Record 19 verified HPC batch manifests

manifests/batches/mathlib/001105.json

@@ -0,0 +1,279 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001105",
+  "batch_fingerprint": "b44db09103d28912d2e15cfe6e93f77e455b1e00151bcfbeffb5bc0fe61c9a47",
+  "evidence": {
+    "batch_fingerprint": "b44db09103d28912d2e15cfe6e93f77e455b1e00151bcfbeffb5bc0fe61c9a47",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffAverage_congr_ring",
+        "id": "birkhoffAverage_congr_ring",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            54
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Average.lean",
+          "full_name": "birkhoffAverage_congr_ring",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Average.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffAverage_one'",
+        "id": "birkhoffAverage_one'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            38
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Average.lean",
+          "full_name": "birkhoffAverage_one'",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Average.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffAverage_zero'",
+        "id": "birkhoffAverage_zero'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            39
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Average.lean",
+          "full_name": "birkhoffAverage_zero'",
+          "start": [
+            47,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Average.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_one",
+        "id": "birkhoffSum_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            18
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_one",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_one'",
+        "id": "birkhoffSum_one'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            32
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_one'",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_succ",
+        "id": "birkhoffSum_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            45,
+            21
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_succ",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_succ'",
+        "id": "birkhoffSum_succ'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            45
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_succ'",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_zero",
+        "id": "birkhoffSum_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            30,
+            19
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_zero",
+          "start": [
+            29,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "birkhoffSum_zero'",
+        "id": "birkhoffSum_zero'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            34,
+            33
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "birkhoffSum_zero'",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "map_birkhoffSum",
+        "id": "map_birkhoffSum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            17
+          ],
+          "file_path": "Mathlib/Dynamics/BirkhoffSum/Basic.lean",
+          "full_name": "map_birkhoffSum",
+          "start": [
+            59,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/BirkhoffSum/Basic.lean"
+      }
+    ],
+    "input_proofs": 19,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 9,
+    "published_transitions": 11,
+    "replay_failures": [],
+    "transitions": 11,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001105-0000-1b31cd0c04a75da3.parquet",
+      "sha256": "1b31cd0c04a75da3da1ba17a4f834086aa0448cf590e961502a4751d42570381",
+      "bytes": 67739,
+      "rows": 9,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/mathlib-001105-0000-10399eba52e5d9c7.parquet",
+      "sha256": "10399eba52e5d9c724cb291c16162054b41134b71ab9b2ed35985ed617bb7d7f",
+      "bytes": 45870,
+      "rows": 2,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Function.IsFixedPt.birkhoffAverage_eq",
+    "mathlib:Function.IsFixedPt.birkhoffSum_eq",
+    "mathlib:birkhoffAverage_apply_sub_birkhoffAverage",
+    "mathlib:birkhoffAverage_congr_ring'",
+    "mathlib:birkhoffAverage_one",
+    "mathlib:birkhoffAverage_zero",
+    "mathlib:birkhoffSum_add",
+    "mathlib:birkhoffSum_apply_sub_birkhoffSum",
+    "mathlib:map_birkhoffAverage"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"Function.IsFixedPt.birkhoffAverage_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"birkhoffAverage_apply_sub_birkhoffAverage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"birkhoffAverage_congr_ring'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"birkhoffAverage_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"birkhoffAverage_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Average.lean\",null,null,\"map_birkhoffAverage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Basic.lean\",null,null,\"Function.IsFixedPt.birkhoffSum_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Basic.lean\",null,null,\"birkhoffSum_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/BirkhoffSum/Basic.lean\",null,null,\"birkhoffSum_apply_sub_birkhoffSum\"]"
+  ]
+}

manifests/batches/mathlib/001108.json

@@ -0,0 +1,616 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001108",
+  "batch_fingerprint": "6394c2322ddea850cb2e251940d2051c12c5ba5b5ba22a07e7a2bc313eb35e68",
+  "evidence": {
+    "batch_fingerprint": "6394c2322ddea850cb2e251940d2051c12c5ba5b5ba22a07e7a2bc313eb35e68",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.ae_empty_or_univ_of_ae_le_preimage",
+        "id": "Ergodic.ae_empty_or_univ_of_ae_le_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            190,
+            70
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "Ergodic.ae_empty_or_univ_of_ae_le_preimage",
+          "start": [
+            188,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [hf.measure_preimage hs]'",
+        "full_name": "Ergodic.ae_empty_or_univ_of_ae_le_preimage'",
+        "id": "Ergodic.ae_empty_or_univ_of_ae_le_preimage'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            89
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "Ergodic.ae_empty_or_univ_of_ae_le_preimage'",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"replace h_fin : \\u03bc (f \\u207b\\u00b9' s) \\u2260 \\u221e := by rwa [hf.measure_preimage hs]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc (f \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 s =\u1da0[ae \u03bc] \u2205 \u2228 s =\u1da0[ae \u03bc] univ",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc s \u2260 \u22a4\n\u22a2 s =\u1da0[ae \u03bc] \u2205 \u2228 s =\u1da0[ae \u03bc] univ",
+              "tactic": "replace h_fin : \u03bc (f \u207b\u00b9' s) \u2260 \u221e := by rwa [hf.measure_preimage hs]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine hf.quasiErgodic.ae_empty_or_univ' hs ?_\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc (f \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 f \u207b\u00b9' s =\u1da0[ae \u03bc] s",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc (f \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 s =\u1da0[ae \u03bc] \u2205 \u2228 s =\u1da0[ae \u03bc] univ",
+              "tactic": "refine hf.quasiErgodic.ae_empty_or_univ' hs ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>ae_eq_of_ae_subset_of_measure_ge<\\/a> hs' (hf.measure_preimage hs).<a>le<\\/a> hs h_fin).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.ae_eq_of_ae_subset_of_measure_ge\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[333,9],\"def_end_pos\":[333,41]},{\"full_name\":\"Eq.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[154,7],\"def_end_pos\":[154,12]},{\"full_name\":\"Filter.EventuallyEq.symm\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1530,9],\"def_end_pos\":[1530,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc (f \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 f \u207b\u00b9' s =\u1da0[ae \u03bc] s",
+              "tactic": "exact (ae_eq_of_ae_subset_of_measure_ge hs' (hf.measure_preimage hs).le hs h_fin).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [hf.measure_preimage hs]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : Ergodic f \u03bc\nhs : MeasurableSet s\nhs' : s \u2264\u1da0[ae \u03bc] f \u207b\u00b9' s\nh_fin : \u03bc s \u2260 \u22a4\n\u22a2 \u03bc (f \u207b\u00b9' s) \u2260 \u22a4",
+              "tactic": "rwa [hf.measure_preimage hs]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.ae_empty_or_univ_of_image_ae_le",
+        "id": "Ergodic.ae_empty_or_univ_of_image_ae_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            67
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "Ergodic.ae_empty_or_univ_of_image_ae_le",
+          "start": [
+            193,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.ae_empty_or_univ_of_preimage_ae_le",
+        "id": "Ergodic.ae_empty_or_univ_of_preimage_ae_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            185,
+            70
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "Ergodic.ae_empty_or_univ_of_preimage_ae_le",
+          "start": [
+            183,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.quasiErgodic",
+        "id": "Ergodic.quasiErgodic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            74
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "Ergodic.quasiErgodic",
+          "start": [
+            150,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]'",
+        "full_name": "MeasureTheory.MeasurePreserving.ergodic_conjugate_iff",
+        "id": "MeasureTheory.MeasurePreserving.ergodic_conjugate_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            79
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.ergodic_conjugate_iff",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : <a>MeasurePreserving<\\/a> (e \\u2218 f \\u2218 e.symm) \\u03bc' \\u03bc' \\u2194 <a>MeasurePreserving<\\/a> f \\u03bc \\u03bc := by\\n    rw [h.comp_left_iff, (<a>MeasurePreserving.symm<\\/a> e h).<a>comp_right_iff<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.MeasurePreserving\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[42,11],\"def_end_pos\":[42,28]},{\"full_name\":\"MeasureTheory.MeasurePreserving\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[42,11],\"def_end_pos\":[42,28]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.MeasurePreserving.comp_right_iff\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[122,19],\"def_end_pos\":[122,33]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nthis : MeasurePreserving (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc\n\u22a2 Ergodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 Ergodic f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\n\u22a2 Ergodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 Ergodic f \u03bc",
+              "tactic": "have : MeasurePreserving (e \u2218 f \u2218 e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc := by\n  rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace h : <a>PreErgodic<\\/a> (e \\u2218 f \\u2218 e.symm) \\u03bc' \\u2194 <a>PreErgodic<\\/a> f \\u03bc := h.preErgodic_conjugate_iff\"",
+                "[{\"full_name\":\"PreErgodic\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/Ergodic.lean\",\"def_pos\":[42,11],\"def_end_pos\":[42,21]},{\"full_name\":\"PreErgodic\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/Ergodic.lean\",\"def_pos\":[42,11],\"def_end_pos\":[42,21]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nthis : MeasurePreserving (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc\nh : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 PreErgodic f \u03bc\n\u22a2 Ergodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 Ergodic f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nthis : MeasurePreserving (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc\n\u22a2 Ergodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 Ergodic f \u03bc",
+              "tactic": "replace h : PreErgodic (e \u2218 f \u2218 e.symm) \u03bc' \u2194 PreErgodic f \u03bc := h.preErgodic_conjugate_iff"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8fun hf => { this.mp hf.toMeasurePreserving, h.mp hf.toPreErgodic with },\\n    fun hf => { this.mpr hf.toMeasurePreserving, h.mpr hf.toPreErgodic with }\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nthis : MeasurePreserving (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc\nh : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 PreErgodic f \u03bc\n\u22a2 Ergodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 Ergodic f \u03bc",
+              "tactic": "exact \u27e8fun hf => { this.mp hf.toMeasurePreserving, h.mp hf.toPreErgodic with },\n  fun hf => { this.mpr hf.toMeasurePreserving, h.mpr hf.toPreErgodic with }\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h.comp_left_iff, (<a>MeasurePreserving.symm<\\/a> e h).<a>comp_right_iff<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.MeasurePreserving.comp_right_iff\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[122,19],\"def_end_pos\":[122,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\n\u22a2 MeasurePreserving (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u03bc' \u2194 MeasurePreserving f \u03bc \u03bc",
+              "tactic": "rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [\u2190 preimage_comp, h_comm, preimage_comp, hs\u2081]'",
+        "full_name": "MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate",
+        "id": "MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            90
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro s hs\\u2080 hs\\u2081\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\n\u22a2 \u2200 \u2983s : Set \u03b2\u2984, MeasurableSet s \u2192 f' \u207b\u00b9' s = s \u2192 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ",
+              "tactic": "intro s hs\u2080 hs\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace hs\\u2081 : f \\u207b\\u00b9' (g \\u207b\\u00b9' s) = g \\u207b\\u00b9' s := by rw [\\u2190 <a>preimage_comp<\\/a>, h_comm, <a>preimage_comp<\\/a>, hs\\u2081]\"",
+                "[{\"full_name\":\"Set.preimage_comp\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,22]},{\"full_name\":\"Set.preimage_comp\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,22]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ",
+              "tactic": "replace hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s := by rw [\u2190 preimage_comp, h_comm, preimage_comp, hs\u2081]"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases' hf.ae_empty_or_univ (hg.measurable hs\\u2080) hs\\u2081 with hs\\u2082 hs\\u2082 <;> [left; right]\"",
+                "[]"
+              ],
+              "state_after": "case inl.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] \u2205\n\u22a2 s =\u1da0[ae \u03bc'] \u2205\n\ncase inr.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] univ\n\u22a2 s =\u1da0[ae \u03bc'] univ",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ",
+              "tactic": "cases' hf.ae_empty_or_univ (hg.measurable hs\u2080) hs\u2081 with hs\u2082 hs\u2082 <;> [left; right]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>preimage_comp<\\/a>, h_comm, <a>preimage_comp<\\/a>, hs\\u2081]\"",
+                "[{\"full_name\":\"Set.preimage_comp\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,22]},{\"full_name\":\"Set.preimage_comp\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s",
+              "tactic": "rw [\u2190 preimage_comp, h_comm, preimage_comp, hs\u2081]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>ae_eq_empty<\\/a>, hg.measure_preimage hs\\u2080] using hs\\u2082\"",
+                "[{\"full_name\":\"MeasureTheory.ae_eq_empty\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[138,9],\"def_end_pos\":[138,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] \u2205\n\u22a2 s =\u1da0[ae \u03bc'] \u2205",
+              "tactic": "simpa only [ae_eq_empty, hg.measure_preimage hs\u2080] using hs\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>ae_eq_univ<\\/a>, \\u2190 <a>preimage_compl<\\/a>, hg.measure_preimage hs\\u2080.compl] using hs\\u2082\"",
+                "[{\"full_name\":\"MeasureTheory.ae_eq_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,19]},{\"full_name\":\"Set.preimage_compl\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[82,9],\"def_end_pos\":[82,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g \u03bc \u03bc'\nhf : PreErgodic f \u03bc\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] univ\n\u22a2 s =\u1da0[ae \u03bc'] univ",
+              "tactic": "simpa only [ae_eq_univ, \u2190 preimage_compl, hg.measure_preimage hs\u2080.compl] using hs\u2082"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PreErgodic.ae_mem_or_ae_nmem",
+        "id": "PreErgodic.ae_mem_or_ae_nmem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            81
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "PreErgodic.ae_mem_or_ae_nmem",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PreErgodic.of_iterate",
+        "id": "PreErgodic.of_iterate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            79
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "PreErgodic.of_iterate",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "QuasiErgodic.ae_empty_or_univ\u2080",
+        "id": "QuasiErgodic.ae_empty_or_univ\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            136,
+            58
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+          "full_name": "QuasiErgodic.ae_empty_or_univ\u2080",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Ergodic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.ae_eq_const_of_ae_eq_comp\u2080",
+        "id": "Ergodic.ae_eq_const_of_ae_eq_comp\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            54
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Function.lean",
+          "full_name": "Ergodic.ae_eq_const_of_ae_eq_comp\u2080",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Function.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ergodic.eq_const_of_compMeasurePreserving_eq",
+        "id": "Ergodic.eq_const_of_compMeasurePreserving_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            65
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Function.lean",
+          "full_name": "Ergodic.eq_const_of_compMeasurePreserving_eq",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Function.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "QuasiErgodic.ae_eq_const_of_ae_eq_comp\u2080",
+        "id": "QuasiErgodic.ae_eq_const_of_ae_eq_comp\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            74
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Function.lean",
+          "full_name": "QuasiErgodic.ae_eq_const_of_ae_eq_comp\u2080",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Function.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "QuasiErgodic.eq_const_of_compQuasiMeasurePreserving_eq",
+        "id": "QuasiErgodic.eq_const_of_compQuasiMeasurePreserving_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            89,
+            62
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/Function.lean",
+          "full_name": "QuasiErgodic.eq_const_of_compQuasiMeasurePreserving_eq",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/Function.lean"
+      }
+    ],
+    "input_proofs": 23,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 7,
+    "published_transitions": 17,
+    "replay_failures": [],
+    "transitions": 24,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+        "proof_id": "mathlib:MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              106,
+              49
+            ],
+            "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+            "full_name": "MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff",
+            "start": [
+              99,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun hf => <a>preErgodic_of_preErgodic_conjugate<\\/a> (h.symm e) hf ?_,\\n      fun hf => <a>preErgodic_of_preErgodic_conjugate<\\/a> h hf ?_\\u27e9\"",
+                  "[{\"full_name\":\"MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/Ergodic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,43]},{\"full_name\":\"MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/Ergodic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,43]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc'\n\u22a2 \u21d1e.symm \u2218 \u21d1e \u2218 f \u2218 \u21d1e.symm = f \u2218 \u21d1e.symm\n\ncase refine_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic f \u03bc\n\u22a2 \u21d1e \u2218 f = (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u2218 \u21d1e",
+                "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\n\u22a2 PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc' \u2194 PreErgodic f \u03bc",
+                "tactic": "refine \u27e8fun hf => preErgodic_of_preErgodic_conjugate (h.symm e) hf ?_,\n    fun hf => preErgodic_of_preErgodic_conjugate h hf ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"change (e.symm \\u2218 e) \\u2218 f \\u2218 e.symm = f \\u2218 e.symm\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc'\n\u22a2 (\u21d1e.symm \u2218 \u21d1e) \u2218 f \u2218 \u21d1e.symm = f \u2218 \u21d1e.symm",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc'\n\u22a2 \u21d1e.symm \u2218 \u21d1e \u2218 f \u2218 \u21d1e.symm = f \u2218 \u21d1e.symm",
+                "tactic": "change (e.symm \u2218 e) \u2218 f \u2218 e.symm = f \u2218 e.symm"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>MeasurableEquiv.symm_comp_self<\\/a>, <a>id_comp<\\/a>]\"",
+                  "[{\"full_name\":\"MeasurableEquiv.symm_comp_self\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[249,9],\"def_end_pos\":[249,23]},{\"full_name\":\"Function.id_comp\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u03bc'\n\u22a2 (\u21d1e.symm \u2218 \u21d1e) \u2218 f \u2218 \u21d1e.symm = f \u2218 \u21d1e.symm",
+                "tactic": "rw [MeasurableEquiv.symm_comp_self, id_comp]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"change e \\u2218 f = e \\u2218 f \\u2218 e.symm \\u2218 e\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic f \u03bc\n\u22a2 \u21d1e \u2218 f = \u21d1e \u2218 f \u2218 \u21d1e.symm \u2218 \u21d1e",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic f \u03bc\n\u22a2 \u21d1e \u2218 f = (\u21d1e \u2218 f \u2218 \u21d1e.symm) \u2218 \u21d1e",
+                "tactic": "change e \u2218 f = e \u2218 f \u2218 e.symm \u2218 e"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>MeasurableEquiv.symm_comp_self<\\/a>, <a>comp_id<\\/a>]\"",
+                  "[{\"full_name\":\"MeasurableEquiv.symm_comp_self\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[249,9],\"def_end_pos\":[249,23]},{\"full_name\":\"Function.comp_id\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\ne : \u03b1 \u2243\u1d50 \u03b2\nh : MeasurePreserving (\u21d1e) \u03bc \u03bc'\nhf : PreErgodic f \u03bc\n\u22a2 \u21d1e \u2218 f = \u21d1e \u2218 f \u2218 \u21d1e.symm \u2218 \u21d1e",
+                "tactic": "rw [MeasurableEquiv.symm_comp_self, comp_id]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.MeasurePreserving.preErgodic_conjugate_iff"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+        "proof_id": "mathlib:QuasiErgodic.ae_mem_or_ae_nmem\u2080",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              143,
+              83
+            ],
+            "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean",
+            "full_name": "QuasiErgodic.ae_mem_or_ae_nmem\u2080",
+            "start": [
+              140,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>mem_ae_iff<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.mem_ae_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[71,9],\"def_end_pos\":[71,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : QuasiErgodic f \u03bc\nhsm : NullMeasurableSet s \u03bc\nhs : f \u207b\u00b9' s =\u1da0[ae \u03bc] s\n\u22a2 s =\u1da0[ae \u03bc] univ \u2192 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s",
+                "tactic": "simp [mem_ae_iff]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>ae_iff<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.ae_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : QuasiErgodic f \u03bc\nhsm : NullMeasurableSet s \u03bc\nhs : f \u207b\u00b9' s =\u1da0[ae \u03bc] s\n\u22a2 s =\u1da0[ae \u03bc] \u2205 \u2192 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2209 s",
+                "tactic": "simp [ae_iff]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "QuasiErgodic.ae_mem_or_ae_nmem\u2080"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001108-0000-2d36c463d0ce7ddd.parquet",
+      "sha256": "2d36c463d0ce7ddd352164132dc0a923a4994e5c61b6cfa1eb6a1e62ed5b2fbe",
+      "bytes": 74907,
+      "rows": 11,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/dev/mathlib-001108-0000-99d401db520b3e09.parquet",
+      "sha256": "99d401db520b3e09d3ec1f2b87ca3be006c08397c606f056f43e0b3d8f78ac65",
+      "bytes": 51011,
+      "rows": 2,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/mathlib-001108-0000-7923140d3c77176a.parquet",
+      "sha256": "7923140d3c77176a837cfe20bf66f7f77dde418cb249cf1e94c738ccbe8e698b",
+      "bytes": 64898,
+      "rows": 4,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Ergodic.ae_empty_or_univ_of_image_ae_le'",
+    "mathlib:Ergodic.ae_empty_or_univ_of_preimage_ae_le'",
+    "mathlib:PreErgodic.ae_eq_const_of_ae_eq_comp",
+    "mathlib:PreErgodic.prob_eq_zero_or_one",
+    "mathlib:QuasiErgodic.ae_empty_or_univ'",
+    "mathlib:QuasiErgodic.ae_eq_const_of_ae_eq_comp_ae",
+    "mathlib:QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range\u2080"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Ergodic.lean\",null,null,\"Ergodic.ae_empty_or_univ_of_image_ae_le'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Ergodic.lean\",null,null,\"Ergodic.ae_empty_or_univ_of_preimage_ae_le'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Ergodic.lean\",null,null,\"PreErgodic.prob_eq_zero_or_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Ergodic.lean\",null,null,\"QuasiErgodic.ae_empty_or_univ'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Function.lean\",null,null,\"PreErgodic.ae_eq_const_of_ae_eq_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Function.lean\",null,null,\"QuasiErgodic.ae_eq_const_of_ae_eq_comp_ae\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/Function.lean\",null,null,\"QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range\u2080\"]"
+  ]
+}

manifests/batches/mathlib/001109.json

@@ -0,0 +1,837 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001109",
+  "batch_fingerprint": "48ecaa5eba0998e3e4613df8d7e3b2e37c1409bce7b156dea0c76fab86dccd14",
+  "evidence": {
+    "batch_fingerprint": "48ecaa5eba0998e3e4613df8d7e3b2e37c1409bce7b156dea0c76fab86dccd14",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.measurePreserving",
+        "id": "Measurable.measurePreserving",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            11
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "Measurable.measurePreserving",
+          "start": [
+            50,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurableEquiv.measurePreserving_symm",
+        "id": "MeasureTheory.MeasurableEquiv.measurePreserving_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            44
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurableEquiv.measurePreserving_symm",
+          "start": [
+            193,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.aemeasurable",
+        "id": "MeasureTheory.MeasurePreserving.aemeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            20
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.aemeasurable",
+          "start": [
+            61,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa [B]'",
+        "full_name": "MeasureTheory.MeasurePreserving.exists_mem_iterate_mem_of_volume_lt_mul_volume",
+        "id": "MeasureTheory.MeasurePreserving.exists_mem_iterate_mem_of_volume_lt_mul_volume",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            58
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.exists_mem_iterate_mem_of_volume_lt_mul_volume",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have A : \\u2200 m, <a>MeasurableSet<\\/a> (f^[m] \\u207b\\u00b9' s) := fun m \\u21a6 (hf.iterate m).<a>measurable<\\/a> hs\"",
+                "[{\"full_name\":\"MeasurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.measurable\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[44,13],\"def_end_pos\":[44,23]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "have A : \u2200 m, MeasurableSet (f^[m] \u207b\u00b9' s) := fun m \u21a6 (hf.iterate m).measurable hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"have B : \\u2200 m, \\u03bc (f^[m] \\u207b\\u00b9' s) = \\u03bc s := fun m \\u21a6 (hf.iterate m).<a>measure_preimage<\\/a> hs\"",
+                "[{\"full_name\":\"MeasureTheory.MeasurePreserving.measure_preimage\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,25]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "have B : \u2200 m, \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s := fun m \u21a6 (hf.iterate m).measure_preimage hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u03bc (<a>univ<\\/a> : <a>Set<\\/a> \\u03b1) < \\u2211 m \\u2208 <a>Finset.range<\\/a> n, \\u03bc (f^[m] \\u207b\\u00b9' s) := by simpa [B]\"",
+                "[{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Finset.range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2925,5],\"def_end_pos\":[2925,10]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "have : \u03bc (univ : Set \u03b1) < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s) := by simpa [B]"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8i, hi, j, hj, hij, x, hxi : f^[i] x \\u2208 s, hxj : f^[j] x \\u2208 s\\u27e9 :\\n      \\u2203 i < n, \\u2203 j < n, i \\u2260 j \\u2227 (f^[i] \\u207b\\u00b9' s \\u2229 f^[j] \\u207b\\u00b9' s).<a>Nonempty<\\/a> := by\\n    simpa using <a>exists_nonempty_inter_of_measure_univ_lt_sum_measure<\\/a> \\u03bc (fun m _ \\u21a6 A m) this\"",
+                "[{\"full_name\":\"Set.Nonempty\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[218,15],\"def_end_pos\":[218,23]},{\"full_name\":\"MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,61]}]"
+              ],
+              "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nx : \u03b1\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "obtain \u27e8i, hi, j, hj, hij, x, hxi : f^[i] x \u2208 s, hxj : f^[j] x \u2208 s\u27e9 :\n    \u2203 i < n, \u2203 j < n, i \u2260 j \u2227 (f^[i] \u207b\u00b9' s \u2229 f^[j] \u207b\u00b9' s).Nonempty := by\n  simpa using exists_nonempty_inter_of_measure_univ_lt_sum_measure \u03bc (fun m _ \u21a6 A m) this"
+            },
+            {
+              "annotated_tactic": [
+                "\"wlog hlt : i < j generalizing i j\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis\u271d : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nx : \u03b1\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nthis : \u2200 i < n, \u2200 j < n, i \u2260 j \u2192 f^[i] x \u2208 s \u2192 f^[j] x \u2208 s \u2192 i < j \u2192 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s\nhlt : \u00aci < j\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\nx : \u03b1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nhlt : i < j\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nx : \u03b1\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "wlog hlt : i < j generalizing i j"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8f^[i] x, hxi, j - i, \\u27e8<a>tsub_pos_of_lt<\\/a> hlt, <a>lt_of_le_of_lt<\\/a> (j.sub_le i) hj\\u27e9, ?_\\u27e9\"",
+                "[{\"full_name\":\"tsub_pos_of_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[362,9],\"def_end_pos\":[362,23]},{\"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": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\nx : \u03b1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nhlt : i < j\n\u22a2 f^[j - i] (f^[i] x) \u2208 s",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\nx : \u03b1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nhlt : i < j\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "refine \u27e8f^[i] x, hxi, j - i, \u27e8tsub_pos_of_lt hlt, lt_of_le_of_lt (j.sub_le i) hj\u27e9, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>iterate_add_apply<\\/a>, <a>tsub_add_cancel_of_le<\\/a> hlt.le]\"",
+                "[{\"full_name\":\"Function.iterate_add_apply\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Iterate.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,26]},{\"full_name\":\"tsub_add_cancel_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[31,9],\"def_end_pos\":[31,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\nx : \u03b1\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nhlt : i < j\n\u22a2 f^[j - i] (f^[i] x) \u2208 s",
+              "tactic": "rwa [\u2190 iterate_add_apply, tsub_add_cancel_of_le hlt.le]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [B]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\n\u22a2 \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)",
+              "tactic": "simpa [B]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using <a>exists_nonempty_inter_of_measure_univ_lt_sum_measure<\\/a> \\u03bc (fun m _ \\u21a6 A m) this\"",
+                "[{\"full_name\":\"MeasureTheory.exists_nonempty_inter_of_measure_univ_lt_sum_measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,61]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\n\u22a2 \u2203 i < n, \u2203 j < n, i \u2260 j \u2227 (f^[i] \u207b\u00b9' s \u2229 f^[j] \u207b\u00b9' s).Nonempty",
+              "tactic": "simpa using exists_nonempty_inter_of_measure_univ_lt_sum_measure \u03bc (fun m _ \u21a6 A m) this"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact this j hj i hi hij.symm hxj hxi (hij.lt_or_lt.resolve_left hlt)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.intro.intro.intro.intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nhvol : \u03bc univ < \u2191n * \u03bc s\nA : \u2200 (m : \u2115), MeasurableSet (f^[m] \u207b\u00b9' s)\nB : \u2200 (m : \u2115), \u03bc (f^[m] \u207b\u00b9' s) = \u03bc s\nthis\u271d : \u03bc univ < \u2211 m \u2208 Finset.range n, \u03bc (f^[m] \u207b\u00b9' s)\ni : \u2115\nhi : i < n\nj : \u2115\nhj : j < n\nhij : i \u2260 j\nx : \u03b1\nhxi : f^[i] x \u2208 s\nhxj : f^[j] x \u2208 s\nthis : \u2200 i < n, \u2200 j < n, i \u2260 j \u2192 f^[i] x \u2208 s \u2192 f^[j] x \u2208 s \u2192 i < j \u2192 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s\nhlt : \u00aci < j\n\u22a2 \u2203 x \u2208 s, \u2203 m \u2208 Ioo 0 n, f^[m] x \u2208 s",
+              "tactic": "exact this j hj i hi hij.symm hxj hxi (hij.lt_or_lt.resolve_left hlt)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.id",
+        "id": "MeasureTheory.MeasurePreserving.id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            26
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.id",
+          "start": [
+            57,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.iterate",
+        "id": "MeasureTheory.MeasurePreserving.iterate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            54
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.iterate",
+          "start": [
+            143,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le",
+        "id": "MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            93
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.measure_symmDiff_preimage_iterate_le",
+          "start": [
+            152,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction' n with n ih\"",
+                "[]"
+              ],
+              "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\n\u22a2 \u03bc (s \u2206 (f^[0] \u207b\u00b9' s)) \u2264 0 \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc (s \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 (n + 1) \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\n\u22a2 \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "induction' n with n ih"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>add_smul<\\/a>, <a>one_smul<\\/a>, \\u2190 n.add_one]\"",
+                "[{\"full_name\":\"add_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc (s \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s)) + \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc (s \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 (n + 1) \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "simp only [add_smul, one_smul, \u2190 n.add_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>measure_symmDiff_le<\\/a> s (f^[n] \\u207b\\u00b9' s) (f^[n+1] \\u207b\\u00b9' s)) (<a>add_le_add<\\/a> ih ?_)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"MeasureTheory.measure_symmDiff_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[144,7],\"def_end_pos\":[144,26]},{\"full_name\":\"add_le_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[205,32],\"def_end_pos\":[205,42]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc ((f^[n] \u207b\u00b9' s) \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc (s \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s)) + \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "refine le_trans (measure_symmDiff_le s (f^[n] \u207b\u00b9' s) (f^[n+1] \u207b\u00b9' s)) (add_le_add ih ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace hs : <a>MeasurableSet<\\/a> (s \\u2206 (f \\u207b\\u00b9' s)) := hs.symmDiff <| hf.measurable hs\"",
+                "[{\"full_name\":\"MeasurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,18]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\nhs : MeasurableSet (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc ((f^[n] \u207b\u00b9' s) \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc ((f^[n] \u207b\u00b9' s) \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "replace hs : MeasurableSet (s \u2206 (f \u207b\u00b9' s)) := hs.symmDiff <| hf.measurable hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>iterate_succ'<\\/a>, <a>preimage_comp<\\/a>, \\u2190 <a>preimage_symmDiff<\\/a>, (hf.iterate n).<a>measure_preimage<\\/a> hs]\"",
+                "[{\"full_name\":\"Function.iterate_succ'\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Iterate.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,22]},{\"full_name\":\"Set.preimage_comp\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,22]},{\"full_name\":\"Set.preimage_symmDiff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[93,7],\"def_end_pos\":[93,24]},{\"full_name\":\"MeasureTheory.MeasurePreserving.measure_preimage\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nn : \u2115\nih : \u03bc (s \u2206 (f^[n] \u207b\u00b9' s)) \u2264 n \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))\nhs : MeasurableSet (s \u2206 (f \u207b\u00b9' s))\n\u22a2 \u03bc ((f^[n] \u207b\u00b9' s) \u2206 (f^[n + 1] \u207b\u00b9' s)) \u2264 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "rw [iterate_succ', preimage_comp, \u2190 preimage_symmDiff, (hf.iterate n).measure_preimage hs]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\ninst\u271d : MeasurableSpace \u03b4\n\u03bca : Measure \u03b1\n\u03bcb : Measure \u03b2\n\u03bcc : Measure \u03b3\n\u03bcd : Measure \u03b4\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\nhf : MeasurePreserving f \u03bc \u03bc\nhs : MeasurableSet s\n\u22a2 \u03bc (s \u2206 (f^[0] \u207b\u00b9' s)) \u2264 0 \u2022 \u03bc (s \u2206 (f \u207b\u00b9' s))",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.of_isEmpty",
+        "id": "MeasureTheory.MeasurePreserving.of_isEmpty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            65
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.of_isEmpty",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.quasiMeasurePreserving",
+        "id": "MeasureTheory.MeasurePreserving.quasiMeasurePreserving",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            36
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.quasiMeasurePreserving",
+          "start": [
+            97,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.MeasurePreserving.trans",
+        "id": "MeasureTheory.MeasurePreserving.trans",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            113,
+            12
+          ],
+          "file_path": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean",
+          "full_name": "MeasureTheory.MeasurePreserving.trans",
+          "start": [
+            109,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Ergodic/MeasurePreserving.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.Injective.isFixedPt_apply_iff",
+        "id": "Function.Injective.isFixedPt_apply_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            38
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.Injective.isFixedPt_apply_iff",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.comp",
+        "id": "Function.IsFixedPt.comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            16
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.comp",
+          "start": [
+            59,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.eq",
+        "id": "Function.IsFixedPt.eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            5
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.eq",
+          "start": [
+            54,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.equiv_symm",
+        "id": "Function.IsFixedPt.equiv_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            108,
+            38
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.equiv_symm",
+          "start": [
+            107,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.image_iterate",
+        "id": "Function.IsFixedPt.image_iterate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            37
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.image_iterate",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.iterate",
+        "id": "Function.IsFixedPt.iterate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            21
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.iterate",
+          "start": [
+            66,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.left_of_comp",
+        "id": "Function.IsFixedPt.left_of_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            17
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.left_of_comp",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.map",
+        "id": "Function.IsFixedPt.map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            30
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.map",
+          "start": [
+            87,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.perm_inv",
+        "id": "Function.IsFixedPt.perm_inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            15
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.perm_inv",
+          "start": [
+            111,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.perm_pow",
+        "id": "Function.IsFixedPt.perm_pow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            95
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.perm_pow",
+          "start": [
+            115,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.perm_zpow",
+        "id": "Function.IsFixedPt.perm_zpow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            52
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.perm_zpow",
+          "start": [
+            118,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.IsFixedPt.to_leftInverse",
+        "id": "Function.IsFixedPt.to_leftInverse",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            17
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.IsFixedPt.to_leftInverse",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.Semiconj.mapsTo_fixedPoints",
+        "id": "Function.Semiconj.mapsTo_fixedPoints",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            161,
+            75
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.Semiconj.mapsTo_fixedPoints",
+          "start": [
+            160,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.bijOn_fixedPoints_comp",
+        "id": "Function.bijOn_fixedPoints_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            97
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.bijOn_fixedPoints_comp",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.fixedPoints_subset_range",
+        "id": "Function.fixedPoints_subset_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            86
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.fixedPoints_subset_range",
+          "start": [
+            155,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.invOn_fixedPoints_comp",
+        "id": "Function.invOn_fixedPoints_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            168,
+            29
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.invOn_fixedPoints_comp",
+          "start": [
+            166,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.mapsTo_fixedPoints_comp",
+        "id": "Function.mapsTo_fixedPoints_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            98
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.mapsTo_fixedPoints_comp",
+          "start": [
+            172,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.mem_fixedPoints",
+        "id": "Function.mem_fixedPoints",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            143,
+            10
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean",
+          "full_name": "Function.mem_fixedPoints",
+          "start": [
+            142,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Basic.lean"
+      }
+    ],
+    "input_proofs": 45,
+    "input_units": 2,
+    "proofs": 17,
+    "published_proofs": 17,
+    "published_transitions": 24,
+    "replay_failures": [],
+    "transitions": 24,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001109-0000-bdc6d201c055daf0.parquet",
+      "sha256": "bdc6d201c055daf0b444dd3e2e0b3c821f7343e4d7a17559860bae208b7e27d2",
+      "bytes": 79960,
+      "rows": 20,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/mathlib-001109-0000-6ba1dca5c8f2e730.parquet",
+      "sha256": "6ba1dca5c8f2e730f9e1fd260b34ca8a0c9a32979d5f4bba8fb78de83598440b",
+      "bytes": 57702,
+      "rows": 4,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Function.Commute.invOn_fixedPoints_comp",
+    "mathlib:Function.Commute.left_bijOn_fixedPoints_comp",
+    "mathlib:Function.Commute.right_bijOn_fixedPoints_comp",
+    "mathlib:Function.IsFixedPt.preimage_iterate",
+    "mathlib:Function.fixedPoints_id",
+    "mathlib:Function.mem_fixedPoints_iff",
+    "mathlib:MeasureTheory.MeasurePreserving.aemeasurable_comp_iff",
+    "mathlib:MeasureTheory.MeasurePreserving.comp",
+    "mathlib:MeasureTheory.MeasurePreserving.comp_left_iff",
+    "mathlib:MeasureTheory.MeasurePreserving.comp_right_iff",
+    "mathlib:MeasureTheory.MeasurePreserving.exists_mem_iterate_mem",
+    "mathlib:MeasureTheory.MeasurePreserving.measure_preimage",
+    "mathlib:MeasureTheory.MeasurePreserving.measure_preimage_emb",
+    "mathlib:MeasureTheory.MeasurePreserving.restrict_image_emb",
+    "mathlib:MeasureTheory.MeasurePreserving.restrict_preimage",
+    "mathlib:MeasureTheory.MeasurePreserving.sigmaFinite",
+    "mathlib:MeasureTheory.MeasurePreserving.symm"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.aemeasurable_comp_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.comp_left_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.comp_right_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.exists_mem_iterate_mem\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.measure_preimage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.measure_preimage_emb\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.restrict_image_emb\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.restrict_preimage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.sigmaFinite\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Ergodic/MeasurePreserving.lean\",null,null,\"MeasureTheory.MeasurePreserving.symm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.Commute.invOn_fixedPoints_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.Commute.left_bijOn_fixedPoints_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.Commute.right_bijOn_fixedPoints_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.IsFixedPt.preimage_iterate\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.fixedPoints_id\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Basic.lean\",null,null,\"Function.mem_fixedPoints_iff\"]"
+  ]
+}

manifests/batches/mathlib/001110.json

@@ -0,0 +1,292 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001110",
+  "batch_fingerprint": "8cf1ccb2cb994ac8668b88e95244ddbaf680a33640bcbe18f1c679a30af39e07",
+  "evidence": {
+    "batch_fingerprint": "8cf1ccb2cb994ac8668b88e95244ddbaf680a33640bcbe18f1c679a30af39e07",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isClosed_fixedPoints",
+        "id": "isClosed_fixedPoints",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            42,
+            31
+          ],
+          "file_path": "Mathlib/Dynamics/FixedPoints/Topology.lean",
+          "full_name": "isClosed_fixedPoints",
+          "start": [
+            41,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/FixedPoints/Topology.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Flow.continuous",
+        "id": "Flow.continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            31
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "Flow.continuous",
+          "start": [
+            116,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Flow.image_eq_preimage",
+        "id": "Flow.image_eq_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            192,
+            49
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "Flow.image_eq_preimage",
+          "start": [
+            191,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Flow.map_add",
+        "id": "Flow.map_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            88
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "Flow.map_add",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Flow.map_zero",
+        "id": "Flow.map_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            50
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "Flow.map_zero",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Flow.map_zero_apply",
+        "id": "Flow.map_zero_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            131,
+            60
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "Flow.map_zero_apply",
+          "start": [
+            131,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsFwInvariant.isInvariant",
+        "id": "IsFwInvariant.isInvariant",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            72
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "IsFwInvariant.isInvariant",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsInvariant.isFwInvariant",
+        "id": "IsInvariant.isFwInvariant",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            66
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "IsInvariant.isFwInvariant",
+          "start": [
+            59,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isFwInvariant_iff_isInvariant",
+        "id": "isFwInvariant_iff_isInvariant",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            57
+          ],
+          "file_path": "Mathlib/Dynamics/Flow.lean",
+          "full_name": "isFwInvariant_iff_isInvariant",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Flow.lean"
+      }
+    ],
+    "input_proofs": 14,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 4,
+    "published_transitions": 10,
+    "replay_failures": [],
+    "transitions": 12,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Dynamics/Flow.lean",
+        "proof_id": "mathlib:Flow.isInvariant_iff_image_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              162,
+              30
+            ],
+            "file_path": "Mathlib/Dynamics/Flow.lean",
+            "full_name": "Flow.isInvariant_iff_image_eq",
+            "start": [
+              158,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [\\u2190 <a>map_add<\\/a>]\"",
+                  "[{\"full_name\":\"Flow.map_add\",\"def_path\":\"Mathlib\\/Dynamics\\/Flow.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03c4 : Type u_1\ninst\u271d\u00b3 : AddCommGroup \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : TopologicalAddGroup \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\ns : Set \u03b1\nh : \u2200 (t : \u03c4), \u03d5.toFun t '' s \u2286 s\nt : \u03c4\nx\u271d : \u03b1\nhx : x\u271d \u2208 s\n\u22a2 \u03d5.toFun t (\u03d5.toFun (-t) x\u271d) = x\u271d",
+                "tactic": "simp [\u2190 map_add]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [h t]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03c4 : Type u_1\ninst\u271d\u00b3 : AddCommGroup \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : TopologicalAddGroup \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\ns : Set \u03b1\nh : \u2200 (t : \u03c4), \u03d5.toFun t '' s = s\nt : \u03c4\n\u22a2 \u03d5.toFun t '' s \u2286 s",
+                "tactic": "rw [h t]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Flow.isInvariant_iff_image_eq"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001110-0000-53711de6535d8b28.parquet",
+      "sha256": "53711de6535d8b28be4a07e262690ad893b06b976c73110555e1bc22c9853739",
+      "bytes": 55043,
+      "rows": 9,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/dev/mathlib-001110-0000-e59b211f5e30add1.parquet",
+      "sha256": "e59b211f5e30add1bbae688c4eee255a2f115d92b1cacbb88f0df0a957f4c7de",
+      "bytes": 31710,
+      "rows": 1,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Flow.continuous_toFun",
+    "mathlib:Flow.ext",
+    "mathlib:isFixedPt_of_tendsto_iterate",
+    "mathlib:isInvariant_iff_image"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/FixedPoints/Topology.lean\",null,null,\"isFixedPt_of_tendsto_iterate\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Flow.lean\",null,null,\"Flow.continuous_toFun\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Flow.lean\",null,null,\"Flow.ext\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Flow.lean\",null,null,\"isInvariant_iff_image\"]"
+  ]
+}

manifests/batches/mathlib/001111.json

@@ -0,0 +1,642 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001111",
+  "batch_fingerprint": "fa7bf1741eeb0f9175c452da9cb37dfffe698d4c38470a2b6be82ceceb035167",
+  "evidence": {
+    "batch_fingerprint": "fa7bf1741eeb0f9175c452da9cb37dfffe698d4c38470a2b6be82ceceb035167",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsCompact.exists_finite_cover_smul",
+        "id": "IsCompact.exists_finite_cover_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            55
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "IsCompact.exists_finite_cover_smul",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsOpen.exists_smul_mem",
+        "id": "IsOpen.exists_smul_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            48
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "IsOpen.exists_smul_mem",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsOpen.iUnion_preimage_smul",
+        "id": "IsOpen.iUnion_preimage_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            59
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "IsOpen.iUnion_preimage_smul",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsOpen.iUnion_smul",
+        "id": "IsOpen.iUnion_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            36
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "IsOpen.iUnion_smul",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulAction.dense_orbit",
+        "id": "MulAction.dense_orbit",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            36
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "MulAction.dense_orbit",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "denseRange_smul",
+        "id": "denseRange_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            28
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "denseRange_smul",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dense_of_nonempty_smul_invariant",
+        "id": "dense_of_nonempty_smul_invariant",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            85
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "dense_of_nonempty_smul_invariant",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "eq_empty_or_univ_of_smul_invariant_closed",
+        "id": "eq_empty_or_univ_of_smul_invariant_closed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            114,
+            78
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "eq_empty_or_univ_of_smul_invariant_closed",
+          "start": [
+            111,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro _ _'",
+        "full_name": "isMinimal_iff_closed_smul_invariant",
+        "id": "isMinimal_iff_closed_smul_invariant",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            126,
+            41
+          ],
+          "file_path": "Mathlib/Dynamics/Minimal.lean",
+          "full_name": "isMinimal_iff_closed_smul_invariant",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 IsMinimal M \u03b1 \u2192 \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\n\ncase mpr\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ) \u2192 IsMinimal M \u03b1",
+              "state_before": "M : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 IsMinimal M \u03b1 \u2194 \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine fun H \\u21a6 \\u27e8fun _ \\u21a6 <a>dense_iff_closure_eq<\\/a>.2 <| (H _ ?_ ?_).<a>resolve_left<\\/a> ?_\\u27e9\"",
+                "[{\"full_name\":\"dense_iff_closure_eq\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[594,9],\"def_end_pos\":[594,29]},{\"full_name\":\"Or.resolve_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,24]}]"
+              ],
+              "state_after": "case mpr.refine_1\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 IsClosed (closure (orbit M x\u271d))\n\ncase mpr.refine_2\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u2200 (c : M), c \u2022 closure (orbit M x\u271d) \u2286 closure (orbit M x\u271d)\n\ncase mpr.refine_3\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u00acclosure (orbit M x\u271d) = \u2205",
+              "state_before": "case mpr\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ) \u2192 IsMinimal M \u03b1",
+              "tactic": "refine fun H \u21a6 \u27e8fun _ \u21a6 dense_iff_closure_eq.2 <| (H _ ?_ ?_).resolve_left ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exacts [<a>isClosed_closure<\\/a>, fun _ \\u21a6 <a>smul_closure_orbit_subset<\\/a> _ _,\\n    (<a>orbit_nonempty<\\/a> _).closure.ne_empty]\"",
+                "[{\"full_name\":\"isClosed_closure\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[399,9],\"def_end_pos\":[399,25]},{\"full_name\":\"smul_closure_orbit_subset\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/ConstMulAction.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,34]},{\"full_name\":\"MulAction.orbit_nonempty\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr.refine_1\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 IsClosed (closure (orbit M x\u271d))\n\ncase mpr.refine_2\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u2200 (c : M), c \u2022 closure (orbit M x\u271d) \u2286 closure (orbit M x\u271d)\n\ncase mpr.refine_3\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u00acclosure (orbit M x\u271d) = \u2205",
+              "tactic": "exacts [isClosed_closure, fun _ \u21a6 smul_closure_orbit_subset _ _,\n  (orbit_nonempty _).closure.ne_empty]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro _ _\"",
+                "[]"
+              ],
+              "state_after": "case mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\na\u271d : IsMinimal M \u03b1\ns\u271d : Set \u03b1\n\u22a2 IsClosed s\u271d \u2192 (\u2200 (c : M), c \u2022 s\u271d \u2286 s\u271d) \u2192 s\u271d = \u2205 \u2228 s\u271d = univ",
+              "state_before": "case mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 IsMinimal M \u03b1 \u2192 \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ",
+              "tactic": "intro _ _"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>eq_empty_or_univ_of_smul_invariant_closed<\\/a> M\"",
+                "[{\"full_name\":\"eq_empty_or_univ_of_smul_invariant_closed\",\"def_path\":\"Mathlib\\/Dynamics\\/Minimal.lean\",\"def_pos\":[111,9],\"def_end_pos\":[111,50]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\na\u271d : IsMinimal M \u03b1\ns\u271d : Set \u03b1\n\u22a2 IsClosed s\u271d \u2192 (\u2200 (c : M), c \u2022 s\u271d \u2286 s\u271d) \u2192 s\u271d = \u2205 \u2228 s\u271d = univ",
+              "tactic": "exact eq_empty_or_univ_of_smul_invariant_closed M"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Minimal.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"rwa [derivative_map, eval_map_algebraMap, eval_map_algebraMap, eval_map_algebraMap,\\n  add_sub_cancel, hr\u2082', hr\u2081', zero_add, pow_two, \u2190 mul_assoc, \u2190 add_mul, eq_comm,\\n  this.mul_right_eq_zero, sub_eq_zero, eq_comm] at hu\"",
+        "full_name": "Polynomial.exists_unique_nilpotent_sub_and_aeval_eq_zero",
+        "id": "Polynomial.exists_unique_nilpotent_sub_and_aeval_eq_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            126,
+            95
+          ],
+          "file_path": "Mathlib/Dynamics/Newton.lean",
+          "full_name": "Polynomial.exists_unique_nilpotent_sub_and_aeval_eq_zero",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp_rw [(<a>neg_sub<\\/a> _ x).<a>symm<\\/a>, <a>isNilpotent_neg_iff<\\/a>]\"",
+                "[{\"full_name\":\"neg_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[553,3],\"def_end_pos\":[553,14]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"isNilpotent_neg_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,28]}]"
+              ],
+              "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 \u2203! r, IsNilpotent (r - x) \u2227 (aeval r) P = 0",
+              "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 \u2203! r, IsNilpotent (x - r) \u2227 (aeval r) P = 0",
+              "tactic": "simp_rw [(neg_sub _ x).symm, isNilpotent_neg_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>exists_unique_of_exists_of_unique<\\/a> ?_ fun r\\u2081 r\\u2082 \\u27e8hr\\u2081, hr\\u2081'\\u27e9 \\u27e8hr\\u2082, hr\\u2082'\\u27e9 \\u21a6 ?_\"",
+                "[{\"full_name\":\"exists_unique_of_exists_of_unique\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[255,9],\"def_end_pos\":[255,42]}]"
+              ],
+              "state_after": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 \u2203 x_1, IsNilpotent (x_1 - x) \u2227 (aeval x_1) P = 0\n\ncase refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\n\u22a2 r\u2081 = r\u2082",
+              "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 \u2203! r, IsNilpotent (r - x) \u2227 (aeval r) P = 0",
+              "tactic": "refine exists_unique_of_exists_of_unique ?_ fun r\u2081 r\u2082 \u27e8hr\u2081, hr\u2081'\u27e9 \u27e8hr\u2082, hr\u2082'\u27e9 \u21a6 ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8n, hn\\u27e9 := <a>id<\\/a> h\"",
+                "[{\"full_name\":\"id\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[33,15],\"def_end_pos\":[33,17]}]"
+              ],
+              "state_after": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 \u2203 x_1, IsNilpotent (x_1 - x) \u2227 (aeval x_1) P = 0",
+              "state_before": "case refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 \u2203 x_1, IsNilpotent (x_1 - x) \u2227 (aeval x_1) P = 0",
+              "tactic": "obtain \u27e8n, hn\u27e9 := id h"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8P.newtonMap^[n] x, <a>isNilpotent_iterate_newtonMap_sub_of_isNilpotent<\\/a> h n, ?_\\u27e9\"",
+                "[{\"full_name\":\"Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,57]}]"
+              ],
+              "state_after": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 (aeval (P.newtonMap^[n] x)) P = 0",
+              "state_before": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 \u2203 x_1, IsNilpotent (x_1 - x) \u2227 (aeval x_1) P = 0",
+              "tactic": "refine \u27e8P.newtonMap^[n] x, isNilpotent_iterate_newtonMap_sub_of_isNilpotent h n, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>zero_dvd_iff<\\/a>, \\u2190 <a>pow_eq_zero_of_le<\\/a> n.lt_two_pow.le hn]\"",
+                "[{\"full_name\":\"zero_dvd_iff\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[34,9],\"def_end_pos\":[34,21]},{\"full_name\":\"pow_eq_zero_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[171,7],\"def_end_pos\":[171,24]}]"
+              ],
+              "state_after": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P",
+              "state_before": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 (aeval (P.newtonMap^[n] x)) P = 0",
+              "tactic": "rw [\u2190 zero_dvd_iff, \u2190 pow_eq_zero_of_le n.lt_two_pow.le hn]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>aeval_pow_two_pow_dvd_aeval_iterate_newtonMap<\\/a> h h' n\"",
+                "[{\"full_name\":\"Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,54]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nhn : (aeval x) P ^ n = 0\n\u22a2 (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P",
+              "tactic": "exact aeval_pow_two_pow_dvd_aeval_iterate_newtonMap h h' n"
+            },
+            {
+              "annotated_tactic": [
+                "\"have \\u27e8u, hu\\u27e9 := <a>binomExpansion<\\/a> (P.map (<a>algebraMap<\\/a> R S)) r\\u2081 (r\\u2082 - r\\u2081)\"",
+                "[{\"full_name\":\"Polynomial.binomExpansion\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Identities.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,19]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
+              ],
+              "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\n\u22a2 r\u2081 = r\u2082",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\n\u22a2 r\u2081 = r\u2082",
+              "tactic": "have \u27e8u, hu\u27e9 := binomExpansion (P.map (algebraMap R S)) r\u2081 (r\u2082 - r\u2081)"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices <a>IsUnit<\\/a> (<a>aeval<\\/a> r\\u2081 (<a>derivative<\\/a> P) + u * (r\\u2082 - r\\u2081)) by\\n      rwa [<a>derivative_map<\\/a>, <a>eval_map_algebraMap<\\/a>, <a>eval_map_algebraMap<\\/a>, <a>eval_map_algebraMap<\\/a>,\\n        <a>add_sub_cancel<\\/a>, hr\\u2082', hr\\u2081', <a>zero_add<\\/a>, <a>pow_two<\\/a>, \\u2190 <a>mul_assoc<\\/a>, \\u2190 <a>add_mul<\\/a>, <a>eq_comm<\\/a>,\\n        this.mul_right_eq_zero, <a>sub_eq_zero<\\/a>, <a>eq_comm<\\/a>] at hu\"",
+                "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]},{\"full_name\":\"Polynomial.derivative_map\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,23]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"add_sub_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"add_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[91,7],\"def_end_pos\":[91,14]},{\"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\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"eq_comm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,16]}]"
+              ],
+              "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - r\u2081))",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\n\u22a2 r\u2081 = r\u2082",
+              "tactic": "suffices IsUnit (aeval r\u2081 (derivative P) + u * (r\u2082 - r\u2081)) by\n  rwa [derivative_map, eval_map_algebraMap, eval_map_algebraMap, eval_map_algebraMap,\n    add_sub_cancel, hr\u2082', hr\u2081', zero_add, pow_two, \u2190 mul_assoc, \u2190 add_mul, eq_comm,\n    this.mul_right_eq_zero, sub_eq_zero, eq_comm] at hu"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>IsUnit<\\/a> (<a>aeval<\\/a> r\\u2081 (<a>derivative<\\/a> P)) :=\\n      <a>isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub<\\/a> h' hr\\u2081\"",
+                "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]},{\"full_name\":\"Polynomial.isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Nilpotent.lean\",\"def_pos\":[202,7],\"def_end_pos\":[202,54]}]"
+              ],
+              "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - r\u2081))",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - r\u2081))",
+              "tactic": "have : IsUnit (aeval r\u2081 (derivative P)) :=\n  isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub h' hr\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>sub_sub_sub_cancel_right<\\/a> r\\u2082 r\\u2081 x]\"",
+                "[{\"full_name\":\"sub_sub_sub_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1064,30],\"def_end_pos\":[1064,54]}]"
+              ],
+              "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - x - (r\u2081 - x)))",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - r\u2081))",
+              "tactic": "rw [\u2190 sub_sub_sub_cancel_right r\u2082 r\u2081 x]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>IsNilpotent.isUnit_add_left_of_commute<\\/a> ?_ this (<a>Commute.all<\\/a> _ _)\"",
+                "[{\"full_name\":\"IsNilpotent.isUnit_add_left_of_commute\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,47]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]}]"
+              ],
+              "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsNilpotent (u * (r\u2082 - x - (r\u2081 - x)))",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - x - (r\u2081 - x)))",
+              "tactic": "refine IsNilpotent.isUnit_add_left_of_commute ?_ this (Commute.all _ _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>Commute.all<\\/a> _ _).<a>isNilpotent_mul_right<\\/a> <| (<a>Commute.all<\\/a> _ _).<a>isNilpotent_sub<\\/a> hr\\u2082 hr\\u2081\"",
+                "[{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_mul_right\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Defs.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,30]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[174,9],\"def_end_pos\":[174,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P))\n\u22a2 IsNilpotent (u * (r\u2082 - x - (r\u2081 - x)))",
+              "tactic": "exact (Commute.all _ _).isNilpotent_mul_right <| (Commute.all _ _).isNilpotent_sub hr\u2082 hr\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>derivative_map<\\/a>, <a>eval_map_algebraMap<\\/a>, <a>eval_map_algebraMap<\\/a>, <a>eval_map_algebraMap<\\/a>,\\n        <a>add_sub_cancel<\\/a>, hr\\u2082', hr\\u2081', <a>zero_add<\\/a>, <a>pow_two<\\/a>, \\u2190 <a>mul_assoc<\\/a>, \\u2190 <a>add_mul<\\/a>, <a>eq_comm<\\/a>,\\n        this.mul_right_eq_zero, <a>sub_eq_zero<\\/a>, <a>eq_comm<\\/a>] at hu\"",
+                "[{\"full_name\":\"Polynomial.derivative_map\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,23]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"add_sub_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1330,3],\"def_end_pos\":[1330,14]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"add_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[91,7],\"def_end_pos\":[91,14]},{\"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\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"eq_comm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr\u2081 r\u2082 : S\nx\u271d\u00b9 : IsNilpotent (r\u2081 - x) \u2227 (aeval r\u2081) P = 0\nx\u271d : IsNilpotent (r\u2082 - x) \u2227 (aeval r\u2082) P = 0\nhr\u2081 : IsNilpotent (r\u2081 - x)\nhr\u2081' : (aeval r\u2081) P = 0\nhr\u2082 : IsNilpotent (r\u2082 - x)\nhr\u2082' : (aeval r\u2082) P = 0\nu : S\nhu :\n  eval (r\u2081 + (r\u2082 - r\u2081)) (map (algebraMap R S) P) =\n    eval r\u2081 (map (algebraMap R S) P) + eval r\u2081 (derivative (map (algebraMap R S) P)) * (r\u2082 - r\u2081) + u * (r\u2082 - r\u2081) ^ 2\nthis : IsUnit ((aeval r\u2081) (derivative P) + u * (r\u2082 - r\u2081))\n\u22a2 r\u2081 = r\u2082",
+              "tactic": "rwa [derivative_map, eval_map_algebraMap, eval_map_algebraMap, eval_map_algebraMap,\n  add_sub_cancel, hr\u2082', hr\u2081', zero_add, pow_two, \u2190 mul_assoc, \u2190 add_mul, eq_comm,\n  this.mul_right_eq_zero, sub_eq_zero, eq_comm] at hu"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Newton.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent",
+        "id": "Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            56
+          ],
+          "file_path": "Mathlib/Dynamics/Newton.lean",
+          "full_name": "Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp\\n  | succ n ih =>\\n    rw [<a>iterate_succ'<\\/a>, <a>comp_apply<\\/a>, <a>newtonMap_apply<\\/a>, <a>sub_right_comm<\\/a>]\\n    refine (<a>Commute.all<\\/a> _ _).<a>isNilpotent_sub<\\/a> ih <| (<a>Commute.all<\\/a> _ _).<a>isNilpotent_mul_right<\\/a> ?_\\n    simpa using <a>Commute.isNilpotent_add<\\/a> (<a>Commute.all<\\/a> _ _)\\n      (<a>isNilpotent_aeval_sub_of_isNilpotent_sub<\\/a> P ih) h\"",
+                "[{\"full_name\":\"Function.iterate_succ'\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Iterate.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,22]},{\"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\":\"Polynomial.newtonMap_apply\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,24]},{\"full_name\":\"sub_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[765,3],\"def_end_pos\":[765,14]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[174,9],\"def_end_pos\":[174,24]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_mul_right\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Defs.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,30]},{\"full_name\":\"Commute.isNilpotent_add\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,24]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Polynomial.isNilpotent_aeval_sub_of_isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Nilpotent.lean\",\"def_pos\":[194,7],\"def_end_pos\":[194,47]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\n\u22a2 IsNilpotent (P.newtonMap^[n] x - x)",
+              "tactic": "induction n with\n| zero => simp\n| succ n ih =>\n  rw [iterate_succ', comp_apply, newtonMap_apply, sub_right_comm]\n  refine (Commute.all _ _).isNilpotent_sub ih <| (Commute.all _ _).isNilpotent_mul_right ?_\n  simpa using Commute.isNilpotent_add (Commute.all _ _)\n    (isNilpotent_aeval_sub_of_isNilpotent_sub P ih) h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\n\u22a2 IsNilpotent (P.newtonMap^[0] x - x)",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>iterate_succ'<\\/a>, <a>comp_apply<\\/a>, <a>newtonMap_apply<\\/a>, <a>sub_right_comm<\\/a>]\"",
+                "[{\"full_name\":\"Function.iterate_succ'\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Iterate.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,22]},{\"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\":\"Polynomial.newtonMap_apply\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,24]},{\"full_name\":\"sub_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[765,3],\"def_end_pos\":[765,14]}]"
+              ],
+              "state_after": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\nih : IsNilpotent (P.newtonMap^[n] x - x)\n\u22a2 IsNilpotent\n    (P.newtonMap^[n] x - x - Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)",
+              "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\nih : IsNilpotent (P.newtonMap^[n] x - x)\n\u22a2 IsNilpotent (P.newtonMap^[n + 1] x - x)",
+              "tactic": "rw [iterate_succ', comp_apply, newtonMap_apply, sub_right_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (<a>Commute.all<\\/a> _ _).<a>isNilpotent_sub<\\/a> ih <| (<a>Commute.all<\\/a> _ _).<a>isNilpotent_mul_right<\\/a> ?_\"",
+                "[{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[174,9],\"def_end_pos\":[174,24]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Commute.isNilpotent_mul_right\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Defs.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,30]}]"
+              ],
+              "state_after": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\nih : IsNilpotent (P.newtonMap^[n] x - x)\n\u22a2 IsNilpotent ((aeval (P.newtonMap^[n] x)) P)",
+              "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\nih : IsNilpotent (P.newtonMap^[n] x - x)\n\u22a2 IsNilpotent\n    (P.newtonMap^[n] x - x - Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)",
+              "tactic": "refine (Commute.all _ _).isNilpotent_sub ih <| (Commute.all _ _).isNilpotent_mul_right ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using <a>Commute.isNilpotent_add<\\/a> (<a>Commute.all<\\/a> _ _)\\n      (<a>isNilpotent_aeval_sub_of_isNilpotent_sub<\\/a> P ih) h\"",
+                "[{\"full_name\":\"Commute.isNilpotent_add\",\"def_path\":\"Mathlib\\/RingTheory\\/Nilpotent\\/Basic.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,24]},{\"full_name\":\"Commute.all\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Commute\\/Defs.lean\",\"def_pos\":[145,19],\"def_end_pos\":[145,22]},{\"full_name\":\"Polynomial.isNilpotent_aeval_sub_of_isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Nilpotent.lean\",\"def_pos\":[194,7],\"def_end_pos\":[194,47]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nn : \u2115\nih : IsNilpotent (P.newtonMap^[n] x - x)\n\u22a2 IsNilpotent ((aeval (P.newtonMap^[n] x)) P)",
+              "tactic": "simpa using Commute.isNilpotent_add (Commute.all _ _)\n  (isNilpotent_aeval_sub_of_isNilpotent_sub P ih) h"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Newton.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Polynomial.newtonMap_apply",
+        "id": "Polynomial.newtonMap_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            6
+          ],
+          "file_path": "Mathlib/Dynamics/Newton.lean",
+          "full_name": "Polynomial.newtonMap_apply",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Dynamics/Newton.lean"
+      }
+    ],
+    "input_proofs": 17,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 4,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 14,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Dynamics/Newton.lean",
+        "proof_id": "mathlib:Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              99,
+              61
+            ],
+            "file_path": "Mathlib/Dynamics/Newton.lean",
+            "full_name": "Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap",
+            "start": [
+              81,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case zero\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\n\u22a2 (aeval x) P ^ 2 ^ 0 \u2223 (aeval (P.newtonMap^[0] x)) P",
+                "tactic": "simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have \\u27e8d, hd\\u27e9 := <a>binomExpansion<\\/a> (P.map (<a>algebraMap<\\/a> R S)) (P.newtonMap^[n] x)\\n      (-<a>Ring.inverse<\\/a> (<a>aeval<\\/a> (P.newtonMap^[n] x) <| <a>derivative<\\/a> P) * <a>aeval<\\/a> (P.newtonMap^[n] x) P)\"",
+                  "[{\"full_name\":\"Polynomial.binomExpansion\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Identities.lean\",\"def_pos\":[79,5],\"def_end_pos\":[79,19]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Ring.inverse\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[93,19],\"def_end_pos\":[93,26]},{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]},{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]}]"
+                ],
+                "state_after": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  eval (P.newtonMap^[n] x + -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)\n      (map (algebraMap R S) P) =\n    eval (P.newtonMap^[n] x) (map (algebraMap R S) P) +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223 (aeval (P.newtonMap^[n + 1] x)) P",
+                "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223 (aeval (P.newtonMap^[n + 1] x)) P",
+                "tactic": "have \u27e8d, hd\u27e9 := binomExpansion (P.map (algebraMap R S)) (P.newtonMap^[n] x)\n  (-Ring.inverse (aeval (P.newtonMap^[n] x) <| derivative P) * aeval (P.newtonMap^[n] x) P)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>eval_map_algebraMap<\\/a>, <a>eval_map_algebraMap<\\/a>] at hd\"",
+                  "[{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]}]"
+                ],
+                "state_after": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223 (aeval (P.newtonMap^[n + 1] x)) P",
+                "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  eval (P.newtonMap^[n] x + -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)\n      (map (algebraMap R S) P) =\n    eval (P.newtonMap^[n] x) (map (algebraMap R S) P) +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223 (aeval (P.newtonMap^[n + 1] x)) P",
+                "tactic": "rw [eval_map_algebraMap, eval_map_algebraMap] at hd"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>iterate_succ'<\\/a>, <a>comp_apply<\\/a>, <a>newtonMap_apply<\\/a>, <a>sub_eq_add_neg<\\/a>, <a>neg_mul_eq_neg_mul<\\/a>, hd]\"",
+                  "[{\"full_name\":\"Function.iterate_succ'\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Iterate.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,22]},{\"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\":\"Polynomial.newtonMap_apply\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,24]},{\"full_name\":\"sub_eq_add_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1057,3],\"def_end_pos\":[1057,14]},{\"full_name\":\"neg_mul_eq_neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,27]}]"
+                ],
+                "state_after": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2",
+                "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223 (aeval (P.newtonMap^[n + 1] x)) P",
+                "tactic": "rw [iterate_succ', comp_apply, newtonMap_apply, sub_eq_add_neg, neg_mul_eq_neg_mul, hd]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>dvd_add<\\/a> ?_ (<a>dvd_mul_of_dvd_right<\\/a> ?_ _)\"",
+                  "[{\"full_name\":\"dvd_add\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Divisibility\\/Basic.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,16]},{\"full_name\":\"dvd_mul_of_dvd_right\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[209,9],\"def_end_pos\":[209,29]}]"
+                ],
+                "state_after": "case succ.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)\n\ncase succ.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2",
+                "state_before": "case succ\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2",
+                "tactic": "refine dvd_add ?_ (dvd_mul_of_dvd_right ?_ _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"convert <a>dvd_zero<\\/a> _\"",
+                  "[{\"full_name\":\"dvd_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Divisibility.lean\",\"def_pos\":[41,9],\"def_end_pos\":[41,17]}]"
+                ],
+                "state_after": "case h.e'_4\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) =\n    0",
+                "state_before": "case succ.refine_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)",
+                "tactic": "convert dvd_zero _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have : <a>IsUnit<\\/a> (<a>aeval<\\/a> (P.newtonMap^[n] x) <| <a>derivative<\\/a> P) :=\\n        <a>isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub<\\/a> h' <|\\n        <a>isNilpotent_iterate_newtonMap_sub_of_isNilpotent<\\/a> h n\"",
+                  "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"Polynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[180,5],\"def_end_pos\":[180,10]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]},{\"full_name\":\"Polynomial.isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Nilpotent.lean\",\"def_pos\":[202,7],\"def_end_pos\":[202,54]},{\"full_name\":\"Polynomial.isNilpotent_iterate_newtonMap_sub_of_isNilpotent\",\"def_path\":\"Mathlib\\/Dynamics\\/Newton.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,57]}]"
+                ],
+                "state_after": "case h.e'_4\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\nthis : IsUnit ((aeval (P.newtonMap^[n] x)) (derivative P))\n\u22a2 (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) =\n    0",
+                "state_before": "case h.e'_4\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) =\n    0",
+                "tactic": "have : IsUnit (aeval (P.newtonMap^[n] x) <| derivative P) :=\n  isUnit_aeval_of_isUnit_aeval_of_isNilpotent_sub h' <|\n  isNilpotent_iterate_newtonMap_sub_of_isNilpotent h n"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>derivative_map<\\/a>, <a>eval_map_algebraMap<\\/a>, \\u2190 <a>mul_assoc<\\/a>, <a>mul_neg<\\/a>, <a>Ring.mul_inverse_cancel<\\/a> _ this,\\n        <a>neg_mul<\\/a>, <a>one_mul<\\/a>, <a>add_right_neg<\\/a>]\"",
+                  "[{\"full_name\":\"Polynomial.derivative_map\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,23]},{\"full_name\":\"Polynomial.eval_map_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Tower.lean\",\"def_pos\":[42,7],\"def_end_pos\":[42,26]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"Ring.mul_inverse_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,27]},{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"add_right_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1237,3],\"def_end_pos\":[1237,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.e'_4\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\nthis : IsUnit ((aeval (P.newtonMap^[n] x)) (derivative P))\n\u22a2 (aeval (P.newtonMap^[n] x)) P +\n      eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n        (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) =\n    0",
+                "tactic": "rw [derivative_map, eval_map_algebraMap, \u2190 mul_assoc, mul_neg, Ring.mul_inverse_cancel _ this,\n  neg_mul, one_mul, add_right_neg]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>neg_mul<\\/a>, even_two.neg_pow, <a>mul_pow<\\/a>, <a>pow_succ<\\/a>, <a>pow_mul<\\/a>]\"",
+                  "[{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]},{\"full_name\":\"mul_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[310,32],\"def_end_pos\":[310,39]},{\"full_name\":\"pow_succ\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[657,9],\"def_end_pos\":[657,17]},{\"full_name\":\"pow_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[713,32],\"def_end_pos\":[713,39]}]"
+                ],
+                "state_after": "case succ.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 ((aeval x) P ^ 2 ^ n) ^ 2 \u2223\n    Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) ^ 2 * (aeval (P.newtonMap^[n] x)) P ^ 2",
+                "state_before": "case succ.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 (aeval x) P ^ 2 ^ (n + 1) \u2223\n    (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2",
+                "tactic": "rw [neg_mul, even_two.neg_pow, mul_pow, pow_succ, pow_mul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>dvd_mul_of_dvd_right<\\/a> (<a>pow_dvd_pow_of_dvd<\\/a> ih 2) _\"",
+                  "[{\"full_name\":\"dvd_mul_of_dvd_right\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[209,9],\"def_end_pos\":[209,29]},{\"full_name\":\"pow_dvd_pow_of_dvd\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[242,9],\"def_end_pos\":[242,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case succ.refine_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nn : \u2115\nih : (aeval x) P ^ 2 ^ n \u2223 (aeval (P.newtonMap^[n] x)) P\nd : S\nhd :\n  (aeval\n        (P.newtonMap^[n] x +\n          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))\n      P =\n    (aeval (P.newtonMap^[n] x)) P +\n        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *\n          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +\n      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2\n\u22a2 ((aeval x) P ^ 2 ^ n) ^ 2 \u2223\n    Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) ^ 2 * (aeval (P.newtonMap^[n] x)) P ^ 2",
+                "tactic": "exact dvd_mul_of_dvd_right (pow_dvd_pow_of_dvd ih 2) _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Polynomial.aeval_pow_two_pow_dvd_aeval_iterate_newtonMap"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001111-0000-7e9475220a89ca04.parquet",
+      "sha256": "7e9475220a89ca0472928ba3b99d3086893b7f8655933f4ee2a11612dd980b1b",
+      "bytes": 50246,
+      "rows": 3,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/mathlib-001111-0000-92dc0566bdd18dff.parquet",
+      "sha256": "92dc0566bdd18dff9712194c2d082f448ec4578c89dece1804b275a4cf2e4483",
+      "bytes": 43315,
+      "rows": 1,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Polynomial.isFixedPt_newtonMap_of_aeval_eq_zero",
+    "mathlib:Polynomial.isFixedPt_newtonMap_of_isUnit_iff",
+    "mathlib:Polynomial.newtonMap_apply_of_isUnit",
+    "mathlib:Polynomial.newtonMap_apply_of_not_isUnit"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Newton.lean\",null,null,\"Polynomial.isFixedPt_newtonMap_of_aeval_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Newton.lean\",null,null,\"Polynomial.isFixedPt_newtonMap_of_isUnit_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Newton.lean\",null,null,\"Polynomial.newtonMap_apply_of_isUnit\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Dynamics/Newton.lean\",null,null,\"Polynomial.newtonMap_apply_of_not_isUnit\"]"
+  ]
+}

manifests/batches/mathlib/001114.json

@@ -0,0 +1,420 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001114",
+  "batch_fingerprint": "1cc09a8d25f49ffe3f635c8530ba3f952e4f2f371dab97a680e5bb03dcec4f62",
+  "evidence": {
+    "batch_fingerprint": "1cc09a8d25f49ffe3f635c8530ba3f952e4f2f371dab97a680e5bb03dcec4f62",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic \"classical\\n  intro v\\n  cases nonempty_fintype \u03b9\\n  \\n  let s : Finset R :=\\n    (Finset.biUnion (univ : Finset \u03b9) fun i => (ps i).support.image fun x => coeff x (ps i)) \u222a\\n      (univ : Finset \u03b9).image v\\n  have hv : \u2200 i, v i \u2208 Algebra.adjoin K (s : Set R) := fun j =>\\n    Algebra.subset_adjoin (mem_union_right _ (mem_image.2 \u27e8j, mem_univ _, rfl\u27e9))\\n  have hs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115),\\n      k \u2208 (ps i).support \u2192 coeff k (ps i) \u2208 Algebra.adjoin K (s : Set R) :=\\n    fun i k hk => Algebra.subset_adjoin\\n      (mem_union_left _ (mem_biUnion.2 \u27e8i, mem_univ _, mem_image_of_mem _ hk\u27e9))\\n  letI := isNoetherian_adjoin_finset s fun x _ => Algebra.IsIntegral.isIntegral (R := K) x\\n  letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))\\n  letI : Finite (Algebra.adjoin K (s : Set R)) :=\\n    FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))\\n  let res : (\u03b9 \u2192 Algebra.adjoin K (s : Set R)) \u2192 \u03b9 \u2192 Algebra.adjoin K (s : Set R) := fun x i =>\\n    \u27e8eval (fun j : \u03b9 => (x j : R)) (ps i), eval_mem (hs\u2081 _) fun i => (x i).2\u27e9\\n  have hres_inj : Injective res := by\\n    intro x y hxy\\n    ext i\\n    simp only [Subtype.ext_iff, funext_iff] at hxy\\n    exact congr_fun (hinj (funext hxy)) i\\n  have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj\\n  cases' hres_surj fun i => \u27e8v i, hv i\u27e9 with w hw\\n  use fun i => w i\\n  simpa only [Subtype.ext_iff, funext_iff] using hw\"",
+        "full_name": "ax_grothendieck_of_locally_finite",
+        "id": "ax_grothendieck_of_locally_finite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            54
+          ],
+          "file_path": "Mathlib/FieldTheory/AxGrothendieck.lean",
+          "full_name": "ax_grothendieck_of_locally_finite",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"classical\\n    intro v\\n    cases <a>nonempty_fintype<\\/a> \\u03b9\\n    \\/- `s` is the set of all coefficients of the polynomial, as well as all of\\n      the coordinates of `v`, the point I am trying to find the preimage of. -\\/\\n    let s : <a>Finset<\\/a> R :=\\n      (<a>Finset.biUnion<\\/a> (<a>univ<\\/a> : <a>Finset<\\/a> \\u03b9) fun i => (ps i).support.image fun x => <a>coeff<\\/a> x (ps i)) \\u222a\\n        (<a>univ<\\/a> : <a>Finset<\\/a> \\u03b9).<a>image<\\/a> v\\n    have hv : \\u2200 i, v i \\u2208 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) := fun j =>\\n      <a>Algebra.subset_adjoin<\\/a> (<a>mem_union_right<\\/a> _ (<a>mem_image<\\/a>.2 \\u27e8j, <a>mem_univ<\\/a> _, <a>rfl<\\/a>\\u27e9))\\n    have hs\\u2081 : \\u2200 (i : \\u03b9) (k : \\u03b9 \\u2192\\u2080 \\u2115),\\n        k \\u2208 (ps i).<a>support<\\/a> \\u2192 <a>coeff<\\/a> k (ps i) \\u2208 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) :=\\n      fun i k hk => <a>Algebra.subset_adjoin<\\/a>\\n        (<a>mem_union_left<\\/a> _ (<a>mem_biUnion<\\/a>.2 \\u27e8i, <a>mem_univ<\\/a> _, <a>mem_image_of_mem<\\/a> _ hk\\u27e9))\\n    letI := <a>isNoetherian_adjoin_finset<\\/a> s fun x _ => <a>Algebra.IsIntegral.isIntegral<\\/a> (R := K) x\\n    letI := <a>Module.IsNoetherian.finite<\\/a> K (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R))\\n    letI : <a>Finite<\\/a> (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R)) :=\\n      <a>FiniteDimensional.finite_of_finite<\\/a> K (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R))\\n    -- The restriction of the polynomial map, `ps`, to the subalgebra generated by `s`\\n    let res : (\\u03b9 \\u2192 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R)) \\u2192 \\u03b9 \\u2192 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) := fun x i =>\\n      \\u27e8<a>eval<\\/a> (fun j : \\u03b9 => (x j : R)) (ps i), <a>eval_mem<\\/a> (hs\\u2081 _) fun i => (x i).2\\u27e9\\n    have hres_inj : <a>Injective<\\/a> res := by\\n      intro x y hxy\\n      ext i\\n      simp only [<a>Subtype.ext_iff<\\/a>, <a>funext_iff<\\/a>] at hxy\\n      exact <a>congr_fun<\\/a> (hinj (<a>funext<\\/a> hxy)) i\\n    have hres_surj : <a>Surjective<\\/a> res := <a>Finite.injective_iff_surjective<\\/a>.1 hres_inj\\n    cases' hres_surj fun i => \\u27e8v i, hv i\\u27e9 with w hw\\n    use fun i => w i\\n    simpa only [<a>Subtype.ext_iff<\\/a>, <a>funext_iff<\\/a>] using hw\"",
+                "[{\"full_name\":\"nonempty_fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,25]},{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"Finset.biUnion\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[118,15],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"MvPolynomial.coeff\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[585,5],\"def_end_pos\":[585,10]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"Finset.image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[340,5],\"def_end_pos\":[340,10]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.subset_adjoin\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,22]},{\"full_name\":\"Finset.mem_union_right\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1417,9],\"def_end_pos\":[1417,24]},{\"full_name\":\"Finset.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[357,9],\"def_end_pos\":[357,18]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"MvPolynomial.support\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[531,5],\"def_end_pos\":[531,12]},{\"full_name\":\"MvPolynomial.coeff\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[585,5],\"def_end_pos\":[585,10]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.subset_adjoin\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,22]},{\"full_name\":\"Finset.mem_union_left\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1413,9],\"def_end_pos\":[1413,23]},{\"full_name\":\"Finset.mem_biUnion\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[129,15],\"def_end_pos\":[129,26]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"Finset.mem_image_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[361,9],\"def_end_pos\":[361,25]},{\"full_name\":\"isNoetherian_adjoin_finset\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,35]},{\"full_name\":\"Algebra.IsIntegral.isIntegral\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[69,3],\"def_end_pos\":[69,13]},{\"full_name\":\"Module.IsNoetherian.finite\",\"def_path\":\"Mathlib\\/RingTheory\\/Noetherian.lean\",\"def_pos\":[162,28],\"def_end_pos\":[162,47]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Finite\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Defs.lean\",\"def_pos\":[81,17],\"def_end_pos\":[81,23]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"FiniteDimensional.finite_of_finite\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FiniteDimensional.lean\",\"def_pos\":[123,9],\"def_end_pos\":[123,25]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"MvPolynomial.eval\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1205,5],\"def_end_pos\":[1205,9]},{\"full_name\":\"MvPolynomial.eval_mem\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1768,9],\"def_end_pos\":[1768,17]},{\"full_name\":\"Function.Injective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[123,5],\"def_end_pos\":[123,14]},{\"full_name\":\"Subtype.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]},{\"full_name\":\"congr_fun\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[78,7],\"def_end_pos\":[78,16]},{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]},{\"full_name\":\"Function.Surjective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[133,5],\"def_end_pos\":[133,15]},{\"full_name\":\"Finite.injective_iff_surjective\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[652,9],\"def_end_pos\":[652,33]},{\"full_name\":\"Subtype.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\n\u22a2 Surjective fun v i => (MvPolynomial.eval v) (ps i)",
+              "tactic": "classical\n  intro v\n  cases nonempty_fintype \u03b9\n  \n  let s : Finset R :=\n    (Finset.biUnion (univ : Finset \u03b9) fun i => (ps i).support.image fun x => coeff x (ps i)) \u222a\n      (univ : Finset \u03b9).image v\n  have hv : \u2200 i, v i \u2208 Algebra.adjoin K (s : Set R) := fun j =>\n    Algebra.subset_adjoin (mem_union_right _ (mem_image.2 \u27e8j, mem_univ _, rfl\u27e9))\n  have hs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115),\n      k \u2208 (ps i).support \u2192 coeff k (ps i) \u2208 Algebra.adjoin K (s : Set R) :=\n    fun i k hk => Algebra.subset_adjoin\n      (mem_union_left _ (mem_biUnion.2 \u27e8i, mem_univ _, mem_image_of_mem _ hk\u27e9))\n  letI := isNoetherian_adjoin_finset s fun x _ => Algebra.IsIntegral.isIntegral (R := K) x\n  letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))\n  letI : Finite (Algebra.adjoin K (s : Set R)) :=\n    FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))\n  let res : (\u03b9 \u2192 Algebra.adjoin K (s : Set R)) \u2192 \u03b9 \u2192 Algebra.adjoin K (s : Set R) := fun x i =>\n    \u27e8eval (fun j : \u03b9 => (x j : R)) (ps i), eval_mem (hs\u2081 _) fun i => (x i).2\u27e9\n  have hres_inj : Injective res := by\n    intro x y hxy\n    ext i\n    simp only [Subtype.ext_iff, funext_iff] at hxy\n    exact congr_fun (hinj (funext hxy)) i\n  have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj\n  cases' hres_surj fun i => \u27e8v i, hv i\u27e9 with w hw\n  use fun i => w i\n  simpa only [Subtype.ext_iff, funext_iff] using hw"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro v\"",
+                "[]"
+              ],
+              "state_after": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\n\u22a2 Surjective fun v i => (MvPolynomial.eval v) (ps i)",
+              "tactic": "intro v"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases <a>nonempty_fintype<\\/a> \\u03b9\"",
+                "[{\"full_name\":\"nonempty_fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,25]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "cases nonempty_fintype \u03b9"
+            },
+            {
+              "annotated_tactic": [
+                "\"let s : <a>Finset<\\/a> R :=\\n      (<a>Finset.biUnion<\\/a> (<a>univ<\\/a> : <a>Finset<\\/a> \\u03b9) fun i => (ps i).support.image fun x => <a>coeff<\\/a> x (ps i)) \\u222a\\n        (<a>univ<\\/a> : <a>Finset<\\/a> \\u03b9).<a>image<\\/a> v\"",
+                "[{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"Finset.biUnion\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[118,15],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"MvPolynomial.coeff\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[585,5],\"def_end_pos\":[585,10]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"Finset.image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[340,5],\"def_end_pos\":[340,10]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "let s : Finset R :=\n  (Finset.biUnion (univ : Finset \u03b9) fun i => (ps i).support.image fun x => coeff x (ps i)) \u222a\n    (univ : Finset \u03b9).image v"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hv : \\u2200 i, v i \\u2208 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) := fun j =>\\n      <a>Algebra.subset_adjoin<\\/a> (<a>mem_union_right<\\/a> _ (<a>mem_image<\\/a>.2 \\u27e8j, <a>mem_univ<\\/a> _, <a>rfl<\\/a>\\u27e9))\"",
+                "[{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.subset_adjoin\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,22]},{\"full_name\":\"Finset.mem_union_right\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1417,9],\"def_end_pos\":[1417,24]},{\"full_name\":\"Finset.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[357,9],\"def_end_pos\":[357,18]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"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 intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "have hv : \u2200 i, v i \u2208 Algebra.adjoin K (s : Set R) := fun j =>\n  Algebra.subset_adjoin (mem_union_right _ (mem_image.2 \u27e8j, mem_univ _, rfl\u27e9))"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hs\\u2081 : \\u2200 (i : \\u03b9) (k : \\u03b9 \\u2192\\u2080 \\u2115),\\n        k \\u2208 (ps i).<a>support<\\/a> \\u2192 <a>coeff<\\/a> k (ps i) \\u2208 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) :=\\n      fun i k hk => <a>Algebra.subset_adjoin<\\/a>\\n        (<a>mem_union_left<\\/a> _ (<a>mem_biUnion<\\/a>.2 \\u27e8i, <a>mem_univ<\\/a> _, <a>mem_image_of_mem<\\/a> _ hk\\u27e9))\"",
+                "[{\"full_name\":\"MvPolynomial.support\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[531,5],\"def_end_pos\":[531,12]},{\"full_name\":\"MvPolynomial.coeff\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[585,5],\"def_end_pos\":[585,10]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.subset_adjoin\",\"def_path\":\"Mathlib\\/RingTheory\\/Adjoin\\/Basic.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,22]},{\"full_name\":\"Finset.mem_union_left\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1413,9],\"def_end_pos\":[1413,23]},{\"full_name\":\"Finset.mem_biUnion\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[129,15],\"def_end_pos\":[129,26]},{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"Finset.mem_image_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Image.lean\",\"def_pos\":[361,9],\"def_end_pos\":[361,25]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "have hs\u2081 : \u2200 (i : \u03b9) (k : \u03b9 \u2192\u2080 \u2115),\n    k \u2208 (ps i).support \u2192 coeff k (ps i) \u2208 Algebra.adjoin K (s : Set R) :=\n  fun i k hk => Algebra.subset_adjoin\n    (mem_union_left _ (mem_biUnion.2 \u27e8i, mem_univ _, mem_image_of_mem _ hk\u27e9))"
+            },
+            {
+              "annotated_tactic": [
+                "\"letI := <a>isNoetherian_adjoin_finset<\\/a> s fun x _ => <a>Algebra.IsIntegral.isIntegral<\\/a> (R := K) x\"",
+                "[{\"full_name\":\"isNoetherian_adjoin_finset\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,35]},{\"full_name\":\"Algebra.IsIntegral.isIntegral\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[69,3],\"def_end_pos\":[69,13]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "letI := isNoetherian_adjoin_finset s fun x _ => Algebra.IsIntegral.isIntegral (R := K) x"
+            },
+            {
+              "annotated_tactic": [
+                "\"letI := <a>Module.IsNoetherian.finite<\\/a> K (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R))\"",
+                "[{\"full_name\":\"Module.IsNoetherian.finite\",\"def_path\":\"Mathlib\\/RingTheory\\/Noetherian.lean\",\"def_pos\":[162,28],\"def_end_pos\":[162,47]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))"
+            },
+            {
+              "annotated_tactic": [
+                "\"letI : <a>Finite<\\/a> (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R)) :=\\n      <a>FiniteDimensional.finite_of_finite<\\/a> K (<a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R))\"",
+                "[{\"full_name\":\"Finite\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Defs.lean\",\"def_pos\":[81,17],\"def_end_pos\":[81,23]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"FiniteDimensional.finite_of_finite\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/FiniteDimensional.lean\",\"def_pos\":[123,9],\"def_end_pos\":[123,25]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "letI : Finite (Algebra.adjoin K (s : Set R)) :=\n  FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))"
+            },
+            {
+              "annotated_tactic": [
+                "\"let res : (\\u03b9 \\u2192 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R)) \\u2192 \\u03b9 \\u2192 <a>Algebra.adjoin<\\/a> K (s : <a>Set<\\/a> R) := fun x i =>\\n      \\u27e8<a>eval<\\/a> (fun j : \\u03b9 => (x j : R)) (ps i), <a>eval_mem<\\/a> (hs\\u2081 _) fun i => (x i).2\\u27e9\"",
+                "[{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Algebra.adjoin\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Subalgebra\\/Basic.lean\",\"def_pos\":[746,5],\"def_end_pos\":[746,11]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"MvPolynomial.eval\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1205,5],\"def_end_pos\":[1205,9]},{\"full_name\":\"MvPolynomial.eval_mem\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1768,9],\"def_end_pos\":[1768,17]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "let res : (\u03b9 \u2192 Algebra.adjoin K (s : Set R)) \u2192 \u03b9 \u2192 Algebra.adjoin K (s : Set R) := fun x i =>\n  \u27e8eval (fun j : \u03b9 => (x j : R)) (ps i), eval_mem (hs\u2081 _) fun i => (x i).2\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hres_inj : <a>Injective<\\/a> res := by\\n      intro x y hxy\\n      ext i\\n      simp only [<a>Subtype.ext_iff<\\/a>, <a>funext_iff<\\/a>] at hxy\\n      exact <a>congr_fun<\\/a> (hinj (<a>funext<\\/a> hxy)) i\"",
+                "[{\"full_name\":\"Function.Injective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[123,5],\"def_end_pos\":[123,14]},{\"full_name\":\"Subtype.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]},{\"full_name\":\"congr_fun\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[78,7],\"def_end_pos\":[78,16]},{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "have hres_inj : Injective res := by\n  intro x y hxy\n  ext i\n  simp only [Subtype.ext_iff, funext_iff] at hxy\n  exact congr_fun (hinj (funext hxy)) i"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hres_surj : <a>Surjective<\\/a> res := <a>Finite.injective_iff_surjective<\\/a>.1 hres_inj\"",
+                "[{\"full_name\":\"Function.Surjective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[133,5],\"def_end_pos\":[133,15]},{\"full_name\":\"Finite.injective_iff_surjective\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[652,9],\"def_end_pos\":[652,33]}]"
+              ],
+              "state_after": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases' hres_surj fun i => \\u27e8v i, hv i\\u27e9 with w hw\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhw : res w = fun i => \u27e8v i, \u22ef\u27e9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "state_before": "case intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "cases' hres_surj fun i => \u27e8v i, hv i\u27e9 with w hw"
+            },
+            {
+              "annotated_tactic": [
+                "\"use fun i => w i\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhw : res w = fun i => \u27e8v i, \u22ef\u27e9\n\u22a2 ((fun v i => (MvPolynomial.eval v) (ps i)) fun i => \u2191(w i)) = v",
+              "state_before": "case intro.intro\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhw : res w = fun i => \u27e8v i, \u22ef\u27e9\n\u22a2 \u2203 a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v",
+              "tactic": "use fun i => w i"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>Subtype.ext_iff<\\/a>, <a>funext_iff<\\/a>] using hw\"",
+                "[{\"full_name\":\"Subtype.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nhres_inj : Injective res\nhres_surj : Surjective res\nw : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhw : res w = fun i => \u27e8v i, \u22ef\u27e9\n\u22a2 ((fun v i => (MvPolynomial.eval v) (ps i)) fun i => \u2191(w i)) = v",
+              "tactic": "simpa only [Subtype.ext_iff, funext_iff] using hw"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x y hxy\"",
+                "[]"
+              ],
+              "state_after": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhxy : res x = res y\n\u22a2 x = y",
+              "state_before": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\n\u22a2 Injective res",
+              "tactic": "intro x y hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext i\"",
+                "[]"
+              ],
+              "state_after": "case h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhxy : res x = res y\ni : \u03b9\n\u22a2 \u2191(x i) = \u2191(y i)",
+              "state_before": "\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhxy : res x = res y\n\u22a2 x = y",
+              "tactic": "ext i"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Subtype.ext_iff<\\/a>, <a>funext_iff<\\/a>] at hxy\"",
+                "[{\"full_name\":\"Subtype.ext_iff\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]}]"
+              ],
+              "state_after": "case h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\ni : \u03b9\nhxy : \u2200 (a : \u03b9), (MvPolynomial.eval fun j => \u2191(x j)) (ps a) = (MvPolynomial.eval fun j => \u2191(y j)) (ps a)\n\u22a2 \u2191(x i) = \u2191(y i)",
+              "state_before": "case h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\nhxy : res x = res y\ni : \u03b9\n\u22a2 \u2191(x i) = \u2191(y i)",
+              "tactic": "simp only [Subtype.ext_iff, funext_iff] at hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>congr_fun<\\/a> (hinj (<a>funext<\\/a> hxy)) i\"",
+                "[{\"full_name\":\"congr_fun\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[78,7],\"def_end_pos\":[78,16]},{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.a\n\u03b9 : Type u_1\nK : Type u_2\nR : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Finite K\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra K R\ninst\u271d : Algebra.IsAlgebraic K R\nps : \u03b9 \u2192 MvPolynomial \u03b9 R\nhinj : Injective fun v i => (MvPolynomial.eval v) (ps i)\nv : \u03b9 \u2192 R\nval\u271d : Fintype \u03b9\ns : Finset R := (univ.biUnion fun i => image (fun x => coeff x (ps i)) (ps i).support) \u222a image v univ\nhv : \u2200 (i : \u03b9), v i \u2208 Algebra.adjoin K \u2191s\nhs\u2081 : \u2200 (i : \u03b9), \u2200 k \u2208 (ps i).support, coeff k (ps i) \u2208 Algebra.adjoin K \u2191s\nthis\u271d\u00b9 : IsNoetherian K \u21a5(Algebra.adjoin K \u2191s) :=\n  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsIntegral.isIntegral x\nthis\u271d : Module.Finite K \u21a5(Algebra.adjoin K \u2191s) := Module.IsNoetherian.finite K \u21a5(Algebra.adjoin K \u2191s)\nthis : Finite \u21a5(Algebra.adjoin K \u2191s) := FiniteDimensional.finite_of_finite K \u21a5(Algebra.adjoin K \u2191s)\nres : (\u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)) \u2192 \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s) :=\n  fun x i => \u27e8(MvPolynomial.eval fun j => \u2191(x j)) (ps i), \u22ef\u27e9\nx y : \u03b9 \u2192 \u21a5(Algebra.adjoin K \u2191s)\ni : \u03b9\nhxy : \u2200 (a : \u03b9), (MvPolynomial.eval fun j => \u2191(x j)) (ps a) = (MvPolynomial.eval fun j => \u2191(y j)) (ps a)\n\u22a2 \u2191(x i) = \u2191(y i)",
+              "tactic": "exact congr_fun (hinj (funext hxy)) i"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/AxGrothendieck.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Fintype.not_isField_of_card_not_prime_pow",
+        "id": "Fintype.not_isField_of_card_not_prime_pow",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            56
+          ],
+          "file_path": "Mathlib/FieldTheory/Cardinality.lean",
+          "full_name": "Fintype.not_isField_of_card_not_prime_pow",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Cardinality.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'obtain \u27e8e\u27e9 := Cardinal.eq.1 this'",
+        "full_name": "Infinite.nonempty_field",
+        "id": "Infinite.nonempty_field",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            9
+          ],
+          "file_path": "Mathlib/FieldTheory/Cardinality.lean",
+          "full_name": "Infinite.nonempty_field",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"letI K := <a>FractionRing<\\/a> (<a>MvPolynomial<\\/a> \\u03b1 <| <a>ULift<\\/a>.{u} \\u211a)\"",
+                "[{\"full_name\":\"FractionRing\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/FractionRing.lean\",\"def_pos\":[289,8],\"def_end_pos\":[289,20]},{\"full_name\":\"MvPolynomial\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[84,5],\"def_end_pos\":[84,17]},{\"full_name\":\"ULift\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[809,11],\"def_end_pos\":[809,16]}]"
+              ],
+              "state_after": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 Nonempty (Field \u03b1)",
+              "state_before": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\n\u22a2 Nonempty (Field \u03b1)",
+              "tactic": "letI K := FractionRing (MvPolynomial \u03b1 <| ULift.{u} \u211a)"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices #\\u03b1 = #K by\\n    obtain \\u27e8e\\u27e9 := <a>Cardinal.eq<\\/a>.1 this\\n    exact \\u27e8e.field\\u27e9\"",
+                "[{\"full_name\":\"Cardinal.eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[143,19],\"def_end_pos\":[143,21]}]"
+              ],
+              "state_after": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 = #K",
+              "state_before": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 Nonempty (Field \u03b1)",
+              "tactic": "suffices #\u03b1 = #K by\n  obtain \u27e8e\u27e9 := Cardinal.eq.1 this\n  exact \u27e8e.field\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>IsLocalization.card<\\/a> (<a>MvPolynomial<\\/a> \\u03b1 <| <a>ULift<\\/a>.{u} \\u211a)\\u2070 K <a>le_rfl<\\/a>]\"",
+                "[{\"full_name\":\"IsLocalization.card\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/Cardinality.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,13]},{\"full_name\":\"MvPolynomial\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[84,5],\"def_end_pos\":[84,17]},{\"full_name\":\"ULift\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[809,11],\"def_end_pos\":[809,16]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+              ],
+              "state_after": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 = #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a))",
+              "state_before": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 = #K",
+              "tactic": "rw [\u2190 IsLocalization.card (MvPolynomial \u03b1 <| ULift.{u} \u211a)\u2070 K le_rfl]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>le_antisymm<\\/a>\"",
+                "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]}]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 \u2264 #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\ncase a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a)) \u2264 #\u03b1",
+              "state_before": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 = #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a))",
+              "tactic": "apply le_antisymm"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8e\\u27e9 := <a>Cardinal.eq<\\/a>.1 this\"",
+                "[{\"full_name\":\"Cardinal.eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[143,19],\"def_end_pos\":[143,21]}]"
+              ],
+              "state_after": "case intro\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\nthis : #\u03b1 = #K\ne : \u03b1 \u2243 K\n\u22a2 Nonempty (Field \u03b1)",
+              "state_before": "\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\nthis : #\u03b1 = #K\n\u22a2 Nonempty (Field \u03b1)",
+              "tactic": "obtain \u27e8e\u27e9 := Cardinal.eq.1 this"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8e.field\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\nthis : #\u03b1 = #K\ne : \u03b1 \u2243 K\n\u22a2 Nonempty (Field \u03b1)",
+              "tactic": "exact \u27e8e.field\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine\\n      \\u27e8\\u27e8fun a => <a>MvPolynomial.monomial<\\/a> (<a>Finsupp.single<\\/a> a 1) (1 : <a>ULift<\\/a>.{u} \\u211a), fun x y h => ?_\\u27e9\\u27e9\"",
+                "[{\"full_name\":\"MvPolynomial.monomial\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[162,5],\"def_end_pos\":[162,13]},{\"full_name\":\"Finsupp.single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[270,5],\"def_end_pos\":[270,11]},{\"full_name\":\"ULift\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[809,11],\"def_end_pos\":[809,16]}]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\nx y : \u03b1\nh :\n  (fun a => (MvPolynomial.monomial (Finsupp.single a 1)) 1) x =\n    (fun a => (MvPolynomial.monomial (Finsupp.single a 1)) 1) y\n\u22a2 x = y",
+              "state_before": "case a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #\u03b1 \u2264 #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a))",
+              "tactic": "refine\n  \u27e8\u27e8fun a => MvPolynomial.monomial (Finsupp.single a 1) (1 : ULift.{u} \u211a), fun x y h => ?_\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>MvPolynomial.monomial_eq_monomial_iff<\\/a>, <a>Finsupp.single_eq_single_iff<\\/a>] using h\"",
+                "[{\"full_name\":\"MvPolynomial.monomial_eq_monomial_iff\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[389,9],\"def_end_pos\":[389,33]},{\"full_name\":\"Finsupp.single_eq_single_iff\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[381,9],\"def_end_pos\":[381,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\nx y : \u03b1\nh :\n  (fun a => (MvPolynomial.monomial (Finsupp.single a 1)) 1) x =\n    (fun a => (MvPolynomial.monomial (Finsupp.single a 1)) 1) y\n\u22a2 x = y",
+              "tactic": "simpa [MvPolynomial.monomial_eq_monomial_iff, Finsupp.single_eq_single_iff] using h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a\n\u03b1 : Type u\ninst\u271d : Infinite \u03b1\nK : Type u := FractionRing (MvPolynomial \u03b1 (ULift.{u, 0} \u211a))\n\u22a2 #(MvPolynomial \u03b1 (ULift.{u, 0} \u211a)) \u2264 #\u03b1",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Cardinality.lean"
+      }
+    ],
+    "input_proofs": 5,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 1,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 8,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/FieldTheory/Cardinality.lean",
+        "proof_id": "mathlib:Field.nonempty_iff",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              85,
+              101
+            ],
+            "file_path": "Mathlib/FieldTheory/Cardinality.lean",
+            "full_name": "Field.nonempty_iff",
+            "start": [
+              80,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Cardinal.isPrimePow_iff<\\/a>]\"",
+                  "[{\"full_name\":\"Cardinal.isPrimePow_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Divisibility.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,23]}]"
+                ],
+                "state_after": "\u03b1 : Type u\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n",
+                "state_before": "\u03b1 : Type u\n\u22a2 Nonempty (Field \u03b1) \u2194 IsPrimePow #\u03b1",
+                "tactic": "rw [Cardinal.isPrimePow_iff]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' <a>fintypeOrInfinite<\\/a> \\u03b1 with h h\"",
+                  "[{\"full_name\":\"fintypeOrInfinite\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[994,19],\"def_end_pos\":[994,36]}]"
+                ],
+                "state_after": "case inl\n\u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n\n\ncase inr\n\u03b1 : Type u\nh : Infinite \u03b1\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n",
+                "state_before": "\u03b1 : Type u\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n",
+                "tactic": "cases' fintypeOrInfinite \u03b1 with h h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>Cardinal.mk_fintype<\\/a>, <a>Nat.cast_inj<\\/a>, <a>exists_eq_left'<\\/a>,\\n      (<a>Cardinal.nat_lt_aleph0<\\/a> _).<a>not_le<\\/a>, <a>false_or_iff<\\/a>] using <a>Fintype.nonempty_field_iff<\\/a>\"",
+                  "[{\"full_name\":\"Cardinal.mk_fintype\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[461,9],\"def_end_pos\":[461,19]},{\"full_name\":\"Nat.cast_inj\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,17]},{\"full_name\":\"exists_eq_left'\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[293,17],\"def_end_pos\":[293,32]},{\"full_name\":\"Cardinal.nat_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1526,9],\"def_end_pos\":[1526,22]},{\"full_name\":\"LT.lt.not_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[341,7],\"def_end_pos\":[341,19]},{\"full_name\":\"false_or_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,21]},{\"full_name\":\"Fintype.nonempty_field_iff\",\"def_path\":\"Mathlib\\/FieldTheory\\/Cardinality.lean\",\"def_pos\":[53,9],\"def_end_pos\":[53,35]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\n\u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n",
+                "tactic": "simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',\n  (Cardinal.nat_lt_aleph0 _).not_le, false_or_iff] using Fintype.nonempty_field_iff"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [\\u2190 <a>Cardinal.infinite_iff<\\/a>, h, <a>true_or_iff<\\/a>, <a>iff_true_iff<\\/a>] using <a>Infinite.nonempty_field<\\/a>\"",
+                  "[{\"full_name\":\"Cardinal.infinite_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1733,9],\"def_end_pos\":[1733,21]},{\"full_name\":\"true_or_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[150,9],\"def_end_pos\":[150,20]},{\"full_name\":\"iff_true_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[163,9],\"def_end_pos\":[163,21]},{\"full_name\":\"Infinite.nonempty_field\",\"def_path\":\"Mathlib\\/FieldTheory\\/Cardinality.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,32]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\n\u03b1 : Type u\nh : Infinite \u03b1\n\u22a2 Nonempty (Field \u03b1) \u2194 \u2135\u2080 \u2264 #\u03b1 \u2228 \u2203 n, #\u03b1 = \u2191n \u2227 IsPrimePow n",
+                "tactic": "simpa only [\u2190 Cardinal.infinite_iff, h, true_or_iff, iff_true_iff] using Infinite.nonempty_field"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Field.nonempty_iff"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001114-0000-bff74fb361a09a4b.parquet",
+      "sha256": "bff74fb361a09a4be0666d95391a86fffeb001718bf1b334fbfd927302ae3b77",
+      "bytes": 58808,
+      "rows": 4,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Fintype.nonempty_field_iff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Cardinality.lean\",null,null,\"Fintype.nonempty_field_iff\"]"
+  ]
+}

manifests/batches/numina/001864.json

@@ -0,0 +1,51 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001864",
+  "batch_fingerprint": "0bd8ca18e17251f3210d43a0a80db18f944b806831ec42000d155d862f055671",
+  "evidence": {
+    "batch_fingerprint": "0bd8ca18e17251f3210d43a0a80db18f944b806831ec42000d155d862f055671",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 54,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 54,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001864-0000-e9da3c52eebddafb.parquet",
+      "sha256": "e9da3c52eebddafbc43eadee8991b3aa8611e4850232859a7f62ceffad672d22",
+      "bytes": 730885,
+      "rows": 40,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/train/numina-001864-0001-cbc9862615dbc918.parquet",
+      "sha256": "cbc9862615dbc9189001fc770c8c67e6651327639d6e8bb7412b57a2e1c74094",
+      "bytes": 473778,
+      "rows": 14,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:59164349-6ccf-5600-a693-4acd8c791cf8",
+    "numina:60565fc7-9216-563a-bb47-a5b11d0f8604",
+    "numina:9e5448ed-a4a1-5a4b-bff4-91ea7b7fefe1",
+    "numina:a71e3fa3-13fd-59af-9754-659461b75a97",
+    "numina:ca95be8a-0be7-5228-8d22-08eb7dace93b"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009320.lean\",\"c1dffbf7e6bdd575d6b28c9a49363d7f35b79ea3ca40137e4ce4c21c3c025a0e\",0,\"algebra_208346\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009321.lean\",\"7011d90cf8d4f68f28900e0aa7e1cb025c81753b5bdd5c0b1cffc9dbe32e549f\",0,\"number_theory_4600\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009322.lean\",\"6df5cb476c8474c99e593ce993bbc459dcd501f6fa96c8096c557b59b44a383a\",0,\"inequalities_110574\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009323.lean\",\"f16ae3a2ce6592cc3e145526592805dd404cb6f8ef63b61b70bb6d3fb9550454\",0,\"number_theory_4530\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009324.lean\",\"151ebd754c5494b2362b06643fb5510b227c89d44859d67e447e67a6c61f0b46\",0,\"algebra_311022\"]"
+  ]
+}

manifests/batches/numina/001865.json

@@ -0,0 +1,51 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001865",
+  "batch_fingerprint": "2bb873ee8fb6bc8cf79a1deff88c901bc54ad35f29fb3e36529211c8906a747c",
+  "evidence": {
+    "batch_fingerprint": "2bb873ee8fb6bc8cf79a1deff88c901bc54ad35f29fb3e36529211c8906a747c",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 55,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 55,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001865-0000-3b584e35c4aae8ac.parquet",
+      "sha256": "3b584e35c4aae8ac54c003e8841a40c225b43264c5f957d64ea258471dace474",
+      "bytes": 476369,
+      "rows": 43,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/dev/numina-001865-0000-5f126c9447e45095.parquet",
+      "sha256": "5f126c9447e450952da8055c55fd1c3f113a9124d4cbbfbec9fda02a2ebeb851",
+      "bytes": 90146,
+      "rows": 12,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:51d5ae20-3178-5943-9a6c-82e9980d8043",
+    "numina:553274a5-bf88-560c-ba0d-0ceacc5a32d3",
+    "numina:5716b25c-46d9-5224-9185-81e4a4f0dd15",
+    "numina:b4af0203-e325-520e-905d-de988b5a9af2",
+    "numina:fdb5f07f-2d4f-5877-a27b-1c8c6e49c8b7"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009325.lean\",\"b587c1b58f2b15d3b5642e26b7819e7535aa543386da8d70f89492ed7145b3df\",0,\"number_theory_318744\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009326.lean\",\"621d4b3c19986b643ee2500de7d958ac3ddcc56eb5019aa8fb69f03c1400d913\",0,\"algebra_20320\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009327.lean\",\"271595c88eaef6a20b316143f6ad199c6b2d93de012a1946240720c2c60881fa\",0,\"algebra_273177\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009328.lean\",\"081d6a67efb69e541b0b253c47d1a3b3e6c98bd5a16fab9668296d4a4ea919c2\",0,\"calculus_110647\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009329.lean\",\"e87a53bfb6c118cde0db113fb5fb99976f8c8861129ad81391bdd97e0b46b3fa\",0,\"number_theory_4531\"]"
+  ]
+}

manifests/batches/numina/001867.json

@@ -0,0 +1,59 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001867",
+  "batch_fingerprint": "f5a87a07c47b81011befef69a9ac99dd9b0cccaf86de6bf27319cfc61c6ad55e",
+  "evidence": {
+    "batch_fingerprint": "f5a87a07c47b81011befef69a9ac99dd9b0cccaf86de6bf27319cfc61c6ad55e",
+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 25,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 25,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001867-0000-f2dda41eb57efbd5.parquet",
+      "sha256": "f2dda41eb57efbd50c66a531fb712f2448afd9e072e1cfc935760e819aa6b96e",
+      "bytes": 449549,
+      "rows": 20,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/train/numina-001867-0001-d9aa81d07180d3de.parquet",
+      "sha256": "d9aa81d07180d3ded0ede992353826e938d3a34694749bef9e44f718fd12bf5c",
+      "bytes": 71374,
+      "rows": 4,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/numina-001867-0000-849b976c1036ec96.parquet",
+      "sha256": "849b976c1036ec96c5f0f9263771101898af56af600f3d70d10565803f47ff7a",
+      "bytes": 32716,
+      "rows": 1,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:11fd7bac-3e3f-5421-8605-3993ee37de72",
+    "numina:55975c66-2108-5ea8-a0fe-699c15f36716",
+    "numina:61415640-d3fc-5de0-a8be-33d248234832",
+    "numina:99ec5d3d-a417-5c47-9e45-1ba0a2c7af44",
+    "numina:b63442ee-d327-5c1b-802f-243e0fdf246d"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009335.lean\",\"f364e28091d411e1b5a0834c78574ffa2a9dc262347a5aa0e366aafd3dc6bea6\",0,\"algebra_227350\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009336.lean\",\"718968df56c25286b2d37eb30608a0c338cc7b330f6b2d6739039be184aacf8d\",1,\"number_theory_4785_2\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009337.lean\",\"a274c46e36af1c52f2ab3a52184d93c5df73d0022302cb0109200e112d27aa7c\",0,\"algebra_610795\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009338.lean\",\"561c097025493c046f3fae04d2d3ddd78beb2ddeaf0bb594562bad4946ebb2ce\",0,\"algebra_311594\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009339.lean\",\"4b5e2a65da81159fe1c0a9175ca7dffdd3970db54f1a10228ac9aab40ee70d5c\",0,\"inequalities_599222\"]"
+  ]
+}

manifests/batches/numina/001868.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001868",
+  "batch_fingerprint": "c84f64648c2ab3225ad8a9278e8d11ed7121ed42d425daf9a55f93442b9eed15",
+  "evidence": {
+    "batch_fingerprint": "c84f64648c2ab3225ad8a9278e8d11ed7121ed42d425daf9a55f93442b9eed15",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 45,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 45,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001868-0000-2ecf5ba6788f4b6a.parquet",
+      "sha256": "2ecf5ba6788f4b6ad986037ab739cad15df4d5c4d9ce75ecabfdd47f3f076f15",
+      "bytes": 371766,
+      "rows": 45,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:28a22216-4c54-5de2-853c-b40542b9fea6",
+    "numina:a8f64147-b778-5efc-b1c6-245220010834",
+    "numina:cc62a034-07cb-5731-840b-cbffb5d778c7",
+    "numina:e2613b27-f626-543a-ad37-91e873759f47",
+    "numina:f3be9113-8517-5292-a82f-26b1c43644be"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009340.lean\",\"809298883e1b21c252f7e6099496507c7954356ee77dc36cfa730f425669b218\",0,\"algebra_247269\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009341.lean\",\"3f14f36143d784489ba6fdc65090bf6c84b8e05bf9802f8e7afa040ebb58af94\",0,\"inequality_606436\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009342.lean\",\"07ba3c1066e19491d32a00c544e13197dea3d64baf34afd863630a7c41251e1d\",0,\"algebra_14190\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009343.lean\",\"05f0532261d7b7d10a9ce91efea91b16754b4d042b917affe3f62fc9b68f7888\",0,\"number_theory_18713\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009344.lean\",\"1e255107e5c3bd7d0c264203064d70ab64f208c7de990b56057d80dcaa90b5e1\",0,\"number_theory_609606\"]"
+  ]
+}

manifests/batches/numina/001869.json

@@ -0,0 +1,65 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001869",
+  "batch_fingerprint": "4d78064d1ae3598445aaf1fd883a61908fd2ef79c617473b3744848417bb7779",
+  "evidence": {
+    "batch_fingerprint": "4d78064d1ae3598445aaf1fd883a61908fd2ef79c617473b3744848417bb7779",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 86,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 86,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001869-0000-6bd81cf4fe7ba6ef.parquet",
+      "sha256": "6bd81cf4fe7ba6efd9c865fdf954bcb675b0a64e879ed788ca9ea127d896595b",
+      "bytes": 777288,
+      "rows": 37,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/train/numina-001869-0001-d411d6795a045a93.parquet",
+      "sha256": "d411d6795a045a93631ce0fe3c5b9c005afc18204e09d3835c8ac9fc3412e700",
+      "bytes": 1317957,
+      "rows": 24,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/train/numina-001869-0002-c49994ac3aa6411b.parquet",
+      "sha256": "c49994ac3aa6411bc29e2acb5a92a46d0a1f55c911dfd2eac8493e1f0e48c3d7",
+      "bytes": 231789,
+      "rows": 16,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/numina-001869-0000-b536adcf8ea78cbb.parquet",
+      "sha256": "b536adcf8ea78cbb8a3dc4917d8e1bdaa64a9758156084db264e03990817433e",
+      "bytes": 72362,
+      "rows": 9,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:0aab6f59-9cfb-57c9-95eb-d2a7b6e5a25a",
+    "numina:5a4c90a1-b451-5bb5-b494-984e6192aa11",
+    "numina:724a190a-bcc6-53e9-bb55-74bc528537b9",
+    "numina:b9b42076-5e49-57c9-b437-c3aa70c58ad1",
+    "numina:e60da079-f76c-5ce0-831c-92561eb56e0a"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009345.lean\",\"1132d8b21ac329c25ba3b9fb77fdd9a66905a7411dfeb31d1e18bd95dbe79068\",0,\"number_theory_4787\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009346.lean\",\"b569bcc51e5be08263eecb166f2d3aded62af1d8a9f199a7878fd8edc9367539\",0,\"inequalities_115477\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009347.lean\",\"d4884ecc1aa804bef206137ad798d5bda8e616b51f0689bbb89d813d536223a9\",0,\"inequalities_237192\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009348.lean\",\"cded3a3f6a4771056ecde4524cbc07c076cfbb2eced94a9bab1e9b3d6231e136\",0,\"number_theory_610797\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009349.lean\",\"93547b058d5d720441357014d35c760c76b701e9e43ce19beaf7174557d6d0f0\",0,\"algebra_20432\"]"
+  ]
+}

manifests/batches/numina/001870.json

@@ -0,0 +1,45 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001870",
+  "batch_fingerprint": "4ce27979b0d70ec1b4fc652dc4b2a46255a9e7ce28903d8aa8d61e3e17bb24ba",
+  "evidence": {
+    "batch_fingerprint": "4ce27979b0d70ec1b4fc652dc4b2a46255a9e7ce28903d8aa8d61e3e17bb24ba",
+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 17,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 17,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001870-0000-7340d84bfb923874.parquet",
+      "sha256": "7340d84bfb92387459e4877c0bbe269ea9f652f54e260fe377d61cb03457685d",
+      "bytes": 502578,
+      "rows": 17,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:59dfd310-f8ed-5c47-bb92-d077d4dce21d",
+    "numina:997e977a-d06a-5e48-868d-0225869488ff",
+    "numina:e88fc2fc-4fba-50df-9395-9109b2e90027",
+    "numina:f4b1f0c5-6441-50d5-8f50-466ada95b77a",
+    "numina:f776e1b3-eb22-51b8-b2e0-590ed6ff1db0"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009350.lean\",\"4023853807f01551ba88a46c7343a09b9a7b234f7aa27afa2054d13bf52719d3\",0,\"algebra_306011\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009351.lean\",\"9af9b37814f1cf663f516460970627d10b337308b48227c4509d14a4bed6ca75\",0,\"inequalities_599246\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009352.lean\",\"c7654fae86bdfc913b350e79b620285ef2b045881acab84b4b8183c5fba50e14\",0,\"algebra_609640\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009353.lean\",\"3bbd33bbbe37cd1e475ec22f599f8d8875d291f08e66b5fc2bae1df89525ae67\",1,\"algebra_24619.algebra_24619\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009354.lean\",\"314fbc69824d4af61b185e3fbe79359f392baa237e7f35da258de7a5af77d8a0\",0,\"number_theory_93865\"]"
+  ]
+}

manifests/batches/numina/001871.json

@@ -0,0 +1,52 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001871",
+  "batch_fingerprint": "76290a3379ed74a5dbfe5aaa0f0c580c9c712a8c87cb40f5583a8e7b78d52628",
+  "evidence": {
+    "batch_fingerprint": "76290a3379ed74a5dbfe5aaa0f0c580c9c712a8c87cb40f5583a8e7b78d52628",
+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 123,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 123,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001871-0000-6e7d7d7eb0b6df47.parquet",
+      "sha256": "6e7d7d7eb0b6df47c31787c52e1f46527c3ac1097cee98a47b86fef9f11bccdf",
+      "bytes": 602796,
+      "rows": 99,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/train/numina-001871-0001-06eb06b565633c8d.parquet",
+      "sha256": "06eb06b565633c8d32d3dcebc5635c48e9b55d8b2a310147677db8c2347f2a4c",
+      "bytes": 316831,
+      "rows": 24,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:32d44a7f-27d3-5e78-9e32-3100a27777bb",
+    "numina:78f6aede-2fbb-5d1a-ad1f-daa9ff7e85ef",
+    "numina:8dc0b943-9f7b-5ef7-8233-d1edd713b547",
+    "numina:9fccba47-3997-58e2-9ea3-c34ce6ddeda4",
+    "numina:a0c09495-7313-56d5-a24d-f7a41c0f5bb8"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009355.lean\",\"bf6b328ea42ecb302740488cc63d60c2113b182cfcd3fb8537dfaa15cd1eec7c\",0,\"algebra_611170\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009356.lean\",\"5393a88074265df4952a9ae8f5d93188700155370a1b7702e7e9910e3ac5b529\",0,\"number_theory_210851\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009357.lean\",\"64fca36ef0334b6b60167c865c9b20e8ab274ee2795d522e9e52ead5e1244dff\",0,\"number_theory_41936\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009358.lean\",\"fcd4b4c2a20680604424852b58e43a70eea6b9a22d4e19d7eea54eae5fa11f3c\",1,\"algebra_9088_2\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009359.lean\",\"ce3f07c1110cb2dfe67f64180e957e1557745bdb39018b7d244dca7530ed6221\",0,\"number_theory_155760\"]"
+  ]
+}

manifests/batches/numina/001872.json

@@ -0,0 +1,51 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001872",
+  "batch_fingerprint": "c5b8c00d50d9dc7df442b26281eed7740cd6a7cd54bfa90480d70ddd27040822",
+  "evidence": {
+    "batch_fingerprint": "c5b8c00d50d9dc7df442b26281eed7740cd6a7cd54bfa90480d70ddd27040822",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 31,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 31,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001872-0000-3d580e33e0b4b2b0.parquet",
+      "sha256": "3d580e33e0b4b2b0e4b45d18fe2f763a6886e6bf811d7a132ce97308b4cf6697",
+      "bytes": 193579,
+      "rows": 29,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/numina-001872-0000-67ebb6a6696c6418.parquet",
+      "sha256": "67ebb6a6696c6418f52de8bea46bf80548510d14546c998f8527d41cba37d453",
+      "bytes": 118238,
+      "rows": 2,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:07640039-8838-56b2-b507-b42ecd6c90be",
+    "numina:34e33ffa-cc46-59d8-87f0-2d31a5cc1a5a",
+    "numina:36a4c8b8-7dfe-5eca-a8db-e8790ea8da21",
+    "numina:a0e4b1be-7b74-5c22-a838-e6c3f2d74ed2",
+    "numina:acae03c3-65f4-5653-ab3e-a5a6a904b02b"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009360.lean\",\"ca095b9842318698c59a7e4522880bdebfde2612f51123513c2babbf53b92a7a\",0,\"number_theory_256292\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009361.lean\",\"53cf6d1958d58edc5113ac5e72176150ea3f09e80bdddfdf45624f84a0dafdb5\",0,\"inequalities_222354\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009362.lean\",\"e31b413868cd75a58af592936fc73ce8c26c264b3697aff135d4f34a5bce716f\",0,\"algebra_20434\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009363.lean\",\"7d7df31b35a678a5548b1f5e392783db9a2cf6ab81138467c98873a8b9065044\",0,\"algebra_599504\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009364.lean\",\"6bc1e06d6bfa16995e0a880c57fa2a2dfe4ab451da4969b94c8c85f6c6f57654\",0,\"algebra_306420\"]"
+  ]
+}

manifests/batches/numina/001873.json

@@ -0,0 +1,59 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001873",
+  "batch_fingerprint": "5276632a29143ba41c04ec1fc6ab92b74f4da96fc4e704303a280e619d2809c5",
+  "evidence": {
+    "batch_fingerprint": "5276632a29143ba41c04ec1fc6ab92b74f4da96fc4e704303a280e619d2809c5",
+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 44,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 44,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001873-0000-5dff7c0154c8a377.parquet",
+      "sha256": "5dff7c0154c8a377922dabf3e3278ddf0e72ee39d3c792cdc0aff9302697d06b",
+      "bytes": 139492,
+      "rows": 24,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/dev/numina-001873-0000-e7720c97ff2ba73b.parquet",
+      "sha256": "e7720c97ff2ba73b8a4955f837bd241d9baea0e2a95f133571826885877cfed1",
+      "bytes": 74636,
+      "rows": 9,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    },
+    {
+      "path": "data/internal_test/numina-001873-0000-2eb682f3a6e36605.parquet",
+      "sha256": "2eb682f3a6e36605ac4fa12dbcb4f1605e272e600793f0f107c78c4d74835445",
+      "bytes": 134209,
+      "rows": 11,
+      "revision": "9bbed15a51444d21da33671bab3239d2f0cfd4c6"
+    }
+  ],
+  "proof_ids": [
+    "numina:000f29a6-873f-509d-b506-61b1d450b061",
+    "numina:04567318-e354-5454-b6ec-f1f4b87a1f8a",
+    "numina:28bb1bc1-edb5-5d1b-9629-277c7263b8e1",
+    "numina:5f2e06a9-27a1-5845-95a8-e70091b5d6b9",
+    "numina:a9f36aa2-03f5-54fe-b339-49470cf1c50c"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009365.lean\",\"fda0b8e29cd278204f165752d0cce51e43e8abcfc412291b43b40ba0d4647def\",0,\"inequalities_609644\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009366.lean\",\"cf8ea87a67a4d3c88e3491b68100de6af632153fdf4586c9b51c5a6ccf8fd5d8\",0,\"algebra_10961\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009367.lean\",\"08d5ee7f56de34cb4dbb6b27d1eb0d42b6a6aced0dce8efec8ba90a39ada5631\",0,\"algebra_210959\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009368.lean\",\"46e61b9ddd319ded628ae04da056d9e92a48dfe06f27d9c7add65a54d653d803\",1,\"algebra_9090_2\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009369.lean\",\"a5d72e24900c9287ef38eb6be5bd4bb755f3a8aae74bd59c647d3737ea50b0a0\",0,\"inequalities_604693\"]"
+  ]
+}