithUpper.ofUpper_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            144,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.ofUpper_inj",
+          "start": [
+            144,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.ofUpper_toUpper",
+        "id": "Topology.WithUpper.ofUpper_toUpper",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            142,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.ofUpper_toUpper",
+          "start": [
+            142,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.of_WithUpper_symm_eq",
+        "id": "Topology.WithUpper.of_WithUpper_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.of_WithUpper_symm_eq",
+          "start": [
+            140,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.toUpper_inj",
+        "id": "Topology.WithUpper.toUpper_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            143,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.toUpper_inj",
+          "start": [
+            143,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.toUpper_ofUpper",
+        "id": "Topology.WithUpper.toUpper_ofUpper",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            141,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.toUpper_ofUpper",
+          "start": [
+            141,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpper.to_WithUpper_symm_eq",
+        "id": "Topology.WithUpper.to_WithUpper_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            139,
+            76
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.WithUpper.to_WithUpper_symm_eq",
+          "start": [
+            139,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.isLower_orderDual",
+        "id": "Topology.isLower_orderDual",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            461,
+            25
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "Topology.isLower_orderDual",
+          "start": [
+            460,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "sSupHom.continuous",
+        "id": "sSupHom.continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            447,
+            66
+          ],
+          "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+          "full_name": "sSupHom.continuous",
+          "start": [
+            446,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/LowerUpperTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_ciInf_of_continuousAt",
+        "id": "Antitone.map_ciInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            278,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_ciInf_of_continuousAt",
+          "start": [
+            275,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_ciSup_of_continuousAt",
+        "id": "Antitone.map_ciSup_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            262,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_ciSup_of_continuousAt",
+          "start": [
+            259,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_csInf_of_continuousAt",
+        "id": "Antitone.map_csInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            270,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_csInf_of_continuousAt",
+          "start": [
+            267,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_csSup_of_continuousAt",
+        "id": "Antitone.map_csSup_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            254,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_csSup_of_continuousAt",
+          "start": [
+            251,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_iInf_of_continuousAt",
+        "id": "Antitone.map_iInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            183,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_iInf_of_continuousAt",
+          "start": [
+            180,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_iSup_of_continuousAt",
+        "id": "Antitone.map_iSup_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_iSup_of_continuousAt",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_sInf_of_continuousAt",
+        "id": "Antitone.map_sInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            9
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_sInf_of_continuousAt",
+          "start": [
+            172,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_sInf_of_continuousAt'",
+        "id": "Antitone.map_sInf_of_continuousAt'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_sInf_of_continuousAt'",
+          "start": [
+            67,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.map_sSup_of_continuousAt'",
+        "id": "Antitone.map_sSup_of_continuousAt'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Antitone.map_sSup_of_continuousAt'",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.csInf_mem",
+        "id": "IsClosed.csInf_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            208,
+            43
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.csInf_mem",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.csSup_mem",
+        "id": "IsClosed.csSup_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            43
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.csSup_mem",
+          "start": [
+            201,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.isGreatest_csSup",
+        "id": "IsClosed.isGreatest_csSup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            44
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.isGreatest_csSup",
+          "start": [
+            215,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.isLeast_csInf",
+        "id": "IsClosed.isLeast_csInf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            44
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.isLeast_csInf",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.sInf_mem",
+        "id": "IsClosed.sInf_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            119,
+            39
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.sInf_mem",
+          "start": [
+            118,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsClosed.sSup_mem",
+        "id": "IsClosed.sSup_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            39
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "IsClosed.sSup_mem",
+          "start": [
+            114,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.map_ciInf_of_continuousAt",
+        "id": "Monotone.map_ciInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            246,
+            72
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.map_ciInf_of_continuousAt",
+          "start": [
+            244,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.map_csInf_of_continuousAt",
+        "id": "Monotone.map_csInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            239,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.map_csInf_of_continuousAt",
+          "start": [
+            237,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.map_iInf_of_continuousAt",
+        "id": "Monotone.map_iInf_of_continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            150,
+            74
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.map_iInf_of_continuousAt",
+          "start": [
+            148,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.map_sInf_of_continuousAt'",
+        "id": "Monotone.map_sInf_of_continuousAt'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            85
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.map_sInf_of_continuousAt'",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [h]'",
+        "full_name": "Monotone.tendsto_nhdsWithin_Iio",
+        "id": "Monotone.tendsto_nhdsWithin_Iio",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            292,
+            74
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.tendsto_nhdsWithin_Iio",
+          "start": [
+            282,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>eq_empty_or_nonempty<\\/a> (<a>Iio<\\/a> x) with (h | h)\"",
+                "[{\"full_name\":\"Set.eq_empty_or_nonempty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[605,9],\"def_end_pos\":[605,29]},{\"full_name\":\"Set.Iio\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,8]}]"
+              ],
+              "state_after": "case inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : Iio x = \u2205\n\u22a2 Tendsto f (\ud835\udcdd[<] x) (\ud835\udcdd (sSup (f '' Iio x)))\n\ncase inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\n\u22a2 Tendsto f (\ud835\udcdd[<] x) (\ud835\udcdd (sSup (f '' Iio x)))",
+              "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\n\u22a2 Tendsto f (\ud835\udcdd[<] x) (\ud835\udcdd (sSup (f '' Iio x)))",
+              "tactic": "rcases eq_empty_or_nonempty (Iio x) with (h | h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>tendsto_order<\\/a>.2 \\u27e8fun l hl => ?_, fun m hm => ?_\\u27e9\"",
+                "[{\"full_name\":\"tendsto_order\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,22]}]"
+              ],
+              "state_after": "case inr.refine_1\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nl : \u03b2\nhl : l < sSup (f '' Iio x)\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, l < f b\n\ncase inr.refine_2\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nm : \u03b2\nhm : m > sSup (f '' Iio x)\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, f b < m",
+              "state_before": "case inr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\n\u22a2 Tendsto f (\ud835\udcdd[<] x) (\ud835\udcdd (sSup (f '' Iio x)))",
+              "tactic": "refine tendsto_order.2 \u27e8fun l hl => ?_, fun m hm => ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : Iio x = \u2205\n\u22a2 Tendsto f (\ud835\udcdd[<] x) (\ud835\udcdd (sSup (f '' Iio x)))",
+              "tactic": "simp [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8z, zx, lz\\u27e9 : \\u2203 a : \\u03b1, a < x \\u2227 l < f a := by\\n      simpa only [<a>mem_image<\\/a>, <a>exists_prop<\\/a>, <a>exists_exists_and_eq_and<\\/a>] using\\n        <a>exists_lt_of_lt_csSup<\\/a> (h.image _) hl\"",
+                "[{\"full_name\":\"Set.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"exists_prop\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[307,17],\"def_end_pos\":[307,28]},{\"full_name\":\"exists_exists_and_eq_and\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[794,17],\"def_end_pos\":[794,41]},{\"full_name\":\"exists_lt_of_lt_csSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[1005,9],\"def_end_pos\":[1005,30]}]"
+              ],
+              "state_after": "case inr.refine_1.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nl : \u03b2\nhl : l < sSup (f '' Iio x)\nz : \u03b1\nzx : z < x\nlz : l < f z\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, l < f b",
+              "state_before": "case inr.refine_1\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nl : \u03b2\nhl : l < sSup (f '' Iio x)\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, l < f b",
+              "tactic": "obtain \u27e8z, zx, lz\u27e9 : \u2203 a : \u03b1, a < x \u2227 l < f a := by\n  simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using\n    exists_lt_of_lt_csSup (h.image _) hl"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mem_of_superset<\\/a> (<a>Ioo_mem_nhdsWithin_Iio'<\\/a> zx) fun y hy => lz.trans_le (Mf hy.1.<a>le<\\/a>)\"",
+                "[{\"full_name\":\"Filter.mem_of_superset\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,24]},{\"full_name\":\"Ioo_mem_nhdsWithin_Iio'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[260,9],\"def_end_pos\":[260,32]},{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.refine_1.intro.intro\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nl : \u03b2\nhl : l < sSup (f '' Iio x)\nz : \u03b1\nzx : z < x\nlz : l < f z\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, l < f b",
+              "tactic": "exact mem_of_superset (Ioo_mem_nhdsWithin_Iio' zx) fun y hy => lz.trans_le (Mf hy.1.le)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>mem_image<\\/a>, <a>exists_prop<\\/a>, <a>exists_exists_and_eq_and<\\/a>] using\\n        <a>exists_lt_of_lt_csSup<\\/a> (h.image _) hl\"",
+                "[{\"full_name\":\"Set.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"exists_prop\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[307,17],\"def_end_pos\":[307,28]},{\"full_name\":\"exists_exists_and_eq_and\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[794,17],\"def_end_pos\":[794,41]},{\"full_name\":\"exists_lt_of_lt_csSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[1005,9],\"def_end_pos\":[1005,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nl : \u03b2\nhl : l < sSup (f '' Iio x)\n\u22a2 \u2203 a < x, l < f a",
+              "tactic": "simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using\n  exists_lt_of_lt_csSup (h.image _) hl"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>mem_of_superset<\\/a> <a>self_mem_nhdsWithin<\\/a> fun _ hy => <a>lt_of_le_of_lt<\\/a> ?_ hm\"",
+                "[{\"full_name\":\"Filter.mem_of_superset\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,24]},{\"full_name\":\"self_mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,28]},{\"full_name\":\"lt_of_le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,23]}]"
+              ],
+              "state_after": "case inr.refine_2\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nm : \u03b2\nhm : m > sSup (f '' Iio x)\nx\u271d : \u03b1\nhy : x\u271d \u2208 Iio x\n\u22a2 f x\u271d \u2264 sSup (f '' Iio x)",
+              "state_before": "case inr.refine_2\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nm : \u03b2\nhm : m > sSup (f '' Iio x)\n\u22a2 \u2200\u1da0 (b : \u03b1) in \ud835\udcdd[<] x, f b < m",
+              "tactic": "refine mem_of_superset self_mem_nhdsWithin fun _ hy => lt_of_le_of_lt ?_ hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_csSup<\\/a> (Mf.map_bddAbove <a>bddAbove_Iio<\\/a>) (<a>mem_image_of_mem<\\/a> _ hy)\"",
+                "[{\"full_name\":\"le_csSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[459,9],\"def_end_pos\":[459,17]},{\"full_name\":\"bddAbove_Iio\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[553,9],\"def_end_pos\":[553,21]},{\"full_name\":\"Set.mem_image_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[132,9],\"def_end_pos\":[132,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.refine_2\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9\u00b2 : ConditionallyCompleteLinearOrder \u03b1\u271d\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2070 : OrderTopology \u03b1\u271d\ninst\u271d\u2079 : ConditionallyCompleteLinearOrder \u03b2\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\u271d\ninst\u271d\u2077 : OrderClosedTopology \u03b2\u271d\ninst\u271d\u2076 : Nonempty \u03b3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nMf : Monotone f\nx : \u03b1\nh : (Iio x).Nonempty\nm : \u03b2\nhm : m > sSup (f '' Iio x)\nx\u271d : \u03b1\nhy : x\u271d \u2208 Iio x\n\u22a2 f x\u271d \u2264 sSup (f '' Iio x)",
+              "tactic": "exact le_csSup (Mf.map_bddAbove bddAbove_Iio) (mem_image_of_mem _ hy)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.tendsto_nhdsWithin_Ioi",
+        "id": "Monotone.tendsto_nhdsWithin_Ioi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            299,
+            66
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "Monotone.tendsto_nhdsWithin_Ioi",
+          "start": [
+            296,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "csInf_mem_closure",
+        "id": "csInf_mem_closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            198,
+            36
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "csInf_mem_closure",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "csSup_mem_closure",
+        "id": "csSup_mem_closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            194,
+            36
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "csSup_mem_closure",
+          "start": [
+            193,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "sInf_mem_closure",
+        "id": "sInf_mem_closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            32
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "sInf_mem_closure",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "sSup_mem_closure",
+        "id": "sSup_mem_closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            32
+          ],
+          "file_path": "Mathlib/Topology/Order/Monotone.lean",
+          "full_name": "sSup_mem_closure",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Monotone.lean"
+      }
+    ],
+    "input_proofs": 70,
+    "input_units": 2,
+    "proofs": 14,
+    "published_proofs": 12,
+    "published_transitions": 26,
+    "replay_failures": [],
+    "transitions": 32,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+        "proof_id": "mathlib:Topology.isUpper_orderDual",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              458,
+              32
+            ],
+            "file_path": "Mathlib/Topology/Order/LowerUpperTopology.lean",
+            "full_name": "Topology.isUpper_orderDual",
+            "start": [
+              455,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : ?m.39958\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsUpper \u03b1\u1d52\u1d48 \u2192 IsLower \u03b1\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : ?m.39958\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsLower \u03b1 \u2192 IsUpper \u03b1\u1d52\u1d48",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : ?m.39958\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsUpper \u03b1\u1d52\u1d48 \u2194 IsLower \u03b1",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>OrderDual.instIsLower<\\/a>\"",
+                  "[{\"full_name\":\"OrderDual.instIsLower\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[335,10],\"def_end_pos\":[335,38]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : ?m.39958\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsUpper \u03b1\u1d52\u1d48 \u2192 IsLower \u03b1",
+                "tactic": "apply OrderDual.instIsLower"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>OrderDual.instIsUpper<\\/a>\"",
+                  "[{\"full_name\":\"OrderDual.instIsUpper\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/LowerUpperTopology.lean\",\"def_pos\":[224,10],\"def_end_pos\":[224,38]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : ?m.39958\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsLower \u03b1 \u2192 IsUpper \u03b1\u1d52\u1d48",
+                "tactic": "apply OrderDual.instIsUpper"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.isUpper_orderDual"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/Monotone.lean",
+        "proof_id": "mathlib:Monotone.map_sSup_of_continuousAt",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              128,
+              44
+            ],
+            "file_path": "Mathlib/Topology/Order/Monotone.lean",
+            "full_name": "Monotone.map_sSup_of_continuousAt",
+            "start": [
+              124,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases s.eq_empty_or_nonempty with h | h\"",
+                  "[]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nCf : ContinuousAt f (sSup s)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\nh : s = \u2205\n\u22a2 f (sSup s) = sSup (f '' s)\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nCf : ContinuousAt f (sSup s)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\nh : s.Nonempty\n\u22a2 f (sSup s) = sSup (f '' s)",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nCf : ContinuousAt f (sSup s)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\n\u22a2 f (sSup s) = sSup (f '' s)",
+                "tactic": "rcases s.eq_empty_or_nonempty with h | h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [h, fbot]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nCf : ContinuousAt f (sSup s)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\nh : s = \u2205\n\u22a2 f (sSup s) = sSup (f '' s)",
+                "tactic": "simp [h, fbot]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact Mf.map_sSup_of_continuousAt' Cf h\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nCf : ContinuousAt f (sSup s)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\nh : s.Nonempty\n\u22a2 f (sSup s) = sSup (f '' s)",
+                "tactic": "exact Mf.map_sSup_of_continuousAt' Cf h"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Monotone.map_sSup_of_continuousAt"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001942-0000-b1d876cd4542da56.parquet",
+      "sha256": "b1d876cd4542da561895b6646012c3b0c0c9274b0639290c24755245dc583091",
+      "bytes": 98932,
+      "rows": 24,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/dev/mathlib-001942-0000-ace933c4c3c07e50.parquet",
+      "sha256": "ace933c4c3c07e50881aad0d060fef9cc1c80bdd5959dc292799132a576e418f",
+      "bytes": 61308,
+      "rows": 2,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Antitone.map_iInf_of_continuousAt'",
+    "mathlib:Antitone.map_iSup_of_continuousAt'",
+    "mathlib:Monotone.map_ciSup_of_continuousAt",
+    "mathlib:Monotone.map_csSup_of_continuousAt",
+    "mathlib:Monotone.map_iInf_of_continuousAt'",
+    "mathlib:Monotone.map_iSup_of_continuousAt",
+    "mathlib:Monotone.map_iSup_of_continuousAt'",
+    "mathlib:Topology.IsLower.continuous_iff_Ici",
+    "mathlib:Topology.IsLower.isClosed_upperClosure",
+    "mathlib:Topology.IsLower.isOpen_iff_generate_Ici_compl",
+    "mathlib:Topology.IsUpper.isOpen_iff_generate_Iic_compl",
+    "mathlib:sInfHom.continuous"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LowerUpperTopology.lean\",null,null,\"Topology.IsLower.continuous_iff_Ici\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LowerUpperTopology.lean\",null,null,\"Topology.IsLower.isClosed_upperClosure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LowerUpperTopology.lean\",null,null,\"Topology.IsLower.isOpen_iff_generate_Ici_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LowerUpperTopology.lean\",null,null,\"Topology.IsUpper.isOpen_iff_generate_Iic_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/LowerUpperTopology.lean\",null,null,\"sInfHom.continuous\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Antitone.map_iInf_of_continuousAt'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Antitone.map_iSup_of_continuousAt'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Monotone.map_ciSup_of_continuousAt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Monotone.map_csSup_of_continuousAt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Monotone.map_iInf_of_continuousAt'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Monotone.map_iSup_of_continuousAt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/Monotone.lean\",null,null,\"Monotone.map_iSup_of_continuousAt'\"]"
+  ]
+}

manifests/batches/mathlib/001943.json

@@ -0,0 +1,1332 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001943",
+  "batch_fingerprint": "b64bc5844fd6bf9a31f319fbb9d593f3a6e153c4c940042b08e3f51c42777a03",
+  "evidence": {
+    "batch_fingerprint": "b64bc5844fd6bf9a31f319fbb9d593f3a6e153c4c940042b08e3f51c42777a03",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.continuous_of_surjective",
+        "id": "Monotone.continuous_of_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            286,
+            52
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "Monotone.continuous_of_surjective",
+          "start": [
+            284,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "OrderIso.coe_toHomeomorph",
+        "id": "OrderIso.coe_toHomeomorph",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            322,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "OrderIso.coe_toHomeomorph",
+          "start": [
+            321,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "OrderIso.coe_toHomeomorph_symm",
+        "id": "OrderIso.coe_toHomeomorph_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            327,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "OrderIso.coe_toHomeomorph_symm",
+          "start": [
+            326,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousAt_of_closure_image_mem_nhds",
+        "id": "StrictMonoOn.continuousAt_of_closure_image_mem_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            229,
+            74
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousAt_of_closure_image_mem_nhds",
+          "start": [
+            222,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousAt_of_exists_between",
+        "id": "StrictMonoOn.continuousAt_of_exists_between",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            216,
+            95
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousAt_of_exists_between",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousAt_of_image_mem_nhds",
+        "id": "StrictMonoOn.continuousAt_of_image_mem_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            88
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousAt_of_image_mem_nhds",
+          "start": [
+            234,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin",
+        "id": "StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            188,
+            78
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin",
+          "start": [
+            185,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_left_of_exists_between",
+        "id": "StrictMonoOn.continuousWithinAt_left_of_exists_between",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            144,
+            23
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_left_of_exists_between",
+          "start": [
+            139,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin",
+        "id": "StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            197,
+            70
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin",
+          "start": [
+            194,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_left_of_surjOn",
+        "id": "StrictMonoOn.continuousWithinAt_left_of_surjOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            205,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_left_of_surjOn",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin",
+        "id": "StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            57
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin",
+        "id": "StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            119,
+            41
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousAt_of_monotoneOn_of_closure_image_mem_nhds",
+        "id": "continuousAt_of_monotoneOn_of_closure_image_mem_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            263,
+            74
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousAt_of_monotoneOn_of_closure_image_mem_nhds",
+          "start": [
+            256,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousAt_of_monotoneOn_of_exists_between",
+        "id": "continuousAt_of_monotoneOn_of_exists_between",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            250,
+            47
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousAt_of_monotoneOn_of_exists_between",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousAt_of_monotoneOn_of_image_mem_nhds",
+        "id": "continuousAt_of_monotoneOn_of_image_mem_nhds",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            271,
+            41
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousAt_of_monotoneOn_of_image_mem_nhds",
+          "start": [
+            268,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin",
+        "id": "continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            21
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousWithinAt_left_of_monotoneOn_of_exists_between",
+        "id": "continuousWithinAt_left_of_monotoneOn_of_exists_between",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            23
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousWithinAt_left_of_monotoneOn_of_exists_between",
+          "start": [
+            153,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin",
+        "id": "continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            179,
+            41
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin",
+        "id": "continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            39
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+          "full_name": "continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneContinuity.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.ge_of_tendsto",
+        "id": "Antitone.ge_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            284,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "Antitone.ge_of_tendsto",
+          "start": [
+            281,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Antitone.le_of_tendsto",
+        "id": "Antitone.le_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            278,
+            35
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "Antitone.le_of_tendsto",
+          "start": [
+            275,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.ge_of_tendsto",
+        "id": "Monotone.ge_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            266,
+            77
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "Monotone.ge_of_tendsto",
+          "start": [
+            262,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Monotone.le_of_tendsto",
+        "id": "Monotone.le_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            272,
+            29
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "Monotone.le_of_tendsto",
+          "start": [
+            269,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "iInf_eq_iInf_subseq_of_antitone",
+        "id": "iInf_eq_iInf_subseq_of_antitone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            353,
+            45
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "iInf_eq_iInf_subseq_of_antitone",
+          "start": [
+            350,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "iInf_eq_iInf_subseq_of_monotone",
+        "id": "iInf_eq_iInf_subseq_of_monotone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            347,
+            45
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "iInf_eq_iInf_subseq_of_monotone",
+          "start": [
+            344,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "iInf_eq_of_tendsto",
+        "id": "iInf_eq_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            323,
+            46
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "iInf_eq_of_tendsto",
+          "start": [
+            320,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "iSup_eq_iSup_subseq_of_antitone",
+        "id": "iSup_eq_iSup_subseq_of_antitone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            341,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "iSup_eq_iSup_subseq_of_antitone",
+          "start": [
+            335,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "iSup_eq_of_tendsto",
+        "id": "iSup_eq_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            317,
+            46
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "iSup_eq_of_tendsto",
+          "start": [
+            314,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isGLB_of_tendsto_atBot",
+        "id": "isGLB_of_tendsto_atBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            299,
+            59
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "isGLB_of_tendsto_atBot",
+          "start": [
+            296,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isGLB_of_tendsto_atTop",
+        "id": "isGLB_of_tendsto_atTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            311,
+            62
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "isGLB_of_tendsto_atTop",
+          "start": [
+            308,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isLUB_of_tendsto_atBot",
+        "id": "isLUB_of_tendsto_atBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            305,
+            62
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "isLUB_of_tendsto_atBot",
+          "start": [
+            302,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_atBot_iInf",
+        "id": "tendsto_atBot_iInf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            51
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_atBot_iInf",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_atBot_iSup",
+        "id": "tendsto_atBot_iSup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            162,
+            51
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_atBot_iSup",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_atTop_iInf",
+        "id": "tendsto_atTop_iInf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            176,
+            51
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_atTop_iInf",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_atTop_iSup",
+        "id": "tendsto_atTop_iSup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            158,
+            51
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_atTop_iSup",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_iff_tendsto_subseq_of_antitone",
+        "id": "tendsto_iff_tendsto_subseq_of_antitone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            252,
+            58
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_iff_tendsto_subseq_of_antitone",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_of_antitone",
+        "id": "tendsto_of_antitone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            234,
+            46
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_of_antitone",
+          "start": [
+            231,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "tendsto_of_monotone",
+        "id": "tendsto_of_monotone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            228,
+            59
+          ],
+          "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+          "full_name": "tendsto_of_monotone",
+          "start": [
+            224,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/MonotoneConvergence.lean"
+      }
+    ],
+    "input_proofs": 53,
+    "input_units": 2,
+    "proofs": 15,
+    "published_proofs": 10,
+    "published_transitions": 16,
+    "replay_failures": [],
+    "transitions": 49,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+        "proof_id": "mathlib:OrderIso.continuous",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              310,
+              21
+            ],
+            "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+            "full_name": "OrderIso.continuous",
+            "start": [
+              304,
+              11
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2039OrderTopology \\u03b2\\u203a.<a>topology_eq_generate_intervals<\\/a>, <a>continuous_generateFrom_iff<\\/a>]\"",
+                  "[{\"full_name\":\"OrderTopology.topology_eq_generate_intervals\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[69,3],\"def_end_pos\":[69,33]},{\"full_name\":\"continuous_generateFrom_iff\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[688,7],\"def_end_pos\":[688,34]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 s \u2208 {s | \u2203 a, s = Ioi a \u2228 s = Iio a}, IsOpen (\u21d1e \u207b\u00b9' s)",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\n\u22a2 Continuous \u21d1e",
+                "tactic": "rw [\u2039OrderTopology \u03b2\u203a.topology_eq_generate_intervals, continuous_generateFrom_iff]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro s \\u27e8a, rfl | rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (\u21d1e \u207b\u00b9' Ioi a)\n\ncase intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (\u21d1e \u207b\u00b9' Iio a)",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\n\u22a2 \u2200 s \u2208 {s | \u2203 a, s = Ioi a \u2228 s = Iio a}, IsOpen (\u21d1e \u207b\u00b9' s)",
+                "tactic": "rintro s \u27e8a, rfl | rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [e.preimage_Ioi]\"",
+                  "[]"
+                ],
+                "state_after": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (Ioi (e.symm a))",
+                "state_before": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (\u21d1e \u207b\u00b9' Ioi a)",
+                "tactic": "rw [e.preimage_Ioi]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>isOpen_lt'<\\/a>\"",
+                  "[{\"full_name\":\"isOpen_lt'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[96,9],\"def_end_pos\":[96,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (Ioi (e.symm a))",
+                "tactic": "apply isOpen_lt'"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [e.preimage_Iio]\"",
+                  "[]"
+                ],
+                "state_after": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (Iio (e.symm a))",
+                "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (\u21d1e \u207b\u00b9' Iio a)",
+                "tactic": "rw [e.preimage_Iio]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>isOpen_gt'<\\/a>\"",
+                  "[{\"full_name\":\"isOpen_gt'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : PartialOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : OrderTopology \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b2\n\u22a2 IsOpen (Iio (e.symm a))",
+                "tactic": "apply isOpen_gt'"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "OrderIso.continuous"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+        "proof_id": "mathlib:StrictMonoOn.continuousWithinAt_right_of_exists_between",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              54,
+              57
+            ],
+            "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+            "full_name": "StrictMonoOn.continuousWithinAt_right_of_exists_between",
+            "start": [
+              42,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"have ha : a \\u2208 <a>Ici<\\/a> a := <a>left_mem_Ici<\\/a>\"",
+                  "[{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]},{\"full_name\":\"Set.left_mem_Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,21]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "have ha : a \u2208 Ici a := left_mem_Ici"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have has : a \\u2208 s := <a>mem_of_mem_nhdsWithin<\\/a> ha hs\"",
+                  "[{\"full_name\":\"mem_of_mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[163,9],\"def_end_pos\":[163,30]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "have has : a \u2208 s := mem_of_mem_nhdsWithin ha hs"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>tendsto_order<\\/a>.2 \\u27e8fun b hb => ?_, fun b hb => ?_\\u27e9\"",
+                  "[{\"full_name\":\"tendsto_order\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,22]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, b < f b_1\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "refine tendsto_order.2 \u27e8fun b hb => ?_, fun b hb => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [hs, @<a>self_mem_nhdsWithin<\\/a> _ _ a (<a>Ici<\\/a> a)] with _ hxs hxa using hb.trans_le\\n      ((h_mono.le_iff_le has hxs).2 hxa)\"",
+                  "[{\"full_name\":\"self_mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,28]},{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, b < f b_1",
+                "tactic": "filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le\n  ((h_mono.le_iff_le has hxs).2 hxa)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases hfs b hb with \\u27e8c, hcs, hac, hcb\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "rcases hfs b hb with \u27e8c, hcs, hac, hcb\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [h_mono.lt_iff_lt has hcs] at hac\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "rw [h_mono.lt_iff_lt has hcs] at hac"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [hs, <a>Ico_mem_nhdsWithin_Ici<\\/a> (<a>left_mem_Ico<\\/a>.2 hac)]\"",
+                  "[{\"full_name\":\"Ico_mem_nhdsWithin_Ici\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[572,9],\"def_end_pos\":[572,31]},{\"full_name\":\"Set.left_mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,21]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200 a_1 \u2208 s, a_1 \u2208 Ico a c \u2192 f a_1 < b",
+                "state_before": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "filter_upwards [hs, Ico_mem_nhdsWithin_Ici (left_mem_Ico.2 hac)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro x hx \\u27e8_, hxc\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200 a_1 \u2208 s, a_1 \u2208 Ico a c \u2192 f a_1 < b",
+                "tactic": "rintro x hx \u27e8_, hxc\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact ((h_mono.lt_iff_lt hx hcs).2 hxc).<a>trans_le<\\/a> hcb\"",
+                  "[{\"full_name\":\"LT.lt.trans_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[143,7],\"def_end_pos\":[143,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b",
+                "tactic": "exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "StrictMonoOn.continuousWithinAt_right_of_exists_between"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+        "proof_id": "mathlib:continuousWithinAt_right_of_monotoneOn_of_exists_between",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              75,
+              46
+            ],
+            "file_path": "Mathlib/Topology/Order/MonotoneContinuity.lean",
+            "full_name": "continuousWithinAt_right_of_monotoneOn_of_exists_between",
+            "start": [
+              63,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"have ha : a \\u2208 <a>Ici<\\/a> a := <a>left_mem_Ici<\\/a>\"",
+                  "[{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]},{\"full_name\":\"Set.left_mem_Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,21]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "have ha : a \u2208 Ici a := left_mem_Ici"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have has : a \\u2208 s := <a>mem_of_mem_nhdsWithin<\\/a> ha hs\"",
+                  "[{\"full_name\":\"mem_of_mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[163,9],\"def_end_pos\":[163,30]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "have has : a \u2208 s := mem_of_mem_nhdsWithin ha hs"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>tendsto_order<\\/a>.2 \\u27e8fun b hb => ?_, fun b hb => ?_\\u27e9\"",
+                  "[{\"full_name\":\"tendsto_order\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,22]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, b < f b_1\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a",
+                "tactic": "refine tendsto_order.2 \u27e8fun b hb => ?_, fun b hb => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [hs, @<a>self_mem_nhdsWithin<\\/a> _ _ a (<a>Ici<\\/a> a)] with _ hxs hxa using hb.trans_le\\n      (h_mono has hxs hxa)\"",
+                  "[{\"full_name\":\"self_mem_nhdsWithin\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,28]},{\"full_name\":\"Set.Ici\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, b < f b_1",
+                "tactic": "filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le\n  (h_mono has hxs hxa)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases hfs b hb with \\u27e8c, hcs, hac, hcb\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "rcases hfs b hb with \u27e8c, hcs, hac, hcb\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have : a < c := <a>not_le<\\/a>.1 fun h => hac.not_le <| h_mono hcs has h\"",
+                  "[{\"full_name\":\"not_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[375,9],\"def_end_pos\":[375,15]}]"
+                ],
+                "state_after": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "state_before": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "have : a < c := not_le.1 fun h => hac.not_le <| h_mono hcs has h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [hs, <a>Ico_mem_nhdsWithin_Ici<\\/a> (<a>left_mem_Ico<\\/a>.2 this)]\"",
+                  "[{\"full_name\":\"Ico_mem_nhdsWithin_Ici\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[572,9],\"def_end_pos\":[572,31]},{\"full_name\":\"Set.left_mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,21]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\n\u22a2 \u2200 a_1 \u2208 s, a_1 \u2208 Ico a c \u2192 f a_1 < b",
+                "state_before": "case refine_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[\u2265] a, f b_1 < b",
+                "tactic": "filter_upwards [hs, Ico_mem_nhdsWithin_Ici (left_mem_Ico.2 this)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro x hx \\u27e8_, hxc\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\n\u22a2 \u2200 a_1 \u2208 s, a_1 \u2208 Ico a c \u2192 f a_1 < b",
+                "tactic": "rintro x hx \u27e8_, hxc\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (h_mono hx hcs hxc.le).<a>trans_lt<\\/a> hcb\"",
+                  "[{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : MonotoneOn f s\nhs : s \u2208 \ud835\udcdd[\u2265] a\nhfs : \u2200 b > f a, \u2203 c \u2208 s, f c \u2208 Ioo (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c < b\nthis : a < c\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b",
+                "tactic": "exact (h_mono hx hcs hxc.le).trans_lt hcb"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "continuousWithinAt_right_of_monotoneOn_of_exists_between"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+        "proof_id": "mathlib:isLUB_of_tendsto_atTop",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              293,
+              75
+            ],
+            "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+            "full_name": "isLUB_of_tendsto_atTop",
+            "start": [
+              287,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 a \u2208 upperBounds (range f)\n\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 a \u2208 lowerBounds (upperBounds (range f))",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 IsLUB (range f) a",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro _ \\u27e8b, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case left.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\nb : \u03b2\n\u22a2 f b \u2264 a",
+                "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 a \u2208 upperBounds (range f)",
+                "tactic": "rintro _ \u27e8b, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hf.ge_of_tendsto ha b\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case left.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\nb : \u03b2\n\u22a2 f b \u2264 a",
+                "tactic": "exact hf.ge_of_tendsto ha b"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun _ hb => <a>le_of_tendsto'<\\/a> ha fun x => hb (<a>Set.mem_range_self<\\/a> x)\"",
+                  "[{\"full_name\":\"le_of_tendsto'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[136,9],\"def_end_pos\":[136,23]},{\"full_name\":\"Set.mem_range_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[163,23],\"def_end_pos\":[163,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhf : Monotone f\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 a \u2208 lowerBounds (upperBounds (range f))",
+                "tactic": "exact fun _ hb => le_of_tendsto' ha fun x => hb (Set.mem_range_self x)"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "isLUB_of_tendsto_atTop"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+        "proof_id": "mathlib:tendsto_iff_tendsto_subseq_of_monotone",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              245,
+              48
+            ],
+            "file_path": "Mathlib/Topology/Order/MonotoneConvergence.lean",
+            "full_name": "tendsto_iff_tendsto_subseq_of_monotone",
+            "start": [
+              237,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor <;> intro h\"",
+                  "[]"
+                ],
+                "state_after": "case mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto f atTop (\ud835\udcdd l)\n\u22a2 Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\n\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\n\u22a2 Tendsto f atTop (\ud835\udcdd l)",
+                "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\n\u22a2 Tendsto f atTop (\ud835\udcdd l) \u2194 Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)",
+                "tactic": "constructor <;> intro h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact h.comp hg\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto f atTop (\ud835\udcdd l)\n\u22a2 Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)",
+                "tactic": "exact h.comp hg"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>tendsto_of_monotone<\\/a> hf with (h' | \\u27e8l', hl'\\u27e9)\"",
+                  "[{\"full_name\":\"tendsto_of_monotone\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/MonotoneConvergence.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,28]}]"
+                ],
+                "state_after": "case mpr.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\nh' : Tendsto f atTop atTop\n\u22a2 Tendsto f atTop (\ud835\udcdd l)\n\ncase mpr.inr.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\nl' : \u03b1\nhl' : Tendsto f atTop (\ud835\udcdd l')\n\u22a2 Tendsto f atTop (\ud835\udcdd l)",
+                "state_before": "case mpr\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\n\u22a2 Tendsto f atTop (\ud835\udcdd l)",
+                "tactic": "rcases tendsto_of_monotone hf with (h' | \u27e8l', hl'\u27e9)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>not_tendsto_atTop_of_tendsto_nhds<\\/a> h (h'.comp hg)).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"not_tendsto_atTop_of_tendsto_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[438,9],\"def_end_pos\":[438,42]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr.inl\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\nh' : Tendsto f atTop atTop\n\u22a2 Tendsto f atTop (\ud835\udcdd l)",
+                "tactic": "exact (not_tendsto_atTop_of_tendsto_nhds h (h'.comp hg)).elim"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>tendsto_nhds_unique<\\/a> h (hl'.comp hg)]\"",
+                  "[{\"full_name\":\"tendsto_nhds_unique\",\"def_path\":\"Mathlib\\/Topology\\/Separation.lean\",\"def_pos\":[1508,9],\"def_end_pos\":[1508,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr.inr.intro\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b9\u2081 : Type u_3\n\u03b9\u2082 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2076 : SemilatticeSup \u03b9\u2081\ninst\u271d\u2075 : Preorder \u03b9\u2082\ninst\u271d\u2074 : Nonempty \u03b9\u2081\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : NoMaxOrder \u03b1\nf : \u03b9\u2082 \u2192 \u03b1\n\u03c6 : \u03b9\u2081 \u2192 \u03b9\u2082\nl : \u03b1\nhf : Monotone f\nhg : Tendsto \u03c6 atTop atTop\nh : Tendsto (f \u2218 \u03c6) atTop (\ud835\udcdd l)\nl' : \u03b1\nhl' : Tendsto f atTop (\ud835\udcdd l')\n\u22a2 Tendsto f atTop (\ud835\udcdd l)",
+                "tactic": "rwa [tendsto_nhds_unique h (hl'.comp hg)]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "tendsto_iff_tendsto_subseq_of_monotone"
+      }
+    ],
+    "validation_rejected_proofs": 5,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001943-0000-c1582bd180c4e2ff.parquet",
+      "sha256": "c1582bd180c4e2ff95e785ddcbca51c0017b98cd01682913ee3dd671dfc6fa5b",
+      "bytes": 75618,
+      "rows": 15,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/internal_test/mathlib-001943-0000-b148e10070d6f52d.parquet",
+      "sha256": "b148e10070d6f52da4b8087302334651ce38ca9fee7451d80b61dd532347d36f",
+      "bytes": 38284,
+      "rows": 1,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Monotone.continuous_of_denseRange",
+    "mathlib:continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin",
+    "mathlib:tendsto_atBot_ciInf",
+    "mathlib:tendsto_atBot_ciSup",
+    "mathlib:tendsto_atBot_isGLB",
+    "mathlib:tendsto_atBot_isLUB",
+    "mathlib:tendsto_atTop_ciInf",
+    "mathlib:tendsto_atTop_ciSup",
+    "mathlib:tendsto_atTop_isGLB",
+    "mathlib:tendsto_atTop_isLUB"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneContinuity.lean\",null,null,\"Monotone.continuous_of_denseRange\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneContinuity.lean\",null,null,\"continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atBot_ciInf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atBot_ciSup\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atBot_isGLB\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atBot_isLUB\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atTop_ciInf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atTop_ciSup\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atTop_isGLB\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/MonotoneConvergence.lean\",null,null,\"tendsto_atTop_isLUB\"]"
+  ]
+}

manifests/batches/mathlib/001945.json

@@ -0,0 +1,282 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001945",
+  "batch_fingerprint": "f745b025ce93773ac53fe37f74f2312136d84afd17c803865acf3ea9947cb866",
+  "evidence": {
+    "batch_fingerprint": "f745b025ce93773ac53fe37f74f2312136d84afd17c803865acf3ea9947cb866",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.partialSups",
+        "id": "Continuous.partialSups",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            99
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "Continuous.partialSups",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.partialSups_apply",
+        "id": "Continuous.partialSups_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "Continuous.partialSups_apply",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.partialSups_apply",
+        "id": "ContinuousAt.partialSups_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            31
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "ContinuousAt.partialSups_apply",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.partialSups",
+        "id": "ContinuousOn.partialSups",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            57
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "ContinuousOn.partialSups",
+          "start": [
+            59,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.partialSups_apply",
+        "id": "ContinuousOn.partialSups_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            63
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "ContinuousOn.partialSups_apply",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.partialSups_apply",
+        "id": "ContinuousWithinAt.partialSups_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            31
+          ],
+          "file_path": "Mathlib/Topology/Order/PartialSups.lean",
+          "full_name": "ContinuousWithinAt.partialSups_apply",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/PartialSups.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "exists_isClopen_lower_of_not_le",
+        "id": "exists_isClopen_lower_of_not_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/Priestley.lean",
+          "full_name": "exists_isClopen_lower_of_not_le",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Priestley.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "exists_isClopen_upper_of_not_le",
+        "id": "exists_isClopen_upper_of_not_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            27
+          ],
+          "file_path": "Mathlib/Topology/Order/Priestley.lean",
+          "full_name": "exists_isClopen_upper_of_not_le",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/Priestley.lean"
+      }
+    ],
+    "input_proofs": 13,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 4,
+    "published_transitions": 6,
+    "replay_failures": [],
+    "transitions": 10,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/Priestley.lean",
+        "proof_id": "mathlib:exists_isClopen_upper_or_lower_of_ne",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              72,
+              38
+            ],
+            "file_path": "Mathlib/Topology/Order/Priestley.lean",
+            "full_name": "exists_isClopen_upper_or_lower_of_ne",
+            "start": [
+              67,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain h | h := h.not_le_or_not_le\"",
+                  "[]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acx \u2264 y\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh : x \u2260 y\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "tactic": "obtain h | h := h.not_le_or_not_le"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>exists_isClopen_upper_of_not_le<\\/a> h).<a>imp<\\/a> fun _ \\u21a6 <a>And.imp_right<\\/a> <| <a>And.imp_left<\\/a> <a>Or.inl<\\/a>\"",
+                  "[{\"full_name\":\"exists_isClopen_upper_of_not_le\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Priestley.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,40]},{\"full_name\":\"Exists.imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[193,9],\"def_end_pos\":[193,19]},{\"full_name\":\"And.imp_right\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[41,9],\"def_end_pos\":[41,22]},{\"full_name\":\"And.imp_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,21]},{\"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": "no goals",
+                "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acx \u2264 y\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "tactic": "exact (exists_isClopen_upper_of_not_le h).imp fun _ \u21a6 And.imp_right <| And.imp_left Or.inl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8U, hU, hU', hy, hx\\u27e9 := <a>exists_isClopen_lower_of_not_le<\\/a> h\"",
+                  "[{\"full_name\":\"exists_isClopen_lower_of_not_le\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Priestley.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,40]}]"
+                ],
+                "state_after": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\nU : Set \u03b1\nhU : IsClopen U\nhU' : IsLowerSet U\nhy : y \u2209 U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "tactic": "obtain \u27e8U, hU, hU', hy, hx\u27e9 := exists_isClopen_lower_of_not_le h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8U, hU, <a>Or.inr<\\/a> hU', hx, hy\\u27e9\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\nU : Set \u03b1\nhU : IsClopen U\nhU' : IsLowerSet U\nhy : y \u2209 U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 y \u2209 U",
+                "tactic": "exact \u27e8U, hU, Or.inr hU', hx, hy\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "exists_isClopen_upper_or_lower_of_ne"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001945-0000-227ce0395ab06688.parquet",
+      "sha256": "227ce0395ab0668840def56d10bd3e4c2fbecfeff969b201ebf57af0585769a3",
+      "bytes": 60535,
+      "rows": 6,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ContinuousAt.partialSups",
+    "mathlib:ContinuousWithinAt.partialSups",
+    "mathlib:Filter.Tendsto.partialSups",
+    "mathlib:Filter.Tendsto.partialSups_apply"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/PartialSups.lean\",null,null,\"ContinuousAt.partialSups\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/PartialSups.lean\",null,null,\"ContinuousWithinAt.partialSups\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/PartialSups.lean\",null,null,\"Filter.Tendsto.partialSups\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/PartialSups.lean\",null,null,\"Filter.Tendsto.partialSups_apply\"]"
+  ]
+}

manifests/batches/mathlib/001946.json

@@ -0,0 +1,904 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001946",
+  "batch_fingerprint": "2780dc7a1ebc4430e1500131c9adbec5cb6effdd948395872be4cbc2ef974afb",
+  "evidence": {
+    "batch_fingerprint": "2780dc7a1ebc4430e1500131c9adbec5cb6effdd948395872be4cbc2ef974afb",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.IccExtend",
+        "id": "Continuous.IccExtend",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            70
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "Continuous.IccExtend",
+          "start": [
+            47,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.Icc_extend'",
+        "id": "Continuous.Icc_extend'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            29
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "Continuous.Icc_extend'",
+          "start": [
+            55,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.IccExtend",
+        "id": "ContinuousAt.IccExtend",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            92
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "ContinuousAt.IccExtend",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.IccExtend",
+        "id": "Filter.Tendsto.IccExtend",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            45
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "Filter.Tendsto.IccExtend",
+          "start": [
+            23,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_IccExtend_iff",
+        "id": "continuous_IccExtend_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            42
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "continuous_IccExtend_iff",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_projIcc",
+        "id": "continuous_projIcc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            33,
+            76
+          ],
+          "file_path": "Mathlib/Topology/Order/ProjIcc.lean",
+          "full_name": "continuous_projIcc",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ProjIcc.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirSupClosed.inter",
+        "id": "DirSupClosed.inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            50
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "DirSupClosed.inter",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLowerSet.dirSupInacc",
+        "id": "IsLowerSet.dirSupInacc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            102,
+            99
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "IsLowerSet.dirSupInacc",
+          "start": [
+            102,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsUpperSet.dirSupClosed",
+        "id": "IsUpperSet.dirSupClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            55
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "IsUpperSet.dirSupClosed",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLower.scottHausdorff_le",
+        "id": "Topology.IsLower.scottHausdorff_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            359,
+            40
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsLower.scottHausdorff_le",
+          "start": [
+            356,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScott.isClosed_Iic",
+        "id": "Topology.IsScott.isClosed_Iic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            246,
+            84
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScott.isClosed_Iic",
+          "start": [
+            245,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScott.isLowerSet_of_isClosed",
+        "id": "Topology.IsScott.isLowerSet_of_isClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            55
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScott.isLowerSet_of_isClosed",
+          "start": [
+            236,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScott.isUpperSet_of_isOpen",
+        "id": "Topology.IsScott.isUpperSet_of_isOpen",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            234,
+            60
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScott.isUpperSet_of_isOpen",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScott.topology_eq",
+        "id": "Topology.IsScott.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            55
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScott.topology_eq",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScottHausdorff.dirSupClosed_of_isClosed",
+        "id": "Topology.IsScottHausdorff.dirSupClosed_of_isClosed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            179,
+            50
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScottHausdorff.dirSupClosed_of_isClosed",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScottHausdorff.isClosed_of_isUpperSet",
+        "id": "Topology.IsScottHausdorff.isClosed_of_isUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            53
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScottHausdorff.isClosed_of_isUpperSet",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScottHausdorff.isOpen_of_isLowerSet",
+        "id": "Topology.IsScottHausdorff.isOpen_of_isLowerSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            101
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScottHausdorff.isOpen_of_isLowerSet",
+          "start": [
+            168,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsScottHausdorff.topology_eq",
+        "id": "Topology.IsScottHausdorff.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            73
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.IsScottHausdorff.topology_eq",
+          "start": [
+            160,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open'",
+        "id": "Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            335,
+            93
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open'",
+          "start": [
+            334,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithScott.ofScott_inj",
+        "id": "Topology.WithScott.ofScott_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            318,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.WithScott.ofScott_inj",
+          "start": [
+            317,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithScott.ofScott_symm_eq",
+        "id": "Topology.WithScott.ofScott_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            310,
+            67
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.WithScott.ofScott_symm_eq",
+          "start": [
+            310,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithScott.ofScott_toScott",
+        "id": "Topology.WithScott.ofScott_toScott",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            312,
+            71
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.WithScott.ofScott_toScott",
+          "start": [
+            312,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithScott.toScott_ofScott",
+        "id": "Topology.WithScott.toScott_ofScott",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            311,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.WithScott.toScott_ofScott",
+          "start": [
+            311,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.scottHausdorff_le_lower",
+        "id": "Topology.scottHausdorff_le_lower",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            344,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.scottHausdorff_le_lower",
+          "start": [
+            342,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.scottHausdorff_le_scott",
+        "id": "Topology.scottHausdorff_le_scott",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            197,
+            75
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.scottHausdorff_le_scott",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.upperSet_le_scott",
+        "id": "Topology.upperSet_le_scott",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            62
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "Topology.upperSet_le_scott",
+          "start": [
+            195,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dirSupClosed_Iic",
+        "id": "dirSupClosed_Iic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+          "full_name": "dirSupClosed_Iic",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/ScottTopology.lean"
+      }
+    ],
+    "input_proofs": 43,
+    "input_units": 2,
+    "proofs": 16,
+    "published_proofs": 13,
+    "published_transitions": 21,
+    "replay_failures": [],
+    "transitions": 39,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+        "proof_id": "mathlib:Topology.IsScott.closure_singleton",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              255,
+              38
+            ],
+            "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+            "full_name": "Topology.IsScott.closure_singleton",
+            "start": [
+              248,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>singleton_subset_iff<\\/a>, <a>mem_Iic<\\/a>]\"",
+                  "[{\"full_name\":\"Set.singleton_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1287,9],\"def_end_pos\":[1287,29]},{\"full_name\":\"Set.mem_Iic\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[136,9],\"def_end_pos\":[136,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 {a} \u2286 Iic a",
+                "tactic": "rw [singleton_subset_iff, mem_Iic]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>LowerSet.coe_Iic<\\/a>, \\u2190 <a>lowerClosure_singleton<\\/a>]\"",
+                  "[{\"full_name\":\"LowerSet.coe_Iic\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[1290,9],\"def_end_pos\":[1290,16]},{\"full_name\":\"lowerClosure_singleton\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[1545,9],\"def_end_pos\":[1545,31]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191(lowerClosure {a}) \u2264 closure {a}",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 Iic a \u2264 closure {a}",
+                "tactic": "rw [\u2190 LowerSet.coe_Iic, \u2190 lowerClosure_singleton]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>lowerClosure_subset_closure<\\/a>\"",
+                  "[{\"full_name\":\"Topology.IsScott.lowerClosure_subset_closure\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[239,7],\"def_end_pos\":[239,34]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191(lowerClosure {a}) \u2264 closure {a}",
+                "tactic": "apply lowerClosure_subset_closure"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.IsScott.closure_singleton"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+        "proof_id": "mathlib:Topology.IsScott.lowerClosure_subset_closure",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              243,
+              24
+            ],
+            "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+            "full_name": "Topology.IsScott.lowerClosure_subset_closure",
+            "start": [
+              239,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"convert <a>closure.mono<\\/a> (@<a>upperSet_le_scott<\\/a> \\u03b1 _)\"",
+                  "[{\"full_name\":\"closure.mono\",\"def_path\":\"Mathlib\\/Topology\\/Order.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,21]},{\"full_name\":\"Topology.upperSet_le_scott\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[195,7],\"def_end_pos\":[195,24]}]"
+                ],
+                "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191(lowerClosure s) = closure s\n\ncase h.e'_4.h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 inst\u271d\u00b9 = scott \u03b1",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191(lowerClosure s) \u2286 closure s",
+                "tactic": "convert closure.mono (@upperSet_le_scott \u03b1 _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [@<a>IsUpperSet.closure_eq_lowerClosure<\\/a> \\u03b1 _ (<a>upperSet<\\/a> \\u03b1) ?_ s]\"",
+                  "[{\"full_name\":\"Topology.IsUpperSet.closure_eq_lowerClosure\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[237,7],\"def_end_pos\":[237,30]},{\"full_name\":\"Topology.upperSet\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[57,5],\"def_end_pos\":[57,13]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 Topology.IsUpperSet \u03b1",
+                "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 \u2191(lowerClosure s) = closure s",
+                "tactic": "rw [@IsUpperSet.closure_eq_lowerClosure \u03b1 _ (upperSet \u03b1) ?_ s]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"infer_instance\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 Topology.IsUpperSet \u03b1",
+                "tactic": "infer_instance"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>topology_eq<\\/a> \\u03b1\"",
+                  "[{\"full_name\":\"Topology.IsScott.topology_eq\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[214,7],\"def_end_pos\":[214,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.e'_4.h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\n\u22a2 inst\u271d\u00b9 = scott \u03b1",
+                "tactic": "exact topology_eq \u03b1"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.IsScott.lowerClosure_subset_closure"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+        "proof_id": "mathlib:Topology.IsScott.scottContinuous_iff_continuous",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              278,
+              27
+            ],
+            "file_path": "Mathlib/Topology/Order/ScottTopology.lean",
+            "full_name": "Topology.IsScott.scottContinuous_iff_continuous",
+            "start": [
+              264,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun h \\u21a6 <a>continuous_def<\\/a>.2 fun u hu \\u21a6 ?_, ?_\\u27e9\"",
+                  "[{\"full_name\":\"continuous_def\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1539,9],\"def_end_pos\":[1539,23]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nh : ScottContinuous f\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 IsOpen (f \u207b\u00b9' u)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Continuous f \u2192 ScottContinuous f",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 ScottContinuous f \u2194 Continuous f",
+                "tactic": "refine \u27e8fun h \u21a6 continuous_def.2 fun u hu \u21a6 ?_, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>isOpen_iff_isUpperSet_and_dirSupInacc<\\/a>]\"",
+                  "[{\"full_name\":\"Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[221,7],\"def_end_pos\":[221,44]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nh : ScottContinuous f\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nh : ScottContinuous f\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 IsOpen (f \u207b\u00b9' u)",
+                "tactic": "rw [isOpen_iff_isUpperSet_and_dirSupInacc]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8(<a>isUpperSet_of_isOpen<\\/a> hu).<a>preimage<\\/a> h.monotone, fun _ hd\\u2081 hd\\u2082 _ hd\\u2083 ha \\u21a6\\n      image_inter_nonempty_iff.mp <| (isOpen_iff_isUpperSet_and_dirSupInacc.mp hu).2 (hd\\u2081.image f)\\n        (directedOn_image.mpr (hd\\u2082.mono @(h.monotone))) (h hd\\u2081 hd\\u2082 hd\\u2083) ha\\u27e9\"",
+                  "[{\"full_name\":\"Topology.IsScott.isUpperSet_of_isOpen\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[233,7],\"def_end_pos\":[233,27]},{\"full_name\":\"IsUpperSet.preimage\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nh : ScottContinuous f\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)",
+                "tactic": "exact \u27e8(isUpperSet_of_isOpen hu).preimage h.monotone, fun _ hd\u2081 hd\u2082 _ hd\u2083 ha \u21a6\n  image_inter_nonempty_iff.mp <| (isOpen_iff_isUpperSet_and_dirSupInacc.mp hu).2 (hd\u2081.image f)\n    (directedOn_image.mpr (hd\u2082.mono @(h.monotone))) (h hd\u2081 hd\u2082 hd\u2083) ha\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine fun hf _ d\\u2081 d\\u2082 _ d\\u2083 \\u21a6 \\u27e8(<a>monotone_of_continuous<\\/a> hf).<a>mem_upperBounds_image<\\/a> d\\u2083.1,\\n      fun b hb \\u21a6 ?_\\u27e9\"",
+                  "[{\"full_name\":\"Topology.IsScott.monotone_of_continuous\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[259,7],\"def_end_pos\":[259,29]},{\"full_name\":\"Monotone.mem_upperBounds_image\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[1283,9],\"def_end_pos\":[1283,30]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\n\u22a2 f x\u271d \u2264 b",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Continuous f \u2192 ScottContinuous f",
+                "tactic": "refine fun hf _ d\u2081 d\u2082 _ d\u2083 \u21a6 \u27e8(monotone_of_continuous hf).mem_upperBounds_image d\u2083.1,\n  fun b hb \u21a6 ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"by_contra h\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\n\u22a2 False",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\n\u22a2 f x\u271d \u2264 b",
+                "tactic": "by_contra h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"let u := (<a>Iic<\\/a> b)\\u1d9c\"",
+                  "[{\"full_name\":\"Set.Iic\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,8]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\n\u22a2 False",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\n\u22a2 False",
+                "tactic": "let u := (Iic b)\u1d9c"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hu : <a>IsOpen<\\/a> (f \\u207b\\u00b9' u) := (<a>isOpen_compl_iff<\\/a>.2 <a>isClosed_Iic<\\/a>).<a>preimage<\\/a> hf\"",
+                  "[{\"full_name\":\"IsOpen\",\"def_path\":\"Mathlib\\/Topology\\/Defs\\/Basic.lean\",\"def_pos\":[85,5],\"def_end_pos\":[85,11]},{\"full_name\":\"isOpen_compl_iff\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[160,17],\"def_end_pos\":[160,33]},{\"full_name\":\"Topology.IsScott.isClosed_Iic\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[245,7],\"def_end_pos\":[245,19]},{\"full_name\":\"IsOpen.preimage\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1546,9],\"def_end_pos\":[1546,24]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsOpen (f \u207b\u00b9' u)\n\u22a2 False",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\n\u22a2 False",
+                "tactic": "have hu : IsOpen (f \u207b\u00b9' u) := (isOpen_compl_iff.2 isClosed_Iic).preimage hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>isOpen_iff_isUpperSet_and_dirSupInacc<\\/a>] at hu\"",
+                  "[{\"full_name\":\"Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/ScottTopology.lean\",\"def_pos\":[221,7],\"def_end_pos\":[221,44]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\n\u22a2 False",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsOpen (f \u207b\u00b9' u)\n\u22a2 False",
+                "tactic": "rw [isOpen_iff_isUpperSet_and_dirSupInacc] at hu"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8c, hcd, hfcb\\u27e9 := hu.2 d\\u2081 d\\u2082 d\\u2083 h\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\nc : \u03b1\nhcd : c \u2208 x\u271d\u00b9\nhfcb : c \u2208 f \u207b\u00b9' u\n\u22a2 False",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\n\u22a2 False",
+                "tactic": "obtain \u27e8c, hcd, hfcb\u27e9 := hu.2 d\u2081 d\u2082 d\u2083 h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>upperBounds<\\/a>] at hb\"",
+                  "[{\"full_name\":\"upperBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[46,5],\"def_end_pos\":[46,16]}]"
+                ],
+                "state_after": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\nc : \u03b1\nhcd : c \u2208 x\u271d\u00b9\nhfcb : c \u2208 f \u207b\u00b9' u\nhb : \u2200 a \u2208 x\u271d\u00b9, f a \u2264 b\n\u22a2 False",
+                "state_before": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nhb : b \u2208 upperBounds (f '' x\u271d\u00b9)\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\nc : \u03b1\nhcd : c \u2208 x\u271d\u00b9\nhfcb : c \u2208 f \u207b\u00b9' u\n\u22a2 False",
+                "tactic": "simp [upperBounds] at hb"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hfcb <| hb _ hcd\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : IsScott \u03b1\ns : Set \u03b1\na : \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : IsScott \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nx\u271d\u00b9 : Set \u03b1\nd\u2081 : x\u271d\u00b9.Nonempty\nd\u2082 : DirectedOn (fun x x_1 => x \u2264 x_1) x\u271d\u00b9\nx\u271d : \u03b1\nd\u2083 : IsLUB x\u271d\u00b9 x\u271d\nb : \u03b2\nh : \u00acf x\u271d \u2264 b\nu : Set \u03b2 := (Iic b)\u1d9c\nhu : IsUpperSet (f \u207b\u00b9' u) \u2227 DirSupInacc (f \u207b\u00b9' u)\nc : \u03b1\nhcd : c \u2208 x\u271d\u00b9\nhfcb : c \u2208 f \u207b\u00b9' u\nhb : \u2200 a \u2208 x\u271d\u00b9, f a \u2264 b\n\u22a2 False",
+                "tactic": "exact hfcb <| hb _ hcd"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.IsScott.scottContinuous_iff_continuous"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001946-0000-6ff3e2aec05489c9.parquet",
+      "sha256": "6ff3e2aec05489c92a69b3c9291a4845f4c113587bfc08d14aa0fb35ad08446c",
+      "bytes": 73296,
+      "rows": 19,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/dev/mathlib-001946-0000-3c1b2af74417db77.parquet",
+      "sha256": "3c1b2af74417db77785040add90aa6041a2011b97b7c14b0096d871a754e2ad8",
+      "bytes": 31437,
+      "rows": 2,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:DirSupInacc.union",
+    "mathlib:Topology.IsScott.isClosed_iff_isLowerSet_and_dirSupClosed",
+    "mathlib:Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc",
+    "mathlib:Topology.IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open",
+    "mathlib:Topology.IsScott.monotone_of_continuous",
+    "mathlib:Topology.IsScott.scottHausdorff_le",
+    "mathlib:Topology.IsScottHausdorff.dirSupInacc_of_isOpen",
+    "mathlib:Topology.IsScottHausdorff.isOpen_iff",
+    "mathlib:dirSupClosed_compl",
+    "mathlib:dirSupClosed_iff_forall_sSup",
+    "mathlib:dirSupInacc_compl",
+    "mathlib:dirSupInacc_iff_forall_sSup",
+    "mathlib:quotientMap_projIcc"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ProjIcc.lean\",null,null,\"quotientMap_projIcc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"DirSupInacc.union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScott.isClosed_iff_isLowerSet_and_dirSupClosed\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScott.isOpen_iff_isUpperSet_and_dirSupInacc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScott.monotone_of_continuous\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScott.scottHausdorff_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScottHausdorff.dirSupInacc_of_isOpen\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"Topology.IsScottHausdorff.isOpen_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"dirSupClosed_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"dirSupClosed_iff_forall_sSup\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"dirSupInacc_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/ScottTopology.lean\",null,null,\"dirSupInacc_iff_forall_sSup\"]"
+  ]
+}

manifests/batches/mathlib/001947.json

@@ -0,0 +1,1138 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001947",
+  "batch_fingerprint": "6ab5985a8d2888151ea5b80dd37f3530e37061fbb5ebe287b490034c1c1bd353",
+  "evidence": {
+    "batch_fingerprint": "6ab5985a8d2888151ea5b80dd37f3530e37061fbb5ebe287b490034c1c1bd353",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'refine mem_nhdsWithin_of_mem_nhds ?_'",
+        "full_name": "Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Ici",
+        "id": "Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Ici",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            52
+          ],
+          "file_path": "Mathlib/Topology/Order/T5.lean",
+          "full_name": "Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Ici",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have hmem : t\\u1d9c \\u2208 \\ud835\\udcdd[\\u2265] a := by\\n    refine <a>mem_nhdsWithin_of_mem_nhds<\\/a> ?_\\n    rw [\\u2190 <a>mem_interior_iff_mem_nhds<\\/a>, <a>interior_compl<\\/a>]\\n    exact <a>disjoint_left<\\/a>.1 hd ha\"",
+                "[{\"full_name\":\"mem_nhdsWithin_of_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,35]},{\"full_name\":\"mem_interior_iff_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[852,9],\"def_end_pos\":[852,34]},{\"full_name\":\"interior_compl\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[546,9],\"def_end_pos\":[546,23]},{\"full_name\":\"Set.disjoint_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1495,9],\"def_end_pos\":[1495,22]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "have hmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a := by\n  refine mem_nhdsWithin_of_mem_nhds ?_\n  rw [\u2190 mem_interior_iff_mem_nhds, interior_compl]\n  exact disjoint_left.1 hd ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>exists_Icc_mem_subset_of_mem_nhdsWithin_Ici<\\/a> hmem with \\u27e8b, hab, hmem', hsub\\u27e9\"",
+                "[{\"full_name\":\"exists_Icc_mem_subset_of_mem_nhdsWithin_Ici\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[420,9],\"def_end_pos\":[420,52]}]"
+              ],
+              "state_after": "case intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici hmem with \u27e8b, hab, hmem', hsub\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases H : <a>Disjoint<\\/a> (<a>Icc<\\/a> a b) (<a>ordConnectedSection<\\/a> <| <a>ordSeparatingSet<\\/a> s t)\"",
+                "[{\"full_name\":\"Disjoint\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,13]},{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"Set.ordConnectedSection\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[138,5],\"def_end_pos\":[138,24]},{\"full_name\":\"Set.ordSeparatingSet\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[170,5],\"def_end_pos\":[170,21]}]"
+              ],
+              "state_after": "case pos\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : Disjoint (Icc a b) (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a\n\ncase neg\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : \u00acDisjoint (Icc a b) (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "by_cases H : Disjoint (Icc a b) (ordConnectedSection <| ordSeparatingSet s t)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>mem_nhdsWithin_of_mem_nhds<\\/a> ?_\"",
+                "[{\"full_name\":\"mem_nhdsWithin_of_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/ContinuousOn.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,35]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 t\u1d9c \u2208 \ud835\udcdd a",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 t\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "refine mem_nhdsWithin_of_mem_nhds ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>mem_interior_iff_mem_nhds<\\/a>, <a>interior_compl<\\/a>]\"",
+                "[{\"full_name\":\"mem_interior_iff_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[852,9],\"def_end_pos\":[852,34]},{\"full_name\":\"interior_compl\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[546,9],\"def_end_pos\":[546,23]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 a \u2208 (closure t)\u1d9c",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 t\u1d9c \u2208 \ud835\udcdd a",
+              "tactic": "rw [\u2190 mem_interior_iff_mem_nhds, interior_compl]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>disjoint_left<\\/a>.1 hd ha\"",
+                "[{\"full_name\":\"Set.disjoint_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1495,9],\"def_end_pos\":[1495,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\n\u22a2 a \u2208 (closure t)\u1d9c",
+              "tactic": "exact disjoint_left.1 hd ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mem_of_superset<\\/a> hmem' (<a>disjoint_left<\\/a>.1 H)\"",
+                "[{\"full_name\":\"Filter.mem_of_superset\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,24]},{\"full_name\":\"Set.disjoint_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1495,9],\"def_end_pos\":[1495,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : Disjoint (Icc a b) (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "exact mem_of_superset hmem' (disjoint_left.1 H)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Set.disjoint_left<\\/a>, <a>not_forall<\\/a>, <a>Classical.not_not<\\/a>] at H\"",
+                "[{\"full_name\":\"Set.disjoint_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1495,9],\"def_end_pos\":[1495,22]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]}]"
+              ],
+              "state_after": "case neg\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : \u2203 x, \u2203 (_ : x \u2208 Icc a b), x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case neg\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : \u00acDisjoint (Icc a b) (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "simp only [Set.disjoint_left, not_forall, Classical.not_not] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases H with \\u27e8c, \\u27e8hac, hcb\\u27e9, hc\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case neg\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nH : \u2203 x, \u2203 (_ : x \u2208 Icc a b), x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "rcases H with \u27e8c, \u27e8hac, hcb\u27e9, hc\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hsub' : <a>Icc<\\/a> a b \\u2286 <a>ordConnectedComponent<\\/a> t\\u1d9c a :=\\n      <a>subset_ordConnectedComponent<\\/a> (<a>left_mem_Icc<\\/a>.2 hab) hsub\"",
+                "[{\"full_name\":\"Set.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[59,5],\"def_end_pos\":[59,8]},{\"full_name\":\"Set.ordConnectedComponent\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[29,5],\"def_end_pos\":[29,26]},{\"full_name\":\"Set.subset_ordConnectedComponent\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,37]},{\"full_name\":\"Set.left_mem_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[191,9],\"def_end_pos\":[191,21]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "have hsub' : Icc a b \u2286 ordConnectedComponent t\u1d9c a :=\n  subset_ordConnectedComponent (left_mem_Icc.2 hab) hsub"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hd : <a>Disjoint<\\/a> s (<a>ordConnectedSection<\\/a> (<a>ordSeparatingSet<\\/a> s t)) :=\\n      disjoint_left_ordSeparatingSet.mono_right <a>ordConnectedSection_subset<\\/a>\"",
+                "[{\"full_name\":\"Disjoint\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,13]},{\"full_name\":\"Set.ordConnectedSection\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[138,5],\"def_end_pos\":[138,24]},{\"full_name\":\"Set.ordSeparatingSet\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[170,5],\"def_end_pos\":[170,21]},{\"full_name\":\"Set.ordConnectedSection_subset\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[152,9],\"def_end_pos\":[152,35]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "have hd : Disjoint s (ordConnectedSection (ordSeparatingSet s t)) :=\n  disjoint_left_ordSeparatingSet.mono_right ordConnectedSection_subset"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace hac : a < c := hac.lt_of_ne <| <a>Ne.symm<\\/a> <| <a>ne_of_mem_of_not_mem<\\/a> hc <|\\n      <a>disjoint_left<\\/a>.1 hd ha\"",
+                "[{\"full_name\":\"Ne.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[704,9],\"def_end_pos\":[704,16]},{\"full_name\":\"ne_of_mem_of_not_mem\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[116,9],\"def_end_pos\":[116,29]},{\"full_name\":\"Set.disjoint_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1495,9],\"def_end_pos\":[1495,22]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhac : a \u2264 c\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "replace hac : a < c := hac.lt_of_ne <| Ne.symm <| ne_of_mem_of_not_mem hc <|\n  disjoint_left.1 hd ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>mem_of_superset<\\/a> (<a>Ico_mem_nhdsWithin_Ici<\\/a> (<a>left_mem_Ico<\\/a>.2 hac)) fun x hx hx' => ?_\"",
+                "[{\"full_name\":\"Filter.mem_of_superset\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,24]},{\"full_name\":\"Ico_mem_nhdsWithin_Ici\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[572,9],\"def_end_pos\":[572,31]},{\"full_name\":\"Set.left_mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,21]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 False",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\n\u22a2 (s.ordSeparatingSet t).ordConnectedSection\u1d9c \u2208 \ud835\udcdd[\u2265] a",
+              "tactic": "refine mem_of_superset (Ico_mem_nhdsWithin_Ici (left_mem_Ico.2 hac)) fun x hx hx' => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine hx.2.<a>ne<\\/a> (<a>eq_of_mem_ordConnectedSection_of_uIcc_subset<\\/a> hx' hc ?_)\"",
+                "[{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]},{\"full_name\":\"Set.eq_of_mem_ordConnectedSection_of_uIcc_subset\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[156,9],\"def_end_pos\":[156,53]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 s.ordSeparatingSet t",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 False",
+              "tactic": "refine hx.2.ne (eq_of_mem_ordConnectedSection_of_uIcc_subset hx' hc ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>subset_inter<\\/a> (<a>subset_iUnion\\u2082_of_subset<\\/a> a ha ?_) ?_\"",
+                "[{\"full_name\":\"Set.subset_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,21]},{\"full_name\":\"Set.subset_iUnion\\u2082_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[312,9],\"def_end_pos\":[312,33]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_1\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 t\u1d9c.ordConnectedComponent a\n\ncase neg.intro.intro.intro.refine_2\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 \u22c3 x \u2208 t, s\u1d9c.ordConnectedComponent x",
+              "state_before": "case neg.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 s.ordSeparatingSet t",
+              "tactic": "refine subset_inter (subset_iUnion\u2082_of_subset a ha ?_) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>OrdConnected.uIcc_subset<\\/a> <a>inferInstance<\\/a> (hsub' \\u27e8hx.1, hx.2.le.trans hcb\\u27e9)\\n        (hsub' \\u27e8hac.le, hcb\\u27e9)\"",
+                "[{\"full_name\":\"Set.OrdConnected.uIcc_subset\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnected.lean\",\"def_pos\":[325,9],\"def_end_pos\":[325,33]},{\"full_name\":\"inferInstance\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[99,8],\"def_end_pos\":[99,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.intro.intro.intro.refine_1\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 t\u1d9c.ordConnectedComponent a",
+              "tactic": "exact OrdConnected.uIcc_subset inferInstance (hsub' \u27e8hx.1, hx.2.le.trans hcb\u27e9)\n  (hsub' \u27e8hac.le, hcb\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>mem_iUnion\\u2082<\\/a>.1 (<a>ordConnectedSection_subset<\\/a> hx').2 with \\u27e8y, hyt, hxy\\u27e9\"",
+                "[{\"full_name\":\"Set.mem_iUnion\\u2082\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[67,9],\"def_end_pos\":[67,20]},{\"full_name\":\"Set.ordConnectedSection_subset\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[152,9],\"def_end_pos\":[152,35]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 [[x, c]] \u2286 \u22c3 x \u2208 t, s\u1d9c.ordConnectedComponent x",
+              "state_before": "case neg.intro.intro.intro.refine_2\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\n\u22a2 [[x, c]] \u2286 \u22c3 x \u2208 t, s\u1d9c.ordConnectedComponent x",
+              "tactic": "rcases mem_iUnion\u2082.1 (ordConnectedSection_subset hx').2 with \u27e8y, hyt, hxy\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>subset_iUnion\\u2082_of_subset<\\/a> y hyt (<a>OrdConnected.uIcc_subset<\\/a> <a>inferInstance<\\/a> hxy ?_)\"",
+                "[{\"full_name\":\"Set.subset_iUnion\\u2082_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[312,9],\"def_end_pos\":[312,33]},{\"full_name\":\"Set.OrdConnected.uIcc_subset\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnected.lean\",\"def_pos\":[325,9],\"def_end_pos\":[325,33]},{\"full_name\":\"inferInstance\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[99,8],\"def_end_pos\":[99,21]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c \u2208 s\u1d9c.ordConnectedComponent y",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 [[x, c]] \u2286 \u22c3 x \u2208 t, s\u1d9c.ordConnectedComponent x",
+              "tactic": "refine subset_iUnion\u2082_of_subset y hyt (OrdConnected.uIcc_subset inferInstance hxy ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>subset_ordConnectedComponent<\\/a> <a>left_mem_uIcc<\\/a> hxy ?_\"",
+                "[{\"full_name\":\"Set.subset_ordConnectedComponent\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/OrdConnectedComponent.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,37]},{\"full_name\":\"Set.left_mem_uIcc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/UnorderedInterval.lean\",\"def_pos\":[99,15],\"def_end_pos\":[99,28]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c \u2208 [[y, x]]",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c \u2208 s\u1d9c.ordConnectedComponent y",
+              "tactic": "refine subset_ordConnectedComponent left_mem_uIcc hxy ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices c < y by\\n        rw [<a>uIcc_of_ge<\\/a> (hx.2.<a>trans<\\/a> this).<a>le<\\/a>]\\n        exact \\u27e8hx.2.<a>le<\\/a>, this.le\\u27e9\"",
+                "[{\"full_name\":\"Set.uIcc_of_ge\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/UnorderedInterval.lean\",\"def_pos\":[75,7],\"def_end_pos\":[75,17]},{\"full_name\":\"LT.lt.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[139,7],\"def_end_pos\":[139,18]},{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]},{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c < y",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c \u2208 [[y, x]]",
+              "tactic": "suffices c < y by\n  rw [uIcc_of_ge (hx.2.trans this).le]\n  exact \u27e8hx.2.le, this.le\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>lt_of_not_le<\\/a> fun hyc => ?_\"",
+                "[{\"full_name\":\"lt_of_not_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[455,9],\"def_end_pos\":[455,21]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nhyc : y \u2264 c\n\u22a2 False",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\n\u22a2 c < y",
+              "tactic": "refine lt_of_not_le fun hyc => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hya : y < a := <a>not_le<\\/a>.1 fun hay => hsub \\u27e8hay, hyc.trans hcb\\u27e9 hyt\"",
+                "[{\"full_name\":\"not_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[375,9],\"def_end_pos\":[375,15]}]"
+              ],
+              "state_after": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nhyc : y \u2264 c\nhya : y < a\n\u22a2 False",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nhyc : y \u2264 c\n\u22a2 False",
+              "tactic": "have hya : y < a := not_le.1 fun hay => hsub \u27e8hay, hyc.trans hcb\u27e9 hyt"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hxy (<a>Icc_subset_uIcc<\\/a> \\u27e8hya.le, hx.1\\u27e9) ha\"",
+                "[{\"full_name\":\"Set.Icc_subset_uIcc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/UnorderedInterval.lean\",\"def_pos\":[94,7],\"def_end_pos\":[94,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.intro.intro.intro.refine_2.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nhyc : y \u2264 c\nhya : y < a\n\u22a2 False",
+              "tactic": "exact hxy (Icc_subset_uIcc \u27e8hya.le, hx.1\u27e9) ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>uIcc_of_ge<\\/a> (hx.2.<a>trans<\\/a> this).<a>le<\\/a>]\"",
+                "[{\"full_name\":\"Set.uIcc_of_ge\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/UnorderedInterval.lean\",\"def_pos\":[75,7],\"def_end_pos\":[75,17]},{\"full_name\":\"LT.lt.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[139,7],\"def_end_pos\":[139,18]},{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nthis : c < y\n\u22a2 c \u2208 Icc x y",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nthis : c < y\n\u22a2 c \u2208 [[y, x]]",
+              "tactic": "rw [uIcc_of_ge (hx.2.trans this).le]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8hx.2.<a>le<\\/a>, this.le\\u27e9\"",
+                "[{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nhd\u271d : Disjoint s (closure t)\nha : a \u2208 s\nhmem : t\u1d9c \u2208 \ud835\udcdd[\u2265] a\nb : X\nhab : a \u2264 b\nhmem' : Icc a b \u2208 \ud835\udcdd[\u2265] a\nhsub : Icc a b \u2286 t\u1d9c\nc : X\nhc : c \u2208 (s.ordSeparatingSet t).ordConnectedSection\nhcb : c \u2264 b\nhsub' : Icc a b \u2286 t\u1d9c.ordConnectedComponent a\nhd : Disjoint s (s.ordSeparatingSet t).ordConnectedSection\nhac : a < c\nx : X\nhx : x \u2208 Ico a c\nhx' : x \u2208 (s.ordSeparatingSet t).ordConnectedSection\ny : X\nhyt : y \u2208 t\nhxy : x \u2208 s\u1d9c.ordConnectedComponent y\nthis : c < y\n\u22a2 c \u2208 Icc x y",
+              "tactic": "exact \u27e8hx.2.le, this.le\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/T5.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLowerSet.closure_eq_upperClosure",
+        "id": "Topology.IsLowerSet.closure_eq_upperClosure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            315,
+            48
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.IsLowerSet.closure_eq_upperClosure",
+          "start": [
+            314,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLowerSet.monotone_to_lowerTopology_continuous",
+        "id": "Topology.IsLowerSet.monotone_to_lowerTopology_continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            342,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.IsLowerSet.monotone_to_lowerTopology_continuous",
+          "start": [
+            340,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsLowerSet.topology_eq",
+        "id": "Topology.IsLowerSet.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            295,
+            69
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.IsLowerSet.topology_eq",
+          "start": [
+            295,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.IsUpperSet.topology_eq",
+        "id": "Topology.IsUpperSet.topology_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            212,
+            69
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.IsUpperSet.topology_eq",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.isLowerSet_toLowerSet_preimage",
+        "id": "Topology.WithLowerSet.isLowerSet_toLowerSet_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            411,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.isLowerSet_toLowerSet_preimage",
+          "start": [
+            410,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.isOpen_ofLowerSet_preimage",
+        "id": "Topology.WithLowerSet.isOpen_ofLowerSet_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            414,
+            84
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.isOpen_ofLowerSet_preimage",
+          "start": [
+            413,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.map_comp",
+        "id": "Topology.WithLowerSet.map_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            400,
+            95
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.map_comp",
+          "start": [
+            400,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.map_id",
+        "id": "Topology.WithLowerSet.map_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            399,
+            78
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.map_id",
+          "start": [
+            399,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.ofLowerSet_inj",
+        "id": "Topology.WithLowerSet.ofLowerSet_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            134,
+            93
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.ofLowerSet_inj",
+          "start": [
+            134,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.ofLowerSet_le_iff",
+        "id": "Topology.WithLowerSet.ofLowerSet_le_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            149,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.ofLowerSet_le_iff",
+          "start": [
+            149,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.ofLowerSet_le_ofLowerSet",
+        "id": "Topology.WithLowerSet.ofLowerSet_le_ofLowerSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            408,
+            82
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.ofLowerSet_le_ofLowerSet",
+          "start": [
+            407,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.ofLowerSet_toLowerSet",
+        "id": "Topology.WithLowerSet.ofLowerSet_toLowerSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            83
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.ofLowerSet_toLowerSet",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.of_WithLowerSet_symm_eq",
+        "id": "Topology.WithLowerSet.of_WithLowerSet_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            130,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.of_WithLowerSet_symm_eq",
+          "start": [
+            130,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.toLowerSet_le_iff",
+        "id": "Topology.WithLowerSet.toLowerSet_le_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            150,
+            83
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.toLowerSet_le_iff",
+          "start": [
+            150,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithLowerSet.to_WithLowerSet_symm_eq",
+        "id": "Topology.WithLowerSet.to_WithLowerSet_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            129,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithLowerSet.to_WithLowerSet_symm_eq",
+          "start": [
+            129,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.isOpen_ofUpperSet_preimage",
+        "id": "Topology.WithUpperSet.isOpen_ofUpperSet_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            386,
+            84
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.isOpen_ofUpperSet_preimage",
+          "start": [
+            385,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.isUpperSet_toUpperSet_preimage",
+        "id": "Topology.WithUpperSet.isUpperSet_toUpperSet_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            383,
+            56
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.isUpperSet_toUpperSet_preimage",
+          "start": [
+            382,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.map_comp",
+        "id": "Topology.WithUpperSet.map_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            372,
+            95
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.map_comp",
+          "start": [
+            372,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.map_id",
+        "id": "Topology.WithUpperSet.map_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            371,
+            78
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.map_id",
+          "start": [
+            371,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.ofUpperSet_inj",
+        "id": "Topology.WithUpperSet.ofUpperSet_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            93
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.ofUpperSet_inj",
+          "start": [
+            88,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.ofUpperSet_le_iff",
+        "id": "Topology.WithUpperSet.ofUpperSet_le_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.ofUpperSet_le_iff",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.ofUpperSet_le_ofUpperSet",
+        "id": "Topology.WithUpperSet.ofUpperSet_le_ofUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            380,
+            82
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.ofUpperSet_le_ofUpperSet",
+          "start": [
+            379,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.ofUpperSet_toUpperSet",
+        "id": "Topology.WithUpperSet.ofUpperSet_toUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            83
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.ofUpperSet_toUpperSet",
+          "start": [
+            86,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.of_WithUpperSet_symm_eq",
+        "id": "Topology.WithUpperSet.of_WithUpperSet_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.of_WithUpperSet_symm_eq",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.toUpperSet_inj",
+        "id": "Topology.WithUpperSet.toUpperSet_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            80
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.toUpperSet_inj",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.toUpperSet_le_iff",
+        "id": "Topology.WithUpperSet.toUpperSet_le_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            83
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.toUpperSet_le_iff",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.toUpperSet_ofUpperSet",
+        "id": "Topology.WithUpperSet.toUpperSet_ofUpperSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            96
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.toUpperSet_ofUpperSet",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.WithUpperSet.to_WithUpperSet_symm_eq",
+        "id": "Topology.WithUpperSet.to_WithUpperSet_symm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.WithUpperSet.to_WithUpperSet_symm_eq",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Topology.isLowerSet_orderDual",
+        "id": "Topology.isLowerSet_orderDual",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            360,
+            81
+          ],
+          "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+          "full_name": "Topology.isLowerSet_orderDual",
+          "start": [
+            359,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Order/UpperLowerSetTopology.lean"
+      }
+    ],
+    "input_proofs": 48,
+    "input_units": 2,
+    "proofs": 18,
+    "published_proofs": 16,
+    "published_transitions": 32,
+    "replay_failures": [],
+    "transitions": 45,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+        "proof_id": "mathlib:Topology.IsUpperSet.monotone_iff_continuous",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              271,
+              63
+            ],
+            "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+            "full_name": "Topology.IsUpperSet.monotone_iff_continuous",
+            "start": [
+              260,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2192 Continuous f\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Continuous f \u2192 Monotone f",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2194 Continuous f",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro hf\"",
+                  "[]"
+                ],
+                "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 Continuous f",
+                "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Monotone f \u2192 Continuous f",
+                "tactic": "intro hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [<a>continuous_def<\\/a>, <a>isOpen_iff_isUpperSet<\\/a>]\"",
+                  "[{\"full_name\":\"continuous_def\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1539,9],\"def_end_pos\":[1539,23]},{\"full_name\":\"Topology.IsUpperSet.isOpen_iff_isUpperSet\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[225,7],\"def_end_pos\":[225,28]}]"
+                ],
+                "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 \u2200 (s : Set \u03b2), IsUpperSet s \u2192 IsUpperSet (f \u207b\u00b9' s)",
+                "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 Continuous f",
+                "tactic": "simp_rw [continuous_def, isOpen_iff_isUpperSet]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun _ hs \\u21a6 <a>IsUpperSet.preimage<\\/a> hs hf\"",
+                  "[{\"full_name\":\"IsUpperSet.preimage\",\"def_path\":\"Mathlib\\/Order\\/UpperLower\\/Basic.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 \u2200 (s : Set \u03b2), IsUpperSet s \u2192 IsUpperSet (f \u207b\u00b9' s)",
+                "tactic": "exact fun _ hs \u21a6 IsUpperSet.preimage hs hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro hf a b hab\"",
+                  "[]"
+                ],
+                "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2264 b\n\u22a2 f a \u2264 f b",
+                "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 Continuous f \u2192 Monotone f",
+                "tactic": "intro hf a b hab"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>mem_Iic<\\/a>, \\u2190 <a>closure_singleton<\\/a>] at hab \\u22a2\"",
+                  "[{\"full_name\":\"Set.mem_Iic\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[136,9],\"def_end_pos\":[136,16]},{\"full_name\":\"Topology.IsUpperSet.closure_singleton\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[248,15],\"def_end_pos\":[248,32]}]"
+                ],
+                "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 f a \u2208 closure {f b}",
+                "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2264 b\n\u22a2 f a \u2264 f b",
+                "tactic": "rw [\u2190 mem_Iic, \u2190 closure_singleton] at hab \u22a2"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>Continuous.closure_preimage_subset<\\/a> hf {f b}\"",
+                  "[{\"full_name\":\"Continuous.closure_preimage_subset\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[1727,9],\"def_end_pos\":[1727,43]}]"
+                ],
+                "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 a \u2208 closure (f \u207b\u00b9' {f b})",
+                "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 f a \u2208 closure {f b}",
+                "tactic": "apply Continuous.closure_preimage_subset hf {f b}"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>mem_of_mem_of_subset<\\/a> hab\"",
+                  "[{\"full_name\":\"Set.mem_of_mem_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,29]}]"
+                ],
+                "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 closure {b} \u2286 closure (f \u207b\u00b9' {f b})",
+                "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 a \u2208 closure (f \u207b\u00b9' {f b})",
+                "tactic": "apply mem_of_mem_of_subset hab"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>closure_mono<\\/a>\"",
+                  "[{\"full_name\":\"closure_mono\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,21]}]"
+                ],
+                "state_after": "case mpr.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 {b} \u2286 f \u207b\u00b9' {f b}",
+                "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 closure {b} \u2286 closure (f \u207b\u00b9' {f b})",
+                "tactic": "apply closure_mono"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>singleton_subset_iff<\\/a>, <a>mem_preimage<\\/a>, <a>mem_singleton_iff<\\/a>]\"",
+                  "[{\"full_name\":\"Set.singleton_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1287,9],\"def_end_pos\":[1287,29]},{\"full_name\":\"Set.mem_preimage\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,21]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : Preorder \u03b1\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Topology.IsUpperSet \u03b1\ninst\u271d : Topology.IsUpperSet \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\na b : \u03b1\nhab : a \u2208 closure {b}\n\u22a2 {b} \u2286 f \u207b\u00b9' {f b}",
+                "tactic": "rw [singleton_subset_iff, mem_preimage, mem_singleton_iff]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.IsUpperSet.monotone_iff_continuous"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+        "proof_id": "mathlib:Topology.isUpperSet_orderDual",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              357,
+              35
+            ],
+            "file_path": "Mathlib/Topology/Order/UpperLowerSetTopology.lean",
+            "full_name": "Topology.isUpperSet_orderDual",
+            "start": [
+              353,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 Topology.IsUpperSet \u03b1\u1d52\u1d48 \u2192 Topology.IsLowerSet \u03b1\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 Topology.IsLowerSet \u03b1 \u2192 Topology.IsUpperSet \u03b1\u1d52\u1d48",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 Topology.IsUpperSet \u03b1\u1d52\u1d48 \u2194 Topology.IsLowerSet \u03b1",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>OrderDual.instIsLowerSet<\\/a>\"",
+                  "[{\"full_name\":\"OrderDual.instIsLowerSet\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[216,10],\"def_end_pos\":[216,41]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 Topology.IsUpperSet \u03b1\u1d52\u1d48 \u2192 Topology.IsLowerSet \u03b1",
+                "tactic": "apply OrderDual.instIsLowerSet"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>OrderDual.instIsUpperSet<\\/a>\"",
+                  "[{\"full_name\":\"OrderDual.instIsUpperSet\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/UpperLowerSetTopology.lean\",\"def_pos\":[299,10],\"def_end_pos\":[299,41]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 Topology.IsLowerSet \u03b1 \u2192 Topology.IsUpperSet \u03b1\u1d52\u1d48",
+                "tactic": "apply OrderDual.instIsUpperSet"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Topology.isUpperSet_orderDual"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001947-0000-60bb7e0740666138.parquet",
+      "sha256": "60bb7e07406661384873359c1c8790b798494c844a5cf4c709a032be6cab4bc2",
+      "bytes": 92794,
+      "rows": 28,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/dev/mathlib-001947-0000-554e04dfa3854ba0.parquet",
+      "sha256": "554e04dfa3854ba08f454704651a0de66f7aba74ac4f0013ecefbce4027d6115",
+      "bytes": 44266,
+      "rows": 3,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    },
+    {
+      "path": "data/internal_test/mathlib-001947-0000-f9f8fae708e4e52b.parquet",
+      "sha256": "f9f8fae708e4e52b89493ab4574f183d06da65b209ca5d8d4c8125ab0159eacf",
+      "bytes": 42645,
+      "rows": 1,
+      "revision": "3b24a1cae9b52e2312093fce28268e8106673ba0"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.compl_section_ordSeparatingSet_mem_nhds",
+    "mathlib:Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Iic",
+    "mathlib:Set.ordConnectedComponent_mem_nhds",
+    "mathlib:Set.ordT5Nhd_mem_nhdsSet",
+    "mathlib:Topology.IsLowerSet.closure_singleton",
+    "mathlib:Topology.IsLowerSet.isClosed_iff_isUpper",
+    "mathlib:Topology.IsLowerSet.isOpen_iff_isLowerSet",
+    "mathlib:Topology.IsLowerSet.lowerSet_le_lower",
+    "mathlib:Topology.IsUpperSet.closure_eq_lowerClosure",
+    "mathlib:Topology.IsUpperSet.closure_singleton",
+    "mathlib:Topology.IsUpperSet.isClosed_iff_isLower",
+    "mathlib:Topology.IsUpperSet.isOpen_iff_isUpperSet",
+    "mathlib:Topology.IsUpperSet.monotone_to_upperTopology_continuous",
+    "mathlib:Topology.IsUpperSet.upperSet_le_upper",
+    "mathlib:Topology.WithLowerSet.toLowerSet_specializes_toLowerSet",
+    "mathlib:Topology.WithUpperSet.toUpperSet_specializes_toUpperSet"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/T5.lean\",null,null,\"Set.compl_section_ordSeparatingSet_mem_nhds\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/T5.lean\",null,null,\"Set.compl_section_ordSeparatingSet_mem_nhdsWithin_Iic\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/T5.lean\",null,null,\"Set.ordConnectedComponent_mem_nhds\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/T5.lean\",null,null,\"Set.ordT5Nhd_mem_nhdsSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsLowerSet.closure_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsLowerSet.isClosed_iff_isUpper\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsLowerSet.isOpen_iff_isLowerSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsLowerSet.lowerSet_le_lower\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.closure_eq_lowerClosure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.closure_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.isClosed_iff_isLower\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.isOpen_iff_isUpperSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.monotone_to_upperTopology_continuous\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.IsUpperSet.upperSet_le_upper\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.WithLowerSet.toLowerSet_specializes_toLowerSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Topology/Order/UpperLowerSetTopology.lean\",null,null,\"Topology.WithUpperSet.toUpperSet_specializes_toUpperSet\"]"
+  ]
+}